Rapid Arithmetic

QUICK AND SPE CIAL METHO DS IN ARITH METICAL CALCULATION TO GETHE R WITH A CO LLE CTION OF PUZZLES AND CURI OSITIES OF NUMBERS

’

D D . T . O CONOR SLOAN E , PH . LL .

“ ’ " “ A or o Anda o Ele ri i ta da rd le i a Di ion uth f mﬁc f ct c ty, S n E ctr c l ct " a Eleme a a i s r n r Elec r a l C lcul on etc . etc y, tp y t ic t , , .

NE W Y ORK D VA N NOS D O M N . TRAN C PA Y

E mu? WAuw Sum 5 PRINTED IN THE UNITED STATES OF AMERICA which receive little c n in ne or but s a t treatment text books . If o o o f doin i e meth d g an operat on is giv n , it is considered n e ough. But it is certainly interesting to know that there e o f a t are a doz n or more methods adding, th there are a of number of ways applying the other three primary rules , and to ﬁnd that it is quite within the reach of anyo ne to io add up two columns simultaneously . The multiplicat n table for some reason stops abruptly at twelve times ; it is not to a on or hard c rry it to at least towards twenty times . n e n o too Taking up the questio of xpone ts , it is not g ing far to assert that many college graduates do not under i a nd stand the meaning o a fractional exponent , as few can tell why any number great or small raised to the zero power is equa l to one when it seems a s if it ought to be

equal to zero . The multiplication table can be made to give the most curious relations between its constituent ﬁgures and quan i o e o for one tit es, and f thes nly a part are given here , can play a regular game o f solitaire with the multiplication

table . ’ The book now in the rea der s hands has been a wonder

r o f fully inte esting work in its preparation . The sources information and the authorities appealed to were many and

in a number o f cases were little known . The collection of the a of m tter presented here was a sort gleaning , picking iv ARITHMETIC

t . the up what o hers had left It was also a selective task , o f o accumulating the best fr m many sources . A reference to the table of contents will show that this preface tells only of- a small part of what is to be found in the book . In a certain sense this work is a supplement to the o ordinary text bo k of arithmetic. But it is more than i o f i on e that ; it w ll be found practical appl cati in r al work , for by applying the methods to be found in its pages a greater command o f a rithrnetica l operations will be a c of quired and quick ways calculating will result . Much that is amusing in the way of oddities a nd recre a tions in the science of numbers will be met with in its pages . The compiler hopes that his mixture of the useful with the lighter phases o f his subject will prove acceptable to the reader . CHAPTER I .

GN NOTATION AN D SI S .

n Th ma o n h um . a c o a o e c . T e Ar bi N t ti . De i l P i t N ber One m Th hm ca h ci a ct o s. Signs in Arit metic. De a l Fr i n e Arit eti l a c n m o s a nd um e s 1 - 10 Co mplement. Pl i g S y b l N b r

CHAPTER II .

ADDITION

h Th a . Notes o n Addition a nd its T eory. e Addition T ble

n ﬁ f n . n Ho t . Di erent a Ca rryi g O e. w o Add W ys O Addi g ’

o e on G ou on. nd tio . B okkeep r s Additi . r p Additi I ex Addi n

d n o om ina o . o Period Ad itio . Additi n by C b ti n Additi n by

a i ca i n o u ca o . c Aver ge Multipl t o . Additi n by M ltipli ti n De i

m l Two a nd h e o umn on. f a Addition. T re C l Additi Le t o w hou a r n f om o umn to Ha nd Addition. Additi n it t C ryi g r C l

umn c ma iz o . S h a n Col . De i l ed Additi n ig t Re di g in Addi m m n Ad n r f a n o . o a tion. I verted o Le t H d Additi n C ple e t ry dition

CHAPTER III .

UBTRA CT O S I N .

Su o nc f a ci iz o u r . ma btra cti n. Pri iples S bt ction De l ed Sub a cti ubt a c on n I o . ut a c o n tr n S r ti in Couples. S b r ti by spec

. c i n o r ef n tion Subtra tion by Add tion. I verted L t Ha d Sub a c m m o . m n u a c on. Su a c o O f S tr ti n Co ple e t S btr ti btr ti n u s. om m o e O f ubtr c C plement Subtra ction o f Su s. A Pr p rty S a vi ARITHMETIC

CHAPTER IV.

c o S o The Multipli ati n a h rt Method Of Additi on. Multi

c o . E i the Multi limtion pli ati n Table xtend ng p Table. Mul ti liea tion o l o r Two t m p by D ub e Digi Nu bers. Multiplica ti liea tion o f Two m ut l c o p Digit Nu bers. M l ip i a ti n o f Three i a nd h m . I m u Dig t Hig er Nu bers ncre ent M ltiplication. o h ho o f m c o An t er Met d Incre ent Multipli ati n. Special Memod h Of Multiplying T ree Digit Numbers . Multiplica o o f o ti n Special Cases f Polynomials . Inverted or Left l c o o r o o Hand Mu tipli ation. Fact rial Prop rti nal Multiplica

l uo o . f tion. A iq t Part Multiplicati n Practical Use o Aliquot

c a c o . T Parts in Multipli ation. F t r Multiplying o Multiply ut T u H To . o un by Nine. M l iply by Eleven M ltiply by One m l m dred a nd Eleven. Co p e ent Multiplicatio n. Multiplica o f um r wi h l Two tion N be s Ending t Five. Mu tiplying by u a n m w Numbers at Once. M ltiplying by y Nu ber bet een h T e s. Twelve a nd Twenty. Multiplying by t een Cross

S ul i c o . ce o f Multiplication. lide M tipl ati n Ex ss Nines h Proo f o f Multiplication. Curiosities o f t e Multiplication

l A C o ho of . in Tab e. uri us Met d Multiplying Oddities Mul

R CHAPTE v.

o D o . Factor Division. Abbreviated L ng ivisi n Italian

c s o f i in s o . Re Method o f Long Division. Ex es N ne Divi i n

Di isibiﬁt o f m s. S meinders in Division. v y Nu ber pecial - i . w C Ca se o f Division . Dividing by Ninety N ne Le is ar te rs Short Cut RA N S F CTIO .

c Cha nging the Value Of a Fraction. Reducing Fra tions to r o the Same Class o to a Common Denominator. Additi n ul o a nd v o a nd Subtraction o f Fra ctions . M tiplicati n Di isi n o o rs o o f o o D m f Fractions. C nve i n Vulgar Fracti ns int eci al 94- 100

CHAPTER VII .

THE Decma r. O T P IN .

o s Due t s c m o o f Err r o Mi pla ed Deci al Point. Additi n ° f o o f c m s. u Decimals. Subtracti n De i al M ltiplica tion o Deci f tor- 108 mals . Division o Decimal s

CHAPTER VIII .

E E T ERCE AGE AL CULATI N S . I NT R ST AN D DiscouN . P NT C O

E c s xpression Of Rate o f Interest. Cal ulating Intere t. h o f S o rt Ways o f Calculating Interest. Reducti n o Interest ’ s . so for a o f Peri ods. One Day s Intere t Divi rs R tes Inter

t. c es Cancellation ih Interest Ca lculations . Per entage Cal cula tions o m u o c J o - 1 1 . Appr xi ate Calc lati ns by Per entages g 7

IX CHAPTER .

P ws M BE o ns or NU RS .

ow a n Roo ow a nd oo s o f c m a nd o f P ers d ts. P ers R t De i al o O f um s a nd u oo Mixed Numbers. Relati ns N ber Sq are R ts o h n S nd f ow s m o f s a m . t eir P er . Ter i al Digits quare Nu bers m o i o f ua s. A Curious Fra ction . Circ l r Nu ber Pr pert es T m ’ s. O f a st Squa re Property of the Square wo. Fer t s La f ff f h f u s. o s o T eorem. Properties o C be Orders Di erence f u s f o ow in o s o . a o s o S o P wers . P ers Pr gres i ns Rel ti n q are u T m h e o s o O f s. o wo Nu bers. T Pr gres i n Cube A Curi s m o f T o ua t T The Su w S s. Pro per y Of wo Squares . q re a ho Appro ximations to Isoceles Right Angle Tri ngles. S rt viii ARITHMETIC

i in Hi h t on o f R t f Wa y Of Ra s g to g Powers. Extra c i oo s o h r ho o o f S of Hig Powe s. S rt Meth d Calculating quares f um Mc a o s s o S . Large Numbers. V ri u Way quaring N bers ' ' m Nexen o Giffert s Methods o f Squaring Nu bers. s Meth d

EX PON E NTS .

o o o w Emo o o s. Z o Exp nents f P ers . ti nal Exp nent er as

P n v . m o . o an Exponent. ri e Exp ents Negati e Exp nents Powers of Ten 1 48- 1 55 XI CHAPTER .

i o m o o f o m m . Squaring the C rcle. Appr xi ati ns F r er Ti es ’ ’ M 7 h u m A r x o f . G o o i etius Value . S aw s Val e e etrical pp m n h u f 1 o m o s s o o z t O . u ati n . Aid t Me ri i g e Val e C ri us De r o e S u the c Lit termination o f by Pr ba biliti s. q aring Cir le ora lly

CHAPTER XII .

m m . o m m s How Pri e Nu bers Pr perties Of Pri e Nu ber . to i m m . m h F nd Pri e Nu bers Amicable Nu bers. Law o f t e ’ S a nd the u The he o o . T quare C be. F ur O clock Symb l um 1 08 in Our o nd ms uomo l N ber S lar a Lunar Syste . A t bi e w . The T a nd a o . Tires o Clerks . Wine Water P rad x Para dox o f u the Sq a res o f Fractions o f a Number. Divining m h e Nu ber T om of. A Para do x o f th Time Card. Re memberi s e i ng Telephone Number . Making th Nine Dig ts in o r u Pr per O der Add up to One H ndred. Curious Multi

c u . u o s pli ation. A Uniq e Number C ri u Multiplication and t N t o f . o o f um Addi i n. Mul ipli ca tion o Nines Pr perties f ’ s o o . oo s o . ber . Pr perties Nine B kkeeper Err r A Mystery m in Su M n . f . h i Money . Divining a o Digits Ot er ystiﬁca t o s Dividing Multiples o f Ten by Eleven I 6I- 184 i n tingushing feature of the Arabic otation, used

placed at the right O f the number— the right f the number is taken as being on the left of

decimal point . The digit placed here ex f t number O units in the quanti y, after the ds and other higher decimals have been taken

the next number to the left gives the tens, n hundreds and so o . t simple and elementa ry . But suppose here tion system Ofﬁ xing values ; then we would o Old R mans, with their clumsy To express the number eight if there were no place or posi e 8 hundr ds, tens

number O f characters required to

Roman and in Arabic systems . In : XXX runs DCCCL VIII , twelve three numerals expressing the same

INTRODUCH‘OKY 3

Any number whatever raised to th e zero power is

It is the only number all O f who se powers are equal ’ ' l I I r o f I I . to itse f . , and any powe is equal to i I of I The powers exceed ng , all numbers greater than , 2’ = ’ are greater than the numbers so raised . 4, 3 o the power is always greater than the riginal number . i I l The powers exceed ng , Of all quantities sma ler than

I are smaller tha n the quantities . ii i The pro duct Obtained by multiplying together any I a l numbers , each greater than , gives a qu ntity arger than either Of the multipliers ; if the quantities are less tha n I their product will be a smaller quantity than either o f them . Thus the number I stands at a dividing point and quan i t or I i tit es greater than uni y , present characterist cs dif i of I fer ng from those quantities less in value than .

S igns are used in arithmetic as a sort Of short hand to indicate Operati ons to be perfo rmed on the numbers or i e o quant ti s in the operati ns . In arithmetic number and quantity are generally to be n taken as syno ymous . The meaning Of a sign is expressed in English or in a

few cases in Latin ; the latter gives a shorter expression .

two tween the numbers to be added . It is almost always ” expressed as plus , the Latin word fo r more ; it I s “ to n perfectly proper re der it, added to . Where more q ﬁ k ﬁ mr

a u b c ﬁ ; it t h —ﬁ r b k d r

‘ ” M fm fi e lfl m i flg mbe subtn cted ; the m o f the opera tion is ca ned the rema inder or

Wha t ha s bem sa id a bofi tb e h rger qm fity being pla ud fim in me expression of subtra cM refers to “ ” ing to be multiplied ; the lower quantity is ca lled the

ul i r n res t s called the product . There is no easo for placing the multipli ca nd above the multiplier ; they ca n cha nge position and r61es without a ﬁecting the result of the t calcula ion . s One Divi ion is indicated in several ways . is by a horizontal bar or to save spa ce a diagonal one . The number above the bar is called the dividend, from the a u L tin , meaning to be divided the n mber with which it is to be divided is placed below the ba r ; it ' is called the

d . the s i c ivisor Thus g, or what is ame th ng , indi ates 6 i f vi d vided by 3 . The result O a di sion is called the i n . quot ent, from the Latin , meani g how Often Another sign Of division is possibly derived from this

' one ; it is a horizonta l bar with one point over its center + 6 a nd one o i 6 e . bel w ts center. 3 indicat s divided by 3 o i n of The c lon , s a sign of ratio and he ce division , m i n but is not used, si ply to ndicate divisio , except in

The horizontal or oblique bar is not always admitted to be the sign o f division ; the claim is sometimes made that there is a difference between such expressions as

and 34 . The latter is taken as indicating a frac

. a n o b tion only But if we take such expressi n as a / , it is ha rd to see howit can be expres sed except as a divided b ” f by . In the case O numbers the alternative fractional “ a nomenclature is always av ilable, as two fourths in e or the xpression fraction , 34 . n m i iven it mea ns a nd indica tes the Wha tever a e s g , % i 2 divis on of by 4. The indicati on Of the higher powe r Of a number is a small ﬁgure placed above and to the right ; it is called an ' ‘ ‘ a o f c 1 6 exponent ; 4 me ns the square 4, whi h is ; 5 1 2 The i o or o f is . means the th rd p wer cube 5, which 5 l 2 o n s n e o f litt e ﬁgures , and 3, are exp ne ts , in the e i stanc s ,

4 and Of 5 respectively . The term square is an abbreviated way o f express ing the second power ; the term cube is the sa me fo r the third power ; there are no other abbreviations of o p wer expressions . o The radical sign indicates the r ot of a number . By itself it indicates the square root ; if any other root is to i e e o the be ndicat d the exponent is plac d ver it to left .

' vmmea ns the square root o f 1 6 which is 4 ; JIOmea ns u ‘ 1 6 2 the fo rth root Of , which is . When there a re several numbers required to express ” o o 2 a quantity the combinati n is called an expressi n .

3 and 3 5 are expressions . The sign o f equality is a pair Of horizontal bars par ” ”

to . allel one another , they read equal or equals

two t Thus to express the result Of adding quanti ies , say ' 2 o 2 2 and 3, we w uld write , 3 5, which reads plus

3 equal 5. f A statement such as the above , a ﬁrming the equality two or o u Of quantities expressi ns, is called an eq ation

and sometimes a formula . Inequality is indicated by a V-shaped symbol placed on its side ; the quantity next the apex is stated to be the

l . 2 e 2 or 2 smal er 7 m ans that 7 is greater than , that si is less than 7 . Both come to the same thing ; the gn f c can a e the other way ; 2 7 mea ns tha t 2 is less than 7. " i n o f a ro ortion a nd a re rea d while in the s g p p as , “ ” same pro po rtion the single colon is read is to ; thus S 2 : 4 : 24 z8 reads 2 is to 4 as 4 is to 8 . ometimes the = l is used a s th centra l s mbol. If this equa ity sign , , e y n r o o : 2 8 is do e m the above p op rtion we w uld have . Taking the colon as the sip of division our last pro n 2 8 portio would read divided by 4 equal 4 divided by ,

which is perfectly correct, and expresses the relation e i xisting between the members of a proport on . The d o on n n f l o ouble c l is ever take a s a sign o equa ity, th ugh it might be A parenthesis expresses the fact that the quantities within the pa renthesi s a re to be ta ken a s a group a nd to be i 2 treated as a single or ind vidual quantity . 7 ( 3) means that the sum o f 2 and 3 is to be subtracted from f 2 7 ; this gives a remainder o . The same numbers with out r e 2 to the pa nthesis , 7 tell us that is be subtracted fro m 7 and that 3 is to be added ; the result is 8 i . d ﬁ The parenthesis brings about a totally erent result . Multiplicatio n and division sip s take precedence o f addition and subtraction signs ; the ﬁrst two signs unite n the umbers between which they are placed, as if they in t 1 2 10 2 were a paren hesis . Thus indicates that 1 0 is to be divided by 2 and the quotient is to be sub 1 2 t l tracted from ; his gives 7, as the result . In ike ma nner 4 +6 x 3 indicates that 3 times 6 are to be to ' h 22 6 added 4 , w ich gives ; X 3 act as if in a paren

A decimal fraction in the broad sense is a frac o e n o f 1 0 ti n whos denomi ator is a power ; $40, 9400 and era fly the termis rm icted to the wrifing the sa me cla ss ma in o f fra cfiws on the hne using the deei l po t. To do this the numera tor o f the fra ction is written to th i h f l oin a nd the denomina tor a s e r g t o the decima p t , such is mi a nd is ex ressed b the rela tion o f the , o tted p y numera tor to the decima l b the us o f ci hers or y e p , in the num era tor . Thus a re too fewto give the p the e n to o a d cimal poi t the numerat r, ciphers are pl ced n to the right of the poi t between it and the numerator. The number of digits or ciphers less one in the denom ina tor of a decima l vulga r fraction gives the numb er of la in h i is wri 1 ces t e e . tten b s p d cimal fract on %o . , eca ue

e i the because there are thre dig ts in denominator.

Complement means anything which ﬁlls up or com 1 0 pletes . If a number is re ferred to the multiple of diﬁ erence t immediately above it, the be ween it and that 1 0 o On 2 multiple o f is its c mplement . this basis would corn lement o f 1 8 l 1 0 x be the p , because the multip e of ne t 1 8 20 to is . O ften the higher number to which it is referred is the of 10 next higher power . On o of 1 8 82 this basis the c mplement would be , for e t o f 1 0 o 1 8 1 00 the n x power , ab ve , is . The arithmetical complement o f a decimal fraction is ff or 1 the di erence between it and unity . lemen Thus the arithmetical m t of . 55 is 4 5. This

' Tbe mtmdmi ion of the minus — Js a ttributed Micmd Sfifi wfi d l sfiy nhe da te o f

' v 1 To im ls him is gr en 3 5 544 . h is a o

c rd a n En i i b . Re o e, gl sh mathemat cian, ( H m- w it n r e st o d i ssz

- Ou t cd En e b . 1 g h , an glish math matician, 574, It a a in i wo rk Cla vis Ma thema tica e pp red h s , , Th v n i a ttributed to a n e sign of di isio , s m 1 1o 1 8 6 d. h u b . 6 . a t a a tieia n . Pe , Dr John ll, , 5

Ia ﬁn wm lu b h lta lia n wo rd i or b h l d p s y t e mm y t e etter,

. S n m p ubtraction was i dicated by minus, ene, or m. e d i The radical sign succ e ed the cap tal letter, R, as the t of sign of a roo a number . The signs of appear ﬁrst in a H E i m posthumous book of Thomas arriot, an ngl sh ath ema ticia n m Ouhtr d , a conte porary of g e . o N The decimal point was intr duced by John apie r, bJ O m S t ( SS . the fa ous co ch mathematician, in of n the beginning the seventee th century . The decimal system of numbers was introduced into E urope in the tenth century . te range o f the usual multiplication is a t ouce without stopping to thiuk. “ i h e w count the one st on t ere a r , if e f o So o f er ont multiplicati ns . me these m rt d such s times a nd ti es . e , a 3 g 9 3 ie products to be remembered 1 32 in erted ones should count as individual be quantities are identical . ﬁrst rule of arithmetical operations for numeration is hardly to be called of all the four basic things in arith i o subtraction, mult plicati n and di i e v s the most trouble , is the most rd f So is the most used o the four.

' ile every bank has one or more adding tively few c onsider it wort h while to a O m chine . wing to the extensive ding machines there is not the same rho t e o are rapid adders, that h re nce have our own adding to do and few i ddin g machine . It is satisfactory to 12

a re a number of diﬁerent wa ys of a dding up cohi mns of u r s in b ﬁg res , there are a numbe of pecial ways use y i th l 8 individuals , of help ng e work a ong 0 that many operators have what may be called their own ways of e l g tting at resu ts .

In the multiplication table there are 1 44 difierent multiplications to be remembered and to be known so perfectly that no time or thought is to be given to them when asked or required . In the corresponding addition table there are only 45 additions to be remem for o to bered, yet some reas n it is fair say that they are not as well kn own in general as are the products of the o multiplicati n table . The following things are to be noted about the addition to t of the nine digits each o her, in twos , that is to say in one for i t the addition of digit to another, all the n ne digi s. t The total number of such additions is for y five. f i O these additions twenty g ve single numbers . Such are O f t these addi ions twenty five give double numbers . Such are The largest number given by the addition of two digits

: 1 8 1 is eighteen , 9 +9 ; the figure on the left is , this is the figure to be carried when we add ; therefore the figure to be carried in addition is never increased in a n ‘ of b r 1 addition to any number a single digit y mo e than . 6 8 to To illustrate this statement suppose ,7, and 9 are 6 r 1 t . a e 1 be added toge her and 7 3 ; this gives to carry . Then comes 3 and 8 giving 1 1 to which is added the 1 a di io ivin a i r ca rried froni the first d t mg g , so tha t the e z a rr N ex l is a dded to ivin with the 2 to is to c y. t g g g, mb o a s the sum0i the four nu er s. ca rry, 3 T r first 1 rr it eould not ossibl be he e wa s to ca y, p y m r tha n l to be ca rried due to o e . The n ext number , n o f e 2 one the additio this time three figur s was , more the r 1 e than fi st figure carried , which was . Th n when the additio n of the four was omnpleted the figure in the ’ te a ns place was one more th n the last figure carried,

it was 3. This leads to the following conclusion : When a column of single digits is added up the successive left hand ﬁgures in the ten places can never exceed each other by more than 1 at a time ; successive ﬁgures in the ten l c p a es may be the same . In the following additions the separate additions are at the right hand side of the ' column of amounts

Cohi mn a is made up o f such as involve the carrying of a n additional 1 for each adding ; this is the most tha t ca n be ca rried a nd it will be observed tha t the

si succes ve sums are, as regards their ﬁgures in the ten v 1 places , successi ely greater by , so that the ten place a idl his more r p y tha n t .

’ ‘ the merea se o f t he ten pla ce figures iS Slower zthfiy mfl

1 1 2 2 . , , and In column c four figures a re a dded giving the ma x imum fi n f he ten la c fi ures in this ca se di ere ce o t p e g , 1 2 I lumn d h m four fi ures a re ta ken . n co t e sa e , and 3 g 8 o On o but an is placed bel w them . additi n we again have the regular maximum succession of ten pb ce s but n the figure , in this case runni g up to 4, because of e 8 c xtra whi h has been added in .

Now o o do c mes the subject of what additi ns not give, “ a nd i to e of which do g ve one carry , both kinds a r given here .

1 1 1 1 1 1 1 1 234567

23456789 456789 89

These a re t he twenty additions which do not give one to carry next come the twenty ﬁve which do involve 1 the carrying of .

i 4 4 4 4 9 6 7 8 9

1 0 1 1 10 1 1 1 2 1 0 1 1 1 2 1 3 5 5 5 5 5 6 6 6 6 7 7 7 8 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9 In the mo dern teaching of reading to teach the pupils to learn to ‘know o e of pr c ss spelling. In adding numbers they added without naming ; thus in adding the 6 o mn a e 1 ou o l e c lu , pag 3, y sh u d say to yours lf u e si o 1 2 1 0 not s cc s n 9, 7 , 4 , 33 , 4 , 5 ; do say to n z 9 a d 8 a re 1 7 a nd 7 ma kes 4 a nd so on. It is interesting to us e the tables just given to test your quicknes s at addition ; if youcannot give the additions of t on n the tables without hesi ati and almost without thi king, fast addition and even accurate addition to say the least

Addition may be done column by column and one digit a t m o f a ti e ; this is probably the most usual way doing it . It is the most obvious and the simplest but is ‘ also the

" — slowest . There are two ways of doing it from above o of the o « To d wn or from the bottom c lumn upwards . prove the accuracy of the operation it is well to do it t n o ﬁrs one way and the the ther .

A minor variati on which can be used to advantage is to o write down the sum of each c lumn separately, one sum ' ‘ under the other and each successive sum set one space

' to i o the left ; then a subs diary additi n gives the total . This is done here

9938 Addition o f ﬁrst column second column third column fourth column

Total 287 1 0 The diﬁerent a dditions of two digits give only q dif f r ma tter to kn a mt mmba a so it is a ve y msy ow them.

° It is n en iv hem without st " i k ot sy to g e t opping to th n . When perfectly knm they lend to another way of t ﬁ in a d hng the rst step o f gr oup a dd g .

m w m r others in th vertienl l a t a ti e to t o or o e e co umns. Thus the followiug a ddition wo uld be done by a t once in in the 8 2nd the number 1 a nd in the 3 see g 7 5 9 12 9 md 3 a b0ve them the number l z ; then 1 2 x ul a dd ivin Thi m 7 a nd s wo d be ed g g ﬂ. s a y 8 1 5 or ma y not involve adding double numbers to ea ch other but this method is ma de sim le b , p y 2 7 the fa ct tha t the sum o f two numbers can

sev ra l columns a re to be a dded b rouin When e y g p g , there will genera lly be a number to be ca rried ; this ma y or be added to the ﬁrst group of the next column , the

I f the di rect adding of such numbers as 1 5 and 1 2 to t too fﬁ e o c e each o her is di cult , th n for rapid w rk pro e d as follows : In the last example add 1 5 to 1 0 giving 25 ; 2 2 1 2 for then to this 5 add the missing of the , youhave 1 0 oﬁ already added its , reading it to yourself in this : 1 2 2 way 5, 5, 7 . Thi s system makes it as simple as i ordinary add ng . The . process of group adding is easier and in r better eve y way than the direct single column way .

A h l i st fi is rea ched. s t e a st or h ghe gures o f the co lumn ive a number in excess o f 10 the fi ure of their sum g g 7 , 1 is Now th ke -fi ures a r d u h 7 , written. e y g e a ded p ; t ey

- t or 1 for c ke immber . Ther a re four e i 4, ea h y e k y so the o for the tens i v s numbers , t tal is , wh ch gi e 6 f 6 o ol . , and the sum the wh e is 5 Suppo se nowthat there had been no eight at the top of the column and that after the last key- number had been written only the 9 had been left ; then there would have t - 8 i 1 8 o been only hree key numbers , 5, 5 and mak ng ; t o i these w uld have been added the remain ng 9, at what ' would then have been the top o f the column ; this would have given 27 ; then there would have been only 2 to carry a nd to go into the ten place in additi on to the 3 o f the key o o o f o numbers ; we w uld have a t tal 57, which w uld be o f o o o o 8 the true sum the c lumn with ut the t pm st ﬁgure .

Period Addition

o The next process is d ne in a similar way . The addi tion o f the figures is continuous this time right up to the 8 to p, except that when the number, just as before, o 20 dot approaches a is placed at the side , and the xi u addition goes on but with omissi on o f the tens . u Thus after the third figure from the bottom is reached we make a dot and go on with 5 as a a o starting figure . Then we have 5 7 ; this gives o 8 do 1 2 . so t c The next figure is , we put a at the s a side and starting with 2 add it to the figu res above 8 6 2 dot o it . , , and , find that a g es here where the 'd G total is 1 8 ; then this 8 is added to the 3 and 7 above it and another dot is put down and for the final figures we have 1 7 for the next to the last 19

dot a nd ca rrying the 7 the top ﬁgure gives 7 +8 or 1 5. Here is the last dot and the unit ﬁgure 5 to be written ix fo r the tc s o the o . e s down, while n c unt d ts Ther are b r 6 so our tcns a re 6 a nd the num e is 5.

A in of the two up the units . s the ﬁrst methods, there th o ma y be l to ca rry a t e t p .

It is so metimes advised that the combinations of num

~ . S bers a dding up to ten be memorized uch are 3, 3, 4 1 6 2 w o ion a r . t o e , 3, ; , 3, 5 The figure c mbinat s so well known that it is to be presumed that everyone knows o w out he f om them. Any ne can rite t dif erent c binations m It o n to e and study the . is also rec mme ded l arn the o o o f o com c mbinati ns m re figures , such as the eight bina tions o f 20 four figures which give . There are nine

o m i o f 0 c b nati ns o four figures which give 3 . The higher o o o f i for o b c mbinati ns are l ttle use , the m re num ers n i volved in a combination the rarer it is . The two and o o r three figure c mbinati ns a e the most useful . E very such combination which ca n be memorized is an ’ o to o o n G additi n one s faculties in d ing gr up addi g . roup adding should not be confined to sets o f only two num

o u t o . bers . All these ways c ntrib te o gr up adding It will often be found that exper t adders have their own favorite

Addition by Aver age Multiplication

m o When nu bers ccur, whose middle ﬁgure is the o f lot m o f average the , multiplying it by the nu ber ﬁgures

. S o will give the sum upp se 5, 4 and 3 are to be added ; 4 is the a vera ge u m n n i t h is the sum. S ch co bi a tio s a s th s a re o be wa tc ed

The following is another way of adding up a column

of single ﬁgures . By adding and subtracting the ﬁgures

are all reduced to the same, and a simple multiplication gives the addition provided the same amounts were added O as were subtracted . therwise a correction must be Some examples

The idea is to substitute a simple multiplication for a n o 8 additi n . In the second case as has been added and 3 on o f subtracted, the net result is an additi 5, which has

to be subtracted from 35 to give the answer.

Decimal Addition i The next method may be termed dec mal adding. To a ll of the quantities add such quantities as will make 1 1 00 o them multiples of 0 or of . Fr m the last of the original quantities subtract the sum of the increments or A nd Add 8 w dd 9. 7. a 4 97. 9. h o 9 +1 = m 97 +3 = too 7 +3 = xo 89 +xr = l oo 4 — 4 = o 49 — r4 = 35

1 In the ﬁrst example the increments are and 3, their n hi is subtra cted fromthe la st of the ori ina l sumis 4, a d t s g quantities , which is also 4. The sum of the numbers in the right hand column gives the sum of the original num exa m e 1 o f bers . In the second m 4 is the sum the increments, and is subtracted from 49, the last of the the original numbers . The sum of right hand column gives the answer .

Two a nd Three Column Addition

1 0 There are ninety combinations of double numbers, , 1 1 1 2 n o f , and so o up to 99. If the sum of any two two these can be given without hesitation at sight , then columns ca n be added simultaneously . This implies who double speed , and there are many computers can do

o s i . this . Some can add three c lumn s multaneously The addition of two columns at once can be done by o e o o e the f llowing m th d by any ne. It is a m thod well m worth acquire ent . The bottom or -top number of the two columns is added to t o f the ens the number next it, and then the units o f o f the next number are added ; this gives the sum the two . Then the tens of the third number are added and then m the units o f the third number giving the su o f the three .

To i o to m v 29 apply th s meth d the exa ple here gi en 1 8 34 proceed as foll ows : Add 70 to 88 ; this gives 5 . T 1 1 To 1 o f 1 1 . o 7 this add the the 7 , giving 59 59 88 0 i 1 8 to o f the add 3 , giv ng 9, and this add the 4 1 T 1 20 2 1 to 34 giving 93. o 93 add , giving 3 and f 2 222 this add the 9 o the 9 and the total is . A variation can be used and the units can be added in 1 = 8 i f before the tens ; thus 88 + 9. Th s takes care o

o f 1 . 0 the unit ﬁgure the 7 Then add 7 , which makes

‘ 1 0 o f to 59. and adding the 3 34 this 20 to i gives 193. Then and added th s 22 2 gives the total as before . Three columns in width can be added by an extension to to 8 of the same method . Thus add 957 75 if we start with the hundreds it goes as follows : 957 1757 70 = I S27 ; 1 827 5 1 832 The three column method is sometimes said to be only useful within the limits o f a limited range o f numbers ; ‘ o it is good practice h wever . The two number adding for o o o f An is quite practical much l nger r ws numbers . o f o o example three c lumn additi n follows . Add the following

1 542 1 002 + 1 00 1 + 80 1 +zoo = 1 00 1 77 1 +30 801 764 +7 77 1

As the addition was started at the bottom o f the columns the sum is reached at the top . 23

Left Hand Addition When a number o f columns are to be added the a d diti on may co mmence with the left hand column at the o o o t d o f b tt m . When c mple ed the a jacent ﬁgure the o not to o next c lumn is annexed , added, it and the additi n o o is carried right down the second c lumn . The additi n or sum o f the two columns is written down at the bottom the two o and next c lumns are treated in the same way . If their sum includes only two digits they are set down to the right o f the sum o f the ﬁrst two columns ; if they commise three or more digits they are set down one line below the others and with a ll except the two right hand digits under the right hand digits o f the a re upper sum . The two then added if there are only the four columns ; if there

o o out. are m re, the sa me pr cess is carried The one thing to be kept in mind is that o f o the sum the ﬁrst, or right hand c lumn not to o is added the sec nd , but is treated as the tens and hundr eds o f a quantity to

which the seco nd column supplies the units . exa m e o In the mthe . ﬁrst c lumn adds up to 2 5 ; assume that we a re adding from to 2 below upwards . Then 5 we annex the to ﬁ r o f n o o o 7, which is the p gu e the seco d c lumn and g d wn 2 2 28 1 i on 2 6 . with the add ti , 57 , 4 , 73, The last is the sum

° o f the first and second co lumns counting from the left n ' and is written down belo w the li e . The same is done for the next two columns ; we get 32 for the next to the o n 8 28 o last c lumn ; we a nex giving 3 , and adding as bef re 2 wo 28 6 . o t we get 3 , 33 , 343 35 As this has m re than o o o digits it is written a line bel w the ther additi n , its left hand figure under the right hand figure o f the other

two . ? sum . The are then added 24 m rmrsrm

An interesting va ria tion on the a bove method is the l lo w Ta ke h sa m n m l to he a dded. fo ing. t e e e p e We sta rt a s before a t the lower left ha nd figure a nd h n i 2 Then a s a dd u the left a d row which ve3 . p , g 5 before a nnexing the 7 a t the top of the next rowwhich ive 2 w 28 1 H g s 57 a s before e finish the rowa nd ge1 . ere h difieren n W ut 28 0nl a nd t e ce wmes i . e p down y a nnex the g a t the foot o f the next colummvvhich gives 1 Th n r e in a s b f h n w a h the o 9. e p oce d g e ore w e e re c t p i l h 2 T his of the th rd co umn fromt e lefuwe ha ve 4 . o t a nnex h 8 a t h t f h ri ht ha nd cohi mn n t e t e op o t e g , maki g

ﬁna l sum f the la st o era ti n 2 which is ut a s the o p o 45 , p in l Th n a down directly p a ce . ere is here o subsidiary d dition th whol sum is ut down in one row com ; e e p , i u. p eting the addition . ( See example ) Ha d there been more than four columns the numbers 28 would have ﬁrst been put down ; then when the second addition was completed the two left hand numbers 45 2 o c would be put down , and the w uld be arried and x m annexed to the next ( See e a me b . )

284534

26 ro i o w is requ red . Three figures in any sum can nly occur when the column is mo re than twelve 1 1 5 the o figures high . Suppose that first c lumn o f o co m e 1 1 6 f ur lu ns add d up , the third ,

22 h 1 2 u 1 2 . 3 , t e second , 3 and the fo rth 5 They would be put down thus : 1 60536 In such a case as this it would be much to u o t o better set them down in reg lar rder. The me h d only applies when practically each sum is less than 100 ;

sum contains less than th ree ﬁgures . Another method o f adding is shown in the next example . S o 38292 5 tarting at the left the c lumn is added 1 o up giving 5, which is put d wn as n o . o 976879 sh wn The next c lum is added up, 2 o 2 it gives 4 ; this is put d wn , the under 1 54778 1 the 5 and the 4 next to the 5 on the same ” U 2 o o line . The ther c lumns are treated in 1 1 1 8 the same way ; their sums are 7 , 7 , 1 758901 and (23 1 All are put down obliquely on o o f two lines and the ﬁnal additi n gives the sum the whole .

Decimalized Addi ti on

For additions o f two small quantities a convenient way is to make a decimal out o f one of them by addition or h o b o a s t e . subtracti n . case may be The ther num er is

two . treated in the exact reverse way, and the are added Suppose 97 is to be added to 86 ; add 3 to the 97 making it 1 00 ; treat the 86 in the reverse way and subtract 3 from 8 1 8 to . it , which gives 3, and the sum is instantly seen be 3 If we only add the 3 to the ﬁrst number let the other a lo o o ne and add , the subtracti n of 3 fr m the sum gives

the desired total . ADDITION 27

86 - 3 83

m o Carrying out the same ethod to a fuller devel pment ,

leads to the following method of performing addition . Reduce ea ch number to a decimal by adding or sub tracting as you please ; add the decimals a nd add or n subtract the increments, addi g such as you subtracted 8 o . s 1 6 and subtracting th se you added Thu to add 34 , 9

02 . to to and 3 Add 59 the first number, add 4 the next one 2 o t i and subtract fr m the las . Th s gives the result fi shown ; the unchanged gures are first given , then the new

ones and then the increments to be added or subtracted .

1 600

As 59 and 4 have been added to decimalize the numbers to o o f e they have be subtracted fr m the sum the d cimals , 2 o and as has been subtra cted it has to be added . S we 6 to 6 s 2 say 59 and 4 are 3, be subtracted and 3 les gives 61 to , the net sum be subtracted .

Sight Reading in Addition

To be able instinctively to name the sum o f several o ﬁgures is what is s metimes called sight reading . Take the two following additi ons : 23 to be added to 45 and 59 to to be added 75 . Treat each case as two separate - m na n 1mm

a dditions a nd to mke this cleer we sha ll write the ﬁgures v ith dots to se a r te them in ul p z co p es.

5 - 9 4 - 5 7 - 5

6 . 8

1 3 4

The eye is supposed to be fixed between the two . columns where the dots a re flo immedh tely see a fi a nd O i course the rea der will see tha t this is not a ca se o f stra ight ddition it i m ll a ddin in the one ca se 6o t0 8 a nd in a ; s y g , me other ca se a dcfing 1 20 to l 4 ; but it is supposed to be i i o f in done nstinct vely . There is a bit psychology adding — the calculator who hesitates is lost— the minute you let youe think and cease to rely on instinct you f il in ra id will a p work.

Inverted or Left Ha nd Addition

When two large numbers are to be a dded to each other O e the peration is usually b gun at the right hand . It is to i the much better beg n at left hand. This is called

inverted or left hand addition . a Add ment lly the left hand figures . If the two next to tha n a dd up less flmn g write down the sum of the f le t hand figures . If the two next add up more than 9,

carry one, that is increase the sum of the left hand ﬁgures b l a nd write it down. I f the two next a dd u y p g, then look at those next to them on their right and see how 29

I f tha n ri d n the a ddition they a dd up. kss g w te ow if 9 is the sum o f the neighboring ﬁgur es see if their I f there is noth next door numbers a re less tha n g. so ing to ca rry ; if over 9 ther e is l to ca rry ; if the sum is g then go one mo re to the right a nd see if there the um e or s is l ss than 9, is equal to 9 is more than 9, and i t proceed accord ngly . It is perfec ly simple when you do it, though it seems a little complicated in the description . l The fo lowing example will make it clear. Add the following numbers :

1 620001 0

e o i o 8 We add m ntally, with ut writ ng it d wn , 7 and , 1 which are 5 . The next numbers on the right add up

1 o 1 6. more than 9, therefore we carry and write d wn The next numbers a re 9 a nd 2 making 1 1 ; next on the o f a m n right is a sum 9, the s e o its right, and then a third sumof 9 a ndthen a sumof 7 a nd 3 which is 1 0 ; this means ’ the carrying of 1 right down the line giving the three o s one r to m e t 2 and to car y be added to the 9 and , making 1 2 so o the 2 1 it , we write d wn having already carried the . We now g o back to the 7 and 3 ; next to them on their r 8 a nd 2 to 1 0 ight are be added ; these give , so we have to carry 1 to the sum of 7 a nd 3 making it 1 1 ; we write ow 1 l 1 d n , having a ready carried the other we then write sumo f 2 8 0 down for the and a cipher, , because we have h 1 already carried t e . This is a most complicated example ; ordinarily it goes u m ch easier, and with a little practice the addition is ou written down as if y had it by heart. It is especially in taking out loga rithms that this method

o s o f a in is rec mmended , but it is the be t way dd g numbers n when only two high . It is ot so pr actical when the l or o i co umn is three m re h gh .

o o o o f co e n The f ll wing is a s rt mplem ntary additio , o which is very practical and useful . It is a meth d of two o o adding c lumns at nce . o : 8 8 6 Add the foll wing quantities 7 , 54 , 9 and 5 .

78 +22 = 1 oo 54 — zz = 3z 1 00 + 89 1 32 '+68 =2oo 200 +2 1 22 1 22 1

o 22 to 8 Its c mplement, , is added the ﬁrst quantity , 7 , 1 00 o e o giving . The same c mplem nt is subtracted fr m the

. to 2 next quantity be added , 54 ; this gives 3 , which is 1 1 2 h to 00 i . t e added , g ving 3 This is sum of the ﬁrst two To 1 2 o 68 quantities . 3 is added its c mplement, , 2 68 00 . o o giving This c mplement, , is subtracted fr m the o f to 8 diﬁ erence third the quantities be added , 9, and the , 2 1 2 i 22 1 to 00 . to , is added , giv ng It is as well here add m e 6 22 1 h the last nu b r, 5, directly to , which gives t e

286. answer, In cases where the complement is larger than the num ff ber next in order, the di erence between the two is r o subt acted fr m the hundreds quantity . As an example 62 1 1 2 1 add , , 3 and . 62 4-3 8 4- 1 00 38 (to be subtracted) 62 100 — 27 : 73 27 — 23 = 4 (to be subtracted) 1 1 73 +27 = roo 23 100 4 : 96 1 96 +1 97

o 8 o f 62 c mplement, 3 , the first number , , is larger o e o sec nd number . Th ref re the process is and the second number is subtracted from the ement and the difierence is subtracted from 1 00 to en f being added it . As the complem t of 73, by th is subtracti on is larger than the third num ro 2 the latter is subtracted f m 7 , and the dif b t o 1 00 o f e is su trac ed fr m , instead being add d 5 6 to gives 9 , which we add the last figure of

and thus arrive at the total , 97 . cticin o o on l g the perati ns paper at fu l length , 1 be no difﬁculty in doing it almost entirely HAPTER III C .

Subtra ction

Arithmetical subtraction is the subtraction of a smaller

number from a larger one. In algebra the reverse may n t i be do e, the smaller quanti y may be the m nuend, and f l n then the remainder will be af ected by a minus s g . But i o this i s outs de the sc pe of arithmetic. Subtraction may be ta ken as the simplest of the four e o f o n elem ntary processes arithmetic ; rdinarily additio , multiplication o r division may be expected to involve l N more comp ication . evertheless there are a number of o f modiﬁcations subtraction , a number of ways of doing o n it , and as we shall see, it may inv lve complicatio s .

Principles of Subtracti on

o 0 u If we c unt the cipher, , as a digit or n mber, there 1 00 will be subtractions of number from number. Thus from 1 we may have to subtract any one of the ten digits : for 2 the same is to be said , and as there are ten digits the f 1 0 1 1 total number of dif erent subtractions is x 0 or 00. “ In some of these subtractions it is necessary to bor ” row ten and to carry one there are forty ﬁve subtractions of single digit from single digit in which

this has to be done . To perform the operation of subtraction these oper a tions o e have to be kn wn p rfectly . 32

gives 32 the same remainder as in the other

Decimalized Subtraction

To utilize these principles we select a number to be o n added to or subtracted fr m the minuend and subtrahe d , which by such addition or subtraction will produce a r i multiple of ten as the subtrahend . In the p eced ng par h o H a ra . g p , it will be observed that this was d ne aving added or subtracted the two quantities fro m minuend t c i and sub rahend alike, we will have an ex eedingly s mple

operation to do to obtain the remainder. 86 8 Let it be required to subtract 3 5 from 97 3. The method could be carried out by adding 1 35 to each num ber o the . This is hardly w rth while ; regular way is about as easy . But we may divide the quantities into couples and add 5 to each o f the right-hand couples and 2 to - o e each of the left hand c upl s , and a very simple o o perati n will give the result, the remainder .

b. c.

97 +z ==99 3865 +1 35 =4ooo 38 +z =4o 6s+s= 7o

The regular method of subtraction is carried out in

. b a . o 1 example ; in the additi n of 35 is employed , and in i c. o the th rd, example , the subtracti n is done in couples , o 2 o f after the additi n of and 5 . e to o The obj ct be attained is obvi us . The subtraction o f a decimal number is simple and easy, and such sub traction is brought about by these methods , SUBTRACTION 85

m i So etimes when work ng by couples, there will be one

i o e fﬁ . to carry . Th s intr duc s no di culty of any moment

' 1 t diﬂer If there is one to carry, is sub racted from the f n ence o the couple next o the left . 6 Subtract 3983 from 97 5 . This is to be done in couples . 1 f - a to Add to the le t h nd couples , and add 7 the right

57 82

i 0 2 1 In subtract ng 9 from 7 we had to borrow , so there ' is one to carry and this is subtracted fro m the diﬁerence o f 8 — 0 8 9 4 , giving 57 instead of 5 . Sometimes it is better to subtract numbers from quan o f o e tities the pr bl m , rather than to add them . If we in u of o work co ples , one pair c uples may be treated by ut o the o s b racti n and ther by addition. The last example o will be d ne in couples ; in example a . 9 will be sub tracted f rom the left -hand couples and 3 from the right . ' 1 hand couples ; in b . will be added to the left couples and t o c 3 will be subtrac ed from the right nes ; in . 9 will be subtracted from the left couples and 7 will be added to on the right es .

57 82 57 82 57 82 33 m o

In a ll h r n o c rr h h second ca ses t e e wa s o e t a y, so t a t t e o in subtracti n gives 57 instead of 58. In writ g out the result the division into couples is dropped— the remainder 82 is given as 57 . o on A variati n the above is, as the case may be, to add f On sub to or subtract from only one o the quantities . tracting a ﬁrst remainder is obtained ; if a quantity was to to the added the subtrahend , it has to be added ﬁrst remainder ; if it was subtracted from the subtrahend it

has to be subtracted from the ﬁrst remainder . If the h or e o minuend is t e number added to subtract d fr m, the

first remainder is treated in the reverse way . A single ffi o t o example will su ce t illustra e this meth d . 86 1 28 . 0 Subtract 287 from 3 3 Add 3 to 7 giving 3 0 . This is subtracted from 3863 and the 1 3 is added to the remainder ; 3863 and 3563 f n 1 This is a case o addi g . Suppose it was 3 3 to be o 6 1 subtracted fr m 357 . Taking 3 from the subtrahend 00 u o 2 leaves 3 ; s btracting this fr m 3576 leaves 3 76. o e 1 Fr m this we hav to subtract the 3 giving . us the final answer 3263. Subtracti on by Inspection

The subtraction of two-ﬁgure numbers can be done by to decimalization by simple inspection . Thus subtract ro out 39 f m 73, cast the 9 and the 3 and say 30 from 70 v 0 lea es 4 . Now the 3 has to be added and the 9 has to u o f e be s btracted, giving a net ﬁgure 9 l ss 3 or 6 to be o 0 subtracted fr m 4 , and we have the answer 34. We give a b c three examples , , and .

0 73 — 39 = 4o b . 97 — = c. 98 37 6o The left hand example shows the regular subtraction and the right hand one the addition metho 6 7 o 28 1 0d. To do it ﬁrst write d wn , 32I 23I

the subtrahend. Put a line under it o 8 and below this line put d wn 9 7, the 706 987 r to minuend. Then p oceed write above the two numbers a number which added to 28 1 will

8 u e 08 . give 9 7 ; this n mb r is 7 , and is the answer N ow subtract 283 from 931 by addition. We proceed as before except that here we have to

carry one in the operation twice . Thus having 28 o th 1 written 3 ver e line and 93 under it, we 8 1 1 o 8 we say 3 and are . We put d wn the over the 3 1 8 1 1 and carrying , we say, and are 9 and 4 are 3. The 4 is written above the 8 and 1 is carried to the 2 ; this 2 1 6 6 gives and are 3 and 3 and are 9. Writing the ov 2 6 8 ab e the we have the remainder, the answer, 4 .

Inverted or Loft Hand Subtraction

t or - Inver ed left hand subtraction is so similar to or, o o o better, anal g us to inverted additi n , that, if the reader one ot o t has acquired the , the her f llows . It is the be ter way to subtract and should always be employed in pref to ence the usual way. S 2 1 8 8 068 2 ubtract 7 93 3 from 9 7 . It is not necessary to write them one above the other ; it can be done by i on s mple inspecti . We will write it out as usual however, n although this is quite u necessary . 72931 38

£5 1 371“

Proceed thus : 7 from 8 leaves 1 -there is nothing to ca rry a s the next two figures to the right subtra ct without ’ o n 2 a s borr wing ten. The from 9 leaves 7, but the next fi u 0 e — so two g res are 9 from , we hav to borrow ten in a f Th next fi m ste d o 7 we write 6. e gures co e reg ula rly until we get to 3 from 7 ; this is followed on the 8 2 so o f i right by from , instead 3 from 7 g ving 4, as 1 n there is to carry we write down 3, and fi ally write 4 as 8 1 2 e one the remainder of from . Aft r a little practice can write the result o ff almost as rapidly as if it were i e co m being cop ed. For work in logarithms compet nt uta tor do o the p s it by simple inspecti n , writing result as if copying it . S t 8 1 H ub ract 999 from 999 . ere we look down the line to the right and see that we have in the first and l to second p ace the right , 9 from 9 ; these involve carrying o 1 one because they are preceded by 9 fr m . If we carry 1 to g from 9 it gives a rema inder of 9 a nd a ga in 1 to 1 carry . The Operation is this : 8 from 9 leaves but as we see that there is 1 to carry the remainder is nothing . 1 so 9 from 9 would give nothing but there is to carry , it becomes 0 from 9 giving 9 as the first figure of the remainder . Then there is another 9 f rom 9 with I o 0 o carried giving theref re fr m as the next figure . now o 1 1 2 We are at the end and say 9 fr m gives . As — 2 we get each figure we write it down 99 . SUBTRACTION 39

The arithmetical co mplement of a number is the t e ma inder found by subtracting it from the next highest

t 1 0 1 . . mul iple of . It 9 abbreviated as a c — = 1 1 . Thus the complement of 1 is 0 9 ; the a . c of 6 1 00 — 6 S 3 is 3 or 37 . uppose it is a decimal, then the 61 The of . 61 same rule applies . a. c . 3 is 3

The universal way of calculating the a . c . of a number is to subtract its right hand figu re from 1 0 and the others ’ f i o f 1 s. . . o from 9. Th s avoids the carrying The a c 69572 is obtained thus : 6 from9 gives 3 write this down o 0 t as the left hand figure . 9 fr m 9 gives ; his is put down on the next place to the right ; 5 from 9 giving 4 2 comes next on the right ; then 7 from 9 giving , and at 2 o 1 0 no o e the extreme right fr m ( t fr m 9, becaus it is the right hand end o f the row) and the final figure is 8 .

f 2 2 . The a . c . o 6957 is therefor e 304 8 The next highest u o f 1 0 1 0000 1 o o m ltiple is , or f ll wed by as many ciphers as ther e are digits in the number . If this is tried once o f u or twice it will be found that the a . c. a n mber can be written out just as quickly a s the number itself .

' n i . e o e The a c . is us d in subtracti ; it is spec ally available where a quantity is to be subtracted from the sum o f several others . It is constantly employed in logarithmic o calculati ns . To r f f e o . o subt act by m ans the a . c add the a . c . the subtrahend and subt ract the multiple o f ten used in ob n r taining the a . c . This merely mea s to th ow out a 1 to the left of the a . c . along with any ciphers which come e n h b twee it and the ﬁrst number of t e sum . To subtract 8 2 . o : 9 7 from 9903 by the a c . proceed a s f llows a c f 8 2 01 28 . o 9 7

and throwing out 1 0000

a o f 1 The 0 is kept before the a. c. to ﬁx the pl ce the '

diﬁerence or r 1 . to be thrown out. The emainder is 3

There is nothing gained by its use in so simple a case .

But now take a more complicated case . Suppo se that 1 1 93 is to be subtracted from the sum o f 8 6 1 2 1 1 1 o f u 0 . 9 3 , 7 and 9 The sum these three amo nts 1 1 1 6 1 20 o 0 0 . is 99, and subtracting 93 fr m this gives 9 T o o o . o do . This involves tw perati ns it by the a c . write the three numbers to be added one under the other and f o . unde r them the a . c . the subtrahend

9836 1 072 1 1 91 c f 1 1 880 a . o 93 7

adding throwing out 1 0000 1 0906

The throwing out process is simmy the subtracting the f 1 0 o . c . or r 1 multiple used in getting the a , subt acting o o fr m the pr per decimal place. The proper decimal place is that occupied by the 1 o f the multiple o f 1 0 used f 1 1 1 . or 1 0000 1 in getting the a c ; 9 we used , so it is in to o ut S 1 the ﬁfth place which is be thr wn o . uppo se 999 8 o 1 1 . To o f 1 is to be subtracted fr m 93 get the a . c . 999 o 1 0000 f it is subtracted fr m . O course what is really done is to subtract the 1 from 1 0 and the other ﬁgures

li Thi ive8 22 fromwhich column a bove the upper ne. s g h sum o f 2 1 a nd 1 then from is to be subtr a cted t e , ; 4 1 gives 1 8 ; the 8 is put down and the 1 is carried to the next column above the line and the process

is carried through to the end . If there is a l a e figure in the ten p ace at the l st , as th re is in the exa mme it is put down and the oper e ation is complete. The next xample shows a case where the carrying 1 process applies to o o o o f subtra the b tt m figures, th se the 8 i 26 hend . First 9, and 9 are added giv ng 2 8 6 0 . keeping this in mind add , and giving Then 6 ( o f the 26) is subtracted fro m s f r 2 o o o o l and the dif e ence , , is put d wn bel w the b tt m ine , and 2 is carried but this time to the l ower figures giving 2 8 6 2 + 7 3. The figures in the upper column add up 1 6 ; 3 ( o f the 23) from 1 6 leaves 1 3 ; put do wn 3 and carry the difflerence between 2 ( o f the 23) and 1 h 1 6 (of t e ) to the next lower column . It is added to the o o n 2 o o l wer c lum because the larger number, , bel ngs bel w in the upper line. Follow g out the same method we next ha ve to subtra ct 27 from 1 3 ; this gives 6 to be put on the row 2 f w lower , and to carry to the sum o the next lo er column and o to be carried to the sum o f the upper column ; the net result is 2 to be carried to the next lower o l to a 1 2 c lumn . Then fina ly we have subtr ct + o fr m the sum of 4 4 9, which gives the last or left o f 1 0 c hand figures the subtraction, , which are ac ordingly set down . the o The process is perfectly simple, principal p int being to keep correct on the carrying ; youmust carry to the proper place. The next method of doing the same kind of subtraction o n colums is a variati n o the last . The figures in the lower are added and their sums are subtracted fromthe multiple o f ten next a bove each n s 1 col o e. Thu if they add up 4 in any 20 e umn this is subtracted from , b cause 1 1 that is the multiple o f 0 next above 4. This gives and the 6 is added to the n S upper colum . uppose the upper col umn with this additi on o f 6 adds up 23 ; then 3 is put down a nd as there were two tens in the number used to give the com plement o f the lower column and also the same number o f tens in the sum of the upper column there is nothing t Ha d o e o e 20 for o carry . the l w r c lumn requir d the i o e subtracti on giv ng the c mplement , and had the upp r

o 2 f o f . . c lumn added up 3, then the dif erence the tens , i e one two o to t ten less tens, w uld have be subtrac ed from o o Ha d c the next l wer c lumn . the tens in the upper olumn additi on exceeded the tens used in getting the complement of the lower column the difference would have to be added to o m the next lower c lu n .

now to o Referring the example, the ﬁrst lower c lumn adds up 1 4 ; its complement is obtained by subtracting 1 4 fro m 20 giving 6 ; this is added to the upper column and the sum is 23 ; 3 is put down and there is nothing to carry 2 o f 2 2 20 h because the the 3 balances the of the , t e latter the number required in getting the complement o f the l ower column addition . The second lower column adds 44 up 9 ; its complement 1 0 — 9 is 1 ; this is added to the column a bove it giving 20 ; 0 is put down a nd the dif ference between i o a nd zo is ma de unita ry by dropping the o a nd a s the la rger ten ﬁgure belonged to the upper f 1 column the dif erence , is subtracted from the third lower column so that its sum becomes 1 7 ; its complement

o i 1 . is 3, which is added int the upper column giv ng 3 The diﬁerence of complement number o f the lower col umn 2o a nd th tens in the sum o f the u er column , , e pp favo rs the lower column ; the difference therefo re is added to the fourth lower column instead of being subtracted f r 1 o i 8 as before . This dif e ence is ; the additi n g ves as of ou o o o e the sum the f rth l wer c lumn , its c mplem nt i 1 s 0 o em i 2 . to number , and its c mpl ent s This is added the upper column and the sum is 2 1 ; 1 is put down and

diﬁerence o f 1 e o ou the the tens , , is add d t the f rth upper c no o o olumn , because there is l wer c lumn to subtract it

. 1 6 set o from This gives , which is d wn , and the Oper o ation is c mpleted .

o in t The f llowing is the line of proper ies of numbers .

Subtract one number from another ; then subtract the e o one s c nd number from the first . In case you will have o o t i ut o to b rr w ten , but disregard h s and p d wn the unit f figure of the dif erence . If single figures have been used of f n i 1 0 o the sum the dif ere ces w ll be . If d uble figures f 1 00 have been used the dif erences added will give , and on so . Some examples follow SUBTRACTION 45

64 36 454 546

m l the e a n e dd 1 In exa e a . s a u o 0 in the n xt p r m i d r p t , e x m l b th e rema i ers a dd u to 100 in e a p e , . nd p , and the I i hi dto I 0 0 0 . t be not ce a t ca i n t r , w ll i d th the rry ng o e s process i s le ft a ide. HA TER C P IV .

Multiplication a Short Method of Additi on

of Multiplication is a short ened method doing addition . to o to If we are multiply 9 by 7, we at once g the multi

6 o . o o e heart, and put d wn 3 But if the perati n is analyz d , it will be f ound that what has been done it this — seven ’ i u 9 s have been added to each other. Putt ng it in form la we have : ’ Addition of the seven 9 s is the same thing in its result n a s multiplyi g 9 by 7 . ’ The reverse holds ; the addition o f the nine 7 s is the

same in result as multiplying 7 by 9.

The Multiplication Table

The ﬁrst and all - essential thing in multiplication is to

know the multiplication table . As almost universally taught the table includes as its

upper limit, twelve times twelve. At ﬁrst sight the multiplication tabl e seems much o o r m re f rmidable than it eally is . We have seen that the addition table for two number addition is a very simple affair as the number o f sums involved in it a re

o . 1 o i c ncerned It contains only 7 sums in 45 c mb nations . The multiplication table up to twelve times seems to involve one hundred and forty four operations to be 45 47 m On b . memorized . analysis it m much si pler We ne times a s not to be lea rned beca use it is may omit o ,

known to anyone who can count up to twelve . This O f leaves us one hundred and thi rty two operations . 1 a re these eleven are one times ; such as 3 times 3, 4 times 1 are 4 ; leaving these out we have left one hundred o and twenty ne operations . But many of these are i f o . s mple reversals each other, such as 3 X 4 and 4 X 3 l o Counting two reversa s as one peration , which is per fectl e y correct , the Operations reduce to sixty sev n and o f h o on re ea ted many products t ese perati s are p , so

that the number of products is only forty nine . They are the following

4 6 8 9 1 0 1 2 1 4 1 5 1 6 18 20 2 1 22 24 25 27 28 30 32 33 35 36 40 42 44 45 48 50 55 56 60 64 66 70 72 77 80 81 84 88 90 99 1 00 1 08 1 1 0 1 20 1 2 1 1 32 1 44

Extending the Multiplicati on Table There is no particular rea son why the multiplication

table should stop at twelve times as it always does . Many

‘ teachers advocate its being continued up to twenty times

or even twenty ﬁve times. Without absolutely learning all the higher multipli o u o r o he cati ns p to one the ther of t se numbers , anyone , oon doing much calculating, will s learn many of the o multiplicati ns . They can be made available too by taking advantage o f the multiplication o f the lower kno wn 2 r products by o by 4. Fourteen times is seven times multiplied by 2 ; sixteen times is eight times multiplied by 2 for o and so ther even multipliers . Then the squares of the higher numbers such as sixteen times sixteen may 48

im ix f which l t is h ur be ta ken a s four t es s ty our , a s t e sq a e 8 of . But when it oomes to the r ime numbers thirteen seven p , , tha is teen and so on , then there is no simple way, t really

o f o . practical , btaining them It follows that if the learning by heart of the mul tiplica tion table for the higher multiplications is to be odd o accomplished, the number multiplications sh uld be

learned ﬁrst.

It is always easier to multiply by a single number than

one . o 2 to 1 by a double Supp se 9 is be multiplied by 4. If twice 29 is multiplied by half o f 1 4 the answer will be

given . 5 0 we ma y proceed thus

There are any number o f variati ons on this method ; the general principle is to multiply or divide by a number which will make a single number out o f one o f the two o one o f given numbers . The trouble is s metimes that the numbers may be indivisible without a remainder.

Multiplication of Two Digit Numbers

Quantities o f two digits and having the sa me ﬁgure in 6 6 6 the tens place, such as 5 and 53, 9 and 4 , can be mul tiplied as follows : Multiply the units together ; multiply the tens digit of one of the numbers by the sum o f the units of the original numbers and annex an ought ; multiply the tens ﬁgures

Suppose there are two numbers of two figures ea ch to be multiplied ; by addi ng an increment to one and sub o the tracting the same increment fr m other, we make a decimal number out o f one of the two . Then to the difference between the two original numbers add the in en re t crem t and multiply this sum by the inc men , and add The be it to the fi rst product . result will the product of two o i e to the rig nal numbers , if the incr ment was added r l e the larger and subt acted from the sma ler numb r . But if the increment was added to the smaller and subtracted from the larger, then subtract the increment from the difie rence between the origina l numbers and multiply it n o o by the i crement , and subtract it fr m the pr duct of the u i o first m lt plicati n . The first process is carried out in the left hand calculati on— the second described one in the on right hand e. In both the number 83 is to be multi b 62 plied y .

2 1 2 1 62 2 83 3

5 146 46 5 146 54

e 2 In the left hand example the increm nt, , is added to the larger number and subtracted from the smaller ; it is f e two 2 1 added to the dif erence betwe n the numbers, , r 2 and this sum is multiplied by the same inc ement, , and o f 8 is added to the product 5 X 60 . In the left hand in e example the cr ment is 3, it is added to the smaller MULTIPLICATION 51

i f n u 2 1 from the d f ere ce between the two n mbers , , and the 1 e n result , 8, is multiplied by the sam i crement, 3, and the 6 80 product is subtracted from the product of 5 by . The method is interesting but perhaps of little practical value .

Another Method of Increment Multiplica ti on

The next method is a variation on the last one . To each number is added an increment such as will produce a decimal in each case . Then one of the original numbers n is multiplied by the other o e increased by its increment .

This is a simple one figure multiplication . The first number increased by its increment is next multiplied by of one the increment the second , which is also a single figure multiplication ; this product is subtracted from the s first product . Then the product of the two increment not is added to this and is the answer. It is easy to put into words but it is rea lly simplicity itself as will be seen

in the examples .

Let the numbers to be multiplied be 687 by 893. The increments will be 1 3 for the ﬁrst number and 7 for the in second . The numbers increased by their respective m 00 cre ents will be 700 and 9 . They may be arranged as follows : 687 +1 3 = 7oo 893 +7 =9oo

Following the rule multiply 687 by 900 ; this gives 61 00 N 83 . ext multiply 700 by the increment of the other

00 . S 1 number, 7 ; this gives 49 ubtracting we get 6 3400 . i e 1 = To th s we add the product of the increm nts, 7 X 3 52

9L Whi€h gim the a nsr en ﬁi e y uh a d y ags 61 1 ir is 349 .

“ below . = 1 Fromthis mbtn ct th e mdnct 8 62 00. 93 X 7oo 5 p , = = T t 900 X 1 3 1 1 700 61 3400 a s before . o his a dd

The ‘numbers to be mnltiplied ma y rednoe to decim ls a ion of wha t a re ll more ea sily by subtr ct ru y dea m ts. Prooeed exa ctly a s before emept tint v e ha ve ma dd 1 2 h o o Ta k 80 a nd a s t e nmnbers. thr ugh ut. e 5 5 The

805 s=80° 5 1 2 then 805 x 800 x a nd 5 x 1 2 = 60. Adding the three products gi ves the product of i the orig nal numbers, Finally we may add to one number and subtr act from 1 2 the other. Take 8 and 395 as the two numbers to be multiplied . It is obvious that it is a case of adding a nd

8 1 2 395 +S=4OO = = then 81 2 X 4oo 324800 ; 800 X 5 X 1 2 60 ; and the sum of the last two products subtra cted from 20 0 the ﬁrst gives the answer, 3 74 . Ha d 800 00 we multiplied the other way, 395 by , and 4 1 2 1 2 by and as before 5 by , it would have been necessary to a dd two the ﬁrst products and to subtract the third . it is advisable therefore to either a dd increments to both original numbers or to subtra ct decrements from both

The following is an interesting way of multiplying e ﬁ ur ‘num 8 6 e bers . thre g . Multiply 37 by 4 9 First mul ti l 8 60 22680 Now 8 p y 37 by this gives . multiply 37 by 0 02 i 4 9 as follows : 9 times 378 are 34 . Wr te down only

u 02 o n . the ﬁg res , , carry the rest in y ur mi d Then start 8 8 a r 2 with 4 times 37 . Four times e 3 ; add this to the

34 and youhave 66 ; write down 6 a ndf' ca rry the other 6 ; then four times 7 are 28 and 6 are 34 ; put down 4 and and carry 3 ; ﬁnally four times 3 a re 1 2 and 3 to carry a re 1 w 1 602 22680 . no e e 5 We hav 54 to which is to be add d , 1 282 giving 77 the product asked for. n a ice! This method may ot be very pr ct , but this way of 0 0 multiplying by such numbers as 4 9, 3 7 and the like is e xcellent practice in mental a rithmetic . o - The method is a special case of cr ss multiplica tion , n which latter is given further o in this book .

Multiplication of Special Cases of Polynomi als

In multiplication by polynomials it is often convenient to derive one product from another by multiplication or of u f division , instead recurring to the m ltiplication o the

origina l multiplicand .

' S 18 to be multi li d 6 uppose a number p e by 3. After it e has been multiplied by 3 in the r gular order, instead of 6 e multiplying it by , the product alr ady obtained may be t 2 mul iplied by , which will give the identical result, as if 6 the original number had been multiplied by , because the successive multiplications by 3 and by 2 are the same in i 6 effect as multiply ng by . set down the multiplica tion of a qua ntity

The ﬁrst product is the result of multiplying the mul tiplica nd by 3 ; the second product may be obtained by u 6 multiplicati on of the same q antity by . But by the de o method un r discussi n , it is obtained by multiplying

o u 1 1 6 2 . the ﬁrst pr d ct, 94 by The operation as written down does not show whether o the work is d ne by the one or by the other method . 6 Suppo se the multiplier had been 3 . Then the ﬁrst o u e 2 8 2 o pr duct wo ld have be n 3 9 , and the sec nd product r 2 would have been obtained by dividing this p oduct by , o and putting it d wn one place to the left, just as in regular multiplication . The products and their sum would be

1 43352

This method may be applied when numbers divisible o s o n by one an ther are cattered ab ut amo g other numbers . ' 2 Thus suppose a quantity was to be multiplied by 7633. First multiply by 3 ; copy this down one place to the left for the seocnd product ; multiply this product by 2 for 55 the thi rd product ; multiply the original multiplicand by 7 for the fourth pr oduct ; and ﬁnally for the last product h divide t e third product by 3. An exa mme

o out o f o d This shows n thing the r inary way, but it will be seen by inspection how it can be done by the processes o n detailed . An ther thi g to be observed is that if the multiplier and multiplicand changed places the process e i could not be appli d . Thus set the quant ties down thus

In this case multiplication can only be done by the regular m we o fo r ethod, if take the l wer quantity the multiplier . Of course the re is nothing to prevent us from multiplying n upwards , usi g the upper quantity as the multiplier ; then o o the meth d in questi n is again applicable . The method lends itself to many variations ; it is an excellent way o f pr oving the correctness of multiplications done by the regular method . ARITHMETIC

to o a ue the following meth d , where the left h nd fig r s are first multiplied and then the next and the whole then to added gether. l 0 i To multiply 79 by 9 first mu tiply 7 by 9, wh ch gives 630 ; then multiply the 9 o f the 79 by 9 which gives 81 ; 6 0 81 1 1 adding, 3 7 . Now take a three number quantity ; multiply 634 by 8.

600 X 8 = 4800 so x8 = 240 4 X 8 = 32

0 2 5 7 which is the answer.

Fa ctorial or Proportional Multiplication Factorial or proportional multiplicati on is when one of the numbers is multiplied and the other is divided by a li S e quantity which will sirnp fy the operation . uppos we 2 1 want to multiply 9 by 4. We can simplify matters by 1 2 ou dividing 4 by . To carry t the proportion we shall i 2 2 . then multiply 9 by the same quant ty, This gives = = 29 x I 4 58 x 7 4oti

Take 2256 X 36. We take 4 as the quantity to multiply 22 and divide by . Then V: X X In the last example 2255 was multiplied by 4 and 36 sim was divided by it . If either of the numbers can be pliﬁed by division we are always free to multiply the no other by the same quantity . The reverse is t always the o l of case . Multiplicati n may simp ify one the numbers where the other is quite intractable to division by the

1 2 for 61 20 is 5, that is the product of 74 by , and = 1 00 x 20 2000. In the statement o f aliquot parts we simply give the e t t o f 100 E quivalen s of aliquo parts as fractions . ach 1 00 aliquot part is given as a fraction of .

1 160

355

$60 1 9 6

1 00 e As aliquot parts refer to , th y are really hun dredths a o w r , and the left h nd c lumns ith a cipher p eceding — 2 0 &c i 0 . them, can be written as decimals , . 4 unt l we get to the double ﬁgure numbers ; these with the decimal o &c p int read at once as hundredths , thousandths . without i — 1 2 1 2 or r . any cipher preced ng them . V3 more p operly 5,

. 1 . &c 333 . to 1 00 As these aliquot parts refer parts as their basis , they are useful in calculations involving dollars ; the third of 20 of a dollar is 33% cts . the ﬁfth a dollar is cts . o f t The above set parts is far from comple e, for there e i are any numb r of aliquot parts, wh ch may be of use. O with various bases . nce the idea of howto use them is grasped thei r use may be greatly extended according to the operations to be done . Aliquot parts then are of constant use in multipli o cations . The following examples will sh w how they can be used . b 0 2 To multiply y 5 annex two ciphers and divide by . 32 X ou divide by 5 . In this case it w ld generally be easier to r e multiply directly, but the e may be r asons for doing it

The operations mav be combined . Thus to multiply a rmex two e by 75, ciphers and divide ﬁrst by 4 ; th n divide the o riginal number with the two ciphers by 2 and add 2 = 2 00 2 00 the quotients . 9 X 75 9 and 9 1 450 and 725

Practical Use of Aliquot Parts in Multiplication

One o f the conveniences of the use of aliquot parts is i that they enable us to d spense with fractions . They are o to a nd o nly applicable, as given here, hundreds ther our o decimal numbers , but as c inage is decimal they have o extensive applicati n in business calculations . If the n metric system is being used , then agai they may be

useful . S e ome xamples of their applications follow . What is the cost o f 55 articles at ea ch ? Add a 5 5 cipher and divide by 4 ; 94 i 6 6 2 Mult ply 3 items by Multiply 3 by , and 6 or i v add 36, which is 7% th s gi es 2 62 a t be . 7 yards % cts . is to charged This is 96 of 2 w o ” as e add t gether 55 and A, which gives 28 1 1 2 Multiply 79 by 5. This may be done in either o f two ways . We may add three ciphers and divide by 8 ; this is becau se the list o f aliquot parts tells that 1 2 5 1 000 v is of ; this gi es 3598875 . O r we may multiply 8 359 875 as before .

Suppose it is asked what the product of 1000 by is . 2 r This rs given directly by aliquot parts ; it is 6 5. O to 00 1 000 o r 5 , which is $6 of , may be added 56 of

1 000 1 2 n 62 . , this is 5, and addi g the two gives 5 as before u There is another thing to be said about aliq ot parts, n o and this applies to many fractio al calculati ns, where a u o s mixed n mber occurs . It may be t ld by example . 1 2 o one Multiply 00 by 54 . This is d ne in way by i 1 00 2 n multiply ng the multiplicand , , by and the by M1 f and adding the two or the answer . Thus we have = = 1 00 x 2 2oo ; 1 00 X V 25 ; 200+ This f o is simple but there is another way o d ing it, which may o e u be more c nv nient in some cases . The original n mber may be multiplied by the integral number of the multi n plier, and then the product may be multiplied by o e half o t the fracti n and the two are then to be added ogether. t 1 00 2 n In the case jus used , would be multiplied by , givi g 200 to 2 , and that would be added which is 5, and 22 the result will be 5, the same as before . e u If the multiplier had been 454 , th n to the ﬁrst prod ct, 00 o o f 00 4 , there sh uld have been added 943 4 . The

result is the same of course in all methods .

This method is of assistance in some fra ction work. S l 2 uppose a number is to be mu tiplied by 36. It is manifestly easier to multiply the number by 2 and add to the product 56 o f the product than it is to add to the same o f o For product 36 the riginal number . to multiply by

56, we have only to divide by 3 ; to multiply by 36 we have to multiply by 2 as well as to divide by 3 ; one is e twic the work of the other. MULTIPLICATION 61

r not t Mixed numbe s , exactly wi hin the scope of this i S e m n b . trea t mt, may ofte e brought with n it uppos it is a question of multiplying some number by 694 ; this mixed number can be written fo r 54 . Then u 1s 6 to multiply by it, the m ltiplicand multiplied by and

to the product 56 of the product . It is o f the product is equa l to 96 of the mul is easier to do this than to proceed in the e and multiply the original numb r, the by 3 and divide the result by 4 and thus nti r ty to be a dded to the ﬁrst p oduct . b = 8 i b 8 e y 836 %o. Multiply ng 77 y giv s 61 6 or add x %o, what is the same thing ,

by 20 and add . This gives the regular way we should have multiplied divided by 5 to get the quantity to be added

e ﬁrst way is the easier. to 1 00 given here, refer as the base. ved in their use may be extended by This method is of such extensive that it can best he s of examples and by the statements of The computator will have no difﬁculty in l n e app yi g it to a new cas , once the idea is grasped .

Factor Multiplying

To do factor multiplying one of the numbers to be multiplied is factored ; this one is taken as the multiplier the t one and o her is multiplied by the factors in succession . To multiply by any even number between 1 2 and under 2 5 . Divide the multiplier by 2 ; multiply the other num her o e 2 by this qu tient and th n multiply this product by . ﬁrst multiplica tion could have been by 2 and the second by 9 A numb r f multi liers u to 6 in va lue ca n be e o p , p 3 brought within the range of the multiplication table by

division by 3. 8 r Multiply 39 6 by 36. In acco dance with the principle we divide 36 by 3 ; this gives the two factors 3 and 1 2 ; successive multiplication by these gives the product ; 3986 x and x 3 = 1 434 96 Division by 4 carries the principle up to 48 for numbers o a s i 1 08 for divisible by 4. In this way we can g h gh as

numbers divisible by 9. By using three successive mul tiplica tions by three factors of the multiplier the method can be greatly extended ; still greater numbers of factors f can be used until the limit o usefulness is passed . 2 Thus it is as easy to multiply by 43 directly , as it is

a nd u e . by 3, by 9 by 9 again in s cc ssion These are three

fa ctors of 243 which could be used for factor multiplying. This method applies well to numbers ending in one 1 80 r . cipher or in seve al ciphers To multiply by , use as 60 2 0 or a nd . successive multipliers and 9 , 3 We have seen that to multiply by 25 we may add two o ciphers and divide by 4 . Supp se a number is to be multiplied by 24 ; it is clear that the two ciphers may be added and the division by 4 will give the product one time

too high . Therefore all that is necessary is to multiply by 2 5 ; this is done by adding two ciphers and dividing by o 2 4 ; then subtract the number . Supp se 43 is to be mul ti li 1 2 p ed by 4 . First multiply by 1 00 ; this takes care of 1 the initial . Then add two ciphers , divide by 4 and add 63

i just obta ned . Finally subtract 243 and

have the correct result . The operation follows :

243 >< wo =z4300 24300 —z-4 6075

subtra ct 243

301 32

e u were the multipli r, the multiplicand wo ld be

To Multiply by Nine

ti l ri p y a number by 9, w te the number down and

cipher placed after it . Then subtract the num

ber from the unwritten one . It amounts to subtracting the right hand figure of the m nd original from o, carrying one 6082 a 47 . nu ber subtracti ng the next figure of the orig1nal one increa sed by the I carried from the first figure and 0 so on . Thus in the example we say 3 from leaves 7 and 1 to carry ; adding this 1 to the second figure of the o i a 8 i e 1 to rig n l , , g ves 9, and 9 from 3 giv s 4 with carry . one the t 6 Carrying the to nex figure , 5, we have from 8 2 leaves with nothing to carry . 7 from 5 gives 8 with 1 1 0 i to carry ; carrying the , 7 from 7 leaves w th nothing c fi 6 to arry, so nally the is put down completing the u Ha d t 1 m ltiplication . here been to carry at the last the 6 would have been 5 in the product at the left hand . 64

To Multiply by Eleven

To mnlti l b n mb r Then sta rt p y y n write down the u e . by putting down its right ha nd figure a s the 67583 f Then a dd first figure o the product. the second figure to the first and put down the 74341 3 right ha nd figure of the sum if there a re 1 t two figures in it and if there are carry . If here

nl fi ur ut tha t down there bein no r to ca rr . is o y one g e p , g y Add the third figure to the second adding 1 if it is to be fi u o carried a nd wr ite the rst fig re down as bef re. In the example 3 is put down ; then gives 1 1 ; 1 is put 1 6 6 down a nd l is ca rried. Then +5 a re a nd +8 a re 1 i r 1 a nd 1 4 ; the 4 is put down a nd s ca ried. +7 a re 8 1 d 1 6 are 1 3 ; 3 is put down a nd is ca rrie . +are 1 d n h 1 rr 7 a nd 7 +7 a re 4 ; 4 is put ow wit to ca y. This e to 6 o is add d the final figure , , and 7 is written d wn as o f the right hand figure the product .

To Multiply by One Hundred a nd Eleven

To multiply by 1 1 1 the operation is the same essentially one t To as the las described . multiply the same number by 1 1 1 ﬁrst put down the 3 for the right hand number : .

Complement Multiplication

Comﬂement multiplication may be done in a way by taking into the operation the true complements of the numbers, and may be made to include numbers varying in the ten places .

66 ARITHMETIC

To m e o e e o f ultiply two numb rs t g th r, each two digits i in e o o : o 2 and both end ng 5, proce d as f ll ws Put d wn 5 in the right hand pla ce ; multiply together the figures to the left o f the fives and put the product next to the 25 cm the left ; then add half the sum o f the figu res to the left

of the ﬁves , taking them in their proper decimal place , 2 2 Multiply 45 by 5 . Put down 5 to the right ; multiply 2 i 8 o f 2 4 by giv ng ; to this add half the sum 4 and , to 8 o 1 1 which added the already f und gives , which is to o f 2 1 1 2 put the left the 5 ; the result is 5 . M H h f ultiply 35 by 45 . ere t e sum o the tens is uneven . Proceeding as before put down 2 5 ; put in f ront o f it the product o f the tens and add half their sum ; the result is 1 575 . The Operation is done as shown here :

5 X 5 = 25 30 >< 4o = 1 200 35

Multiplying by Two Numbers at Once

To multiply by two numbers at once multiply the two figures o f the multiplier by each figure o f the multipli o r cand , writing d wn the last figu e and carrying the one or o s n m re left hand figure . There is o object in doing o o t it, if the wh le operati n has o be done at length . In the example the work placed on the right should be done e m ntally in practice . 67

1 575 a == 1 1 5 23 7 X 23 = 1 6! 5 X 23 = n s 3622 5 1 X 23 = 23

Next as a development o f the proces s we will put down

o o to the f our pr ducts btained , with the numbers be carried added in then taking the last ﬁgures o f the set a nd putting the last left hand ﬁgure obtained in front we again get the t od o o result . If the me h is applied these f ur pr ducts will : 1 1 1 2 1 2 6 i in e o be 5, 7 , 3 , 3 , and the last d gits rev rse rder, f 6 e with the 3 o the 3 last obtained in front , gives the answ r

or o o . pr duct , as bef re

Multiplying by Any Number between Twelve a nd Twenty

To 1 2 20 multiply by any number between and , the o l f f l owing method is o interest . Multiply the figures o f the multiplicand in the regular o one o ne f rder by by the unit figure o the multiplier . n to a s o If there is anythi g carry add it usual , and add als to each product thus obtained the figure of the multipli on th f cand e right o the figure multiplied . Put down the right hand figu re o f the number thus obtained and one carry the left hand figure if there is . The tens figure o f the multiplier does not enter into the calculation n of directly ; it is take care in the process as described .

An example follows, with the successive steps in detail .

397 1 2 1 7 w n a 1 Put do n 0 a d c rry . c (7 X 7) = d 1 a n a rr . +1 +1 5 L Pil t own d c y 5 4 . (7 X 9) — 75 Put down 5 a nd ca rry 7 3 ( 7 X 3) Put down 7 a nd ca rrying 3 a dd it to the left hand figure of the multiplicand and put down the sum as the last figure . In ea rrying out this method it is importa nt to a ttend to the a ddition of the left ha nd figure to the la st figur e

” Multi plying by the Teens

n e 20 This operatio , when the multiplier is b low in value ” 1 0 m i and exceeds is called ultiply ng by the teens . It on o f o - o to is a modiﬁcati cr ss multiplicati n , the extent that cross-multiplication includes the possibility of any use the il sized multiplier. Its is only limited by ab ity of o i l the computat r . Cross mult p ication is an excellent test ’ object to determine a computator s powers . If the mul tiplier is a large number it is very diﬁ cult to cross multiply successfully .

Cross -multiplying is a method of multiplying by a of one o number or quantity more than digit, with ut o putting d wn the partial products . It will be best ex exa m s The plained by me . following pri nciples apply to its execution . Call the right hand digit o f any quantity the ﬁrst number o f the quantity ; the next digit to the left the e so s cond and on . Then if a quantity is multiplied by n another. any si gle number or digit of the multiplicand multiplied by the ﬁrst digit of the multiplier, will have 69

the first figure of this product directly under itself . If multiplied by the second digit the first figure of the product will be shifted one place to the left ; if multiplied by the third digit it will have its first figure shi fted two f o o places to the left . I the pr duct of a digit c ntains two r on o to figu es , the sec d one will fall int its regular place n the left of the first o e. We will now proceed to give o s me examples . 2 6 6 Multiply 7 by 3. Call 3 the multiplier. Then : 2 x the first figu re o f the product o f the e N : 2 6 = 1 2 to two original numb rs . ext X added 7 X 3 gives 1 2 +2 1 = 33 and 3 is the second number of the 6 product . We have therefore 3 to carry . Finally 7 x = 2 the o 4 and carrying the 3 gives 45 , third and f urth o f 6 o . figures the pr duct, which is 453 i 8 1 o Mult ply by 37, calling 37 the multiplier. It is d ne o in f rmulas . 8 1 1 = X 37 . 7 X 7 ; the first number o f the product . = (7 X 8) (3 X 1 ) 59 ; 9 is the seco nd number with

5 to carry . = (8 x 3) 5 29 ; the third and fourth ﬁgures o f

the product . e o 2 The ntire pr duct is 997 . 6 8 n Multiply 73 by 4 ; the operati o s are in formulas . = 736 X & 6 X 4 24 ; 4 is the first figure ; 2 o t carry . 2 the second figure ; 6 t o carry . (3 X 8) +(7 X 4) 8 the third figure ; 5 to a c rry . the fourth and fifth ues fig r . ut 6182 The entire prod c is 4 . It will be observ ed tha t every figure in the multiplica nd is m lti li d b ev r fi ur in the multi lie r a nd tha t p e y e y g e p , h i i u f l in i r l & c d g t o f the prod cts a ls to ts pr ope p a ce . l f l - Mfllfip y m by fitlIi flhe opera tions a re in ormua zz. 429 x 64s 3 X 9 = 27 ; 7 the first fi u 2 g re ; to ca rry . 4 the second figure mto ca r ry 8 the third

fi re gu ; 7 to ea rry. 5 the fourth fi ure o c rr g ; 3 t a y . (4 X 6) +3 = z7 ; the fifth and sixth fi gures. The entire product is

l 8 2 Mu tiply 39 7 by 49 6. The operations are in for mulas . 7 X 6 2 the first figure ; 4 to carry . 66 ; 6 n fi ure 6 the seco d g ; to carry . 1 39 ; 1 to 9 the third figure ; 3 carry .

u 1 9 the fo rth figure ; 4 to carry . (8 X 4) 1 ’ 3 the fifth figure ; 3 to carry . (9 X 4) + 76; 6 to the sixth figure ; 7 carry . ( 3 X 4) 7 = 1 9; a nd the seventh eighth figure. The entire product is Cross-multiplicati on is the 718 plus ultra of this oper tion ; if youcan do it with a four o r five digit multipli i l and an equally long or longer mult p icand , you may co i I sider yourself a profic ent . It has been explained o ! examples ; after these are gone thr ugh carefully , t i to b princ ple will be understood . It is hard give a ver

o f u for r rule any val e the p ocess . In using it the operations should not be put dow1 o the e u ﬁ ur n thing but r sult sho ld appear, ﬁgure by g i S : By pract ce great expertness can be attained . ome c do o f or o it with numbers twelve m re digits .

S li de Multiplication

Cross -multiplicati on can be done in an other way ca lh out the sliding method. The multiplier is written on . separate slip o f paper but in inverted or reversed orde It is placed over or under the multiplicand and is shiftc constantly one place to the left in succession after tl i i Vi mult plications indicated by its posit ons are done . will repeat some o f the examples already done by tl

o o - o o I regular meth d of cr ss multiplicati n , d ing them o the sliding meth d . 2 6 Multiply 7 by 3. Write the multiplier reversed up< o f : 6 i r a slip paper, thus 3 . Place it over the mult plica with its left-hand ﬁgure over the right -hand ﬁgure of tl multiplicand , and multiply as indicated .

36 2 o f 7 3 X the ﬁrst ﬁgure the product .

Now slide the paper slip one place towards the left ; tu now multiplications will be indicated . 36 “ 800004 72 put do 3, the ﬁ i r gure ; 3 s to w y.

pla ce to the left a nd

36 72 four th ﬁgures of

the product.

The entire product is msﬁ ti 2 6 6 n Mul ply 4 9 by 43. Write 34 o the slip of paper, x a nd prooeed a s in the la st e a mple.

345 429 3 X 9 = 27 ; 7 thc ﬁrst ﬁgnre ; z to ca rry

The sli of a er is shi fted one la ce to the le f p p p p t. 346 429 (9 X 4) (2 X 3) 4 the second ﬁgure

4 to carry .

e one The paper is shift d mor e place to the left. 346 m 8 the

third ﬁgure ; 7 to carry .

t one l The paper is shif ed more place to the eft . 346 429 5 the fourth ﬁg

ure ; 3 to carry.

to the The paper is shifted the left for last time . 346 6 429 (4 X ) the ﬁfth and sixth ﬁgures .

The entire Pd UCt is

' t i the r0dnct is 1 a nd the exocss of nines in pin . s 3 ; p 5, 1 Tb e excess of nirrmin the r duct of the ori ina l 5 is 6. p o g n

There are a number of curious things about the multi li l o f p cation tab e, which things are more interest than

i o f v . ut lity . Some them will be gi en here o t o f o f on If the pr duc s any the divisi s of the table, m o r o i a re o n such as three ti es f ur t mes , written d w , and if their unit figures are added up; the sum o f the figures on o n i i 0 in questi , st ppi g at the n nth place , w ll be either 4 ' exce t in o f or 45 p the case five times . For even number multiplications such as four times or six times the sum of e 0 for or th se units will be 4 ; the odd times , three times A w . s seven times , the sum ill be 45 number of example are given below

Three Seven MULTIPLICATION 75

The left -hand columns a re not added up ; the right-hand i columns a r e the only ones a dded . The ﬁve t mes section is not put down ; this gives as the sum o f its left-hand 2 n o f odd digits only 5 . If e w the sum the numbers in v o or any une en number secti n , such as three times seven 2 fo e i . o r times are add d , the r sum will be 5 It is d ne here i i three t mes, seven times and n ne times

Three times 3 Seven times 7 Nine times 9 9 2 1 27 I S 35 45 21 49 63 27 63 8 1

o on - It is , as bef re , ly the right hand numbers which have the of been added up , and their sum is the same as sum all the right-hand digits o f the ﬁve times division o f the i o mult plicati n table . N ow f take any o the columns and add, this time o o o o n o f ff o h riz ntally , the c mp ne t digits the di erent pr d ucts the o e o ; taking three times c lumn giv n ab ve , its digits : 6 6 6 o add up thus 3, , 9, 3, , 9, 3, , 9. The next c lumn : 8 2 6 1 0 o adds up thus 4 , , 3, 7, , , , 5, 9. The next c lumn , e : 6 6 6 1 2 a nd six tim s , gives , 3, 9, , 3, 9, , , 9, if the digits o f the anomalous looking 1 2 are added together they give the missing 3 . Various degrees of regularity can be traced out for these additions ; the tracing and foll owing them out may be to r o of left the eader . D ing this constitutes a sort

Arithmetic Solitaire . o be Write d wn the nine digits in a vertical column , 76

with 1 a nd ending with 9 Then to the right of this column write another one made up of the

same nine digits , beginning this time with 9, one space above the 1 o f the other column ; by the side o f the 1 of the ﬁr st column place the 8 of the new column and so until all nine

ﬁgures a re written out . Then the full nine times of the multiplication table will be seen

from nine times one up to nine times nine . The two columns are given at the side of the

A Curious Method of Multiplying

It is po ssible to multiply any numbers together and use

o o 2 2 . nly simple additi on, multiplicati n by and division by D Put the two numbers down side by side . ivide one o f t 2 o hem by , put the qu tient under the same number 2 no to re and divide this quotient by . Pay attention e o no m inders . Repea t this until you can g further, or i u i 1 l e unt l the q ot ent is obtained . Mu tiply the oth r e 2 o i of o numb r by , put it al ngs de the ﬁrst qu tient , multi 2 l f o ply this by and put it a ongside o the second qu tient . Keep this up until you have a multiple for each o f the o o n quotients . The quotients can g nly a deﬁ ite distance m O f he o oh and are the limiting ele ent . t pr ducts thus ta ined strike out each one that is opposite an even number quotient ; the sum o f the products remaining will gi ve the o two o pr duct of the riginal numbers . 68 1 Multiply by 9 . It is immaterial which number is successively divided

. o e and which multiplied It is d n in both ways here, t b indica ed as a and . The quantities opposite the odd number quotients have to be added to give the pr oduct of the two original numbers . This is done below in each case below its own calculation .

Oddities in Multiplication

If the digits with the omission of 1 are in l reverse order and multip ied by 9, the product will be a ’ of 8 s succession nine .

N ow write the nine digits including the 1 this tima ’ and multiply by 9 and the product will be the nine 8 s as

the . 1 8 before with a 9 on right end If multiplied by , the of 1 i left hand figure the product will be a , then w ll come ’ s nine 7 and as the right hand figure there will be an 8 . If multi lied b z the left ha nd fi ure will be a 2 the p y y, g , ’ m l fi r il b 6 s nd th ri ht ha nd fi ure nine idd e gues w l e , a e g g I h h the multi les of used a s a 7 . t goes thus t roug p g multi liers until we fin ll et for nine times nine or 8 1 p , a y g fi 8 as a multiplier, the left hand gure, , the right hand 1 a nd i o e o f n figure, c phers t the regular numb r ine , as the

In ea ch product the left and right hand ﬁgures give o r e i r u in rep at the multipl e , and the intermediate ﬁg res run ’ e 8 S o o f r gular order fro m s to ciphers . me the mul

O f course for the left hand figure o f the product of e no to the multiplication by 9, th re is left hand figure be i or i i suppl ed ; the final right hand 9 g ves the mult plier . The number 1 5873 multiplied by 7 gives as product ’ 1 six s .

1 8 : 1 1 1 1 1 1 5 73 X 7 .

N ow multiply this same number by 9 and the product 1 28 i i is 4 57, and th s number multiplied by 7 g ves as ’ o s pr duct six 9 .

1 5873 >< 9 = t42857 a nd 79

s u The following is an odd eries of products . As us al it is the number 9 that is the main factor in the operations .

9 X 9 = 8 1 and 9 X 98 = 882 and 88m+6= 888

The last two in the series are

9 X 9876543 = 88888887 and 9 X 9876543z = 88888888 and 88888888 +o = 88883888

The nunibﬁr l1 53846 multiplied by 1 3 produces 1 as ’ e 8 s the l ft hand digit, as the right hand digit, with ﬁve 9 between them . I 3 = I 999998

In many cases these curious multiplications can be out i so carried by further multipl cations , as to give other nesul ts. We have seen that i f we multiply 1 53846 by 1 3 1 8 f the product is 99999 . If we add one half o itself to b o b 6 2 o 2 0 6 the a ve num er, namely 7 9 3 we btain 3 7 9, and 1 2 n this multiplied by 3 gives 999997 . Addi g it again to 0 6 2 the last sum we obtain 3 7 9 , and multiplying this by I we 6 o 3 get 399999 . This pr cess can be carried down to od 1 8 6 the eighth pr uct, which will be 53 4 multiplied by 5, for that is what the successive additi ons o f 7692 3 leads to ; the product o f 1 53846 by 5 is 7692 30 a nd the product 1 0 N ow of this number by 3 gives 999999 . these products o r e may be placed in c lumns ; the ﬁ st three are giv n in full .

1 3 = 1 999998 5999994 6999993 7999992 s846s 8999991 The left ha nd digits run from 1 to 8 ; the right ha nd ones run from8 to 1 ; the left and right hand digits give 2 &c the products of 9 by , 3 . Other odd multiplications are the following

37037037037 X 9 33333333333 1 37 1 742 1 x9 1 23456789 98765432 1 x9 8888888889

The following way of multiplying two numbers each o 1 0 e m re than 5 and less than , is mor curious than useful , for everyone is supposed to know the multiplication

table . 8 H Suppose we are to multiply 9 by . old up the e f e fingers o f both hands op n . Take the dif erenc between one o f the numbers and 1 0 and bend down a finger or in to f Do fingers correspond g in number that dif erence . this on one hand fo r one of the numbers and on the other f not o hand or the other number. The fingers bent d wn give the tens and the product of the fingers bent down on on a i e h nd by those on the other hand g ve the units . In the case o f 9 and 8 two fingers will be bent do wn on one hand and one on the other ; their product is 2 ; there are seven fingers left standing ; these are the tens ; putting o 2 2 them bef re the we have 7 , the answer . This operation can be carried out very well on an abacus . to o e 1 If the fingers be held d wn exc ed the number 0, o of subtract the excess fr m the tens the product .

82 rema inder comes from the division of lﬁth the number b T n i y 3. his rema i der s multiplied therefore by s a nd b 2 or b 6 a nd is a dded to ust determined ivin y . y , s j , g g

O r t i l d start at the bot om and mult ply the ast remain er, by 3 and add it to the next remainder above it ; this gives ( 2 x 3) Multiply 8 by 2 and add the ﬁrst r i 1 o m ema nder, , and the t tal re ainder is obtained , 1 namely 7 . Abbreviated Long Division

To abbreviate to a certain extent the work of long i out o f the m d vision the writing products may be o itted , a nd the subtractions may be made mentally and the

2 81 1 Divide 7 5 by 3 . 1 2 81 8 f 3 7 5 ( 9795‘ It is put down as or long division . Inspection shows that the first figure 8 8 of the quotient is . Then we say times 1 are 8 and subtracting this first figure o f the product from the dividend we put down 0 N 8 2 to . e with nothing carry ext tim s 3 are 4, which 2 1 subtracted from 7 leaves 3 which is put down . The is taken down mentally from the dividend and it is seen 1 0 1 e i that 3 goes into 3 9 tim s . Proceed ng we have 9 times 1 are 9 and subtracting this 9 from the 1 of the dividend leaves 2 which is put down with 1 to carry . Then 9 times 3 are 27 and 1 to carry makes 28 which o 0 2 t o f re subtracted fr m 3 leaves , the las figure the o 1 22 ma inder . Taking d wn 5 mentally, 3 goes into 5 7 1 2 1 a 22 times. 7 times 3 are 7, which subtr cted from 5 o f 8 ha ves a final remainder , which is written in the o of quotient as numerat r a fraction %1 . DIVISI ON 83

What is termed the Italian method of doing long di visi on consists in putting both diviso r and quotient on the right hand o f the dividend. It is just as " M” ° good t0 put them on the left hand. The va n example shows the method . The a d go 42 tage is that the divisor and quotient are in positi on to be multiplied to prove the correctness of the o i operati n . The operat on is done by the short method s o 2 2 de cribed ab ve . 7 is the divisor and 4 is the quotient .

The proof of the correctness of a multiplication by o f i excess nines has been expla ned . The same can be o o f applied t division. The excess nines in the divisor of n e multiplied by the excess ni es in the quoti nt, and the excess of nines in this product added to the excess of nines in the remainder will give the excess of nines in the o o o e dividend if the perati n has been d ne corr ctly . Take the division o f 9763 by 28 1 ; the quotient is 34 and the m 20 . o f re ainder is 9 The excess nines in the divisor, 28 1 2 s n o is ; the exces of ines in the qu tient , 34, is 7 ; the o o f 2 1 e e pr duct these , x 7 is 4 ; the xcess of nin s in this o c o f n e e i 20 pr duct is 5 ; the ex ess in s in the r ma nder, 9, 2 f 2 is ; the sum o 5 and is 7, and we ﬁnd the proof that the division is in all probability correct in the excess of nines in the dividend , which is also 7 . While it is quite conceivable that the excess o f nines would come out right im rob when the division was wrong, it is exceedingly p able ; if they do come out wrong the division is certainly o inc rrect . If the prowss be compa red with the simila r opera tion f o f multi liea tion it for the proo p , be the ll w f m h omer one fo o s ro t e .

’ 28 1 2 Excess of 9 s in the divisor, is ’ E s o 2 xcess of 9 in the qu tient , 34, is 7 ; 7 X 5 ’ E s 20 2 xcess of 9 in the remainder, 9 is , to be added to above

7 ’ E o f in 6 xcess 9 s the dividend, 97 3 is 7

It follows that the division has been correctly performed because of the agreement in excess of nines .

Remainders in Division

When a number is used as a divisor of numbers indi visible by it without a remainder, the possible remainders on are e less in number than itself . o d Thus for 3, used as a divis r, when it ivides a quantity r giving a remainder, the e can only be two possible re ma inders 1 2 . e , and For the numb r four, used as a o 1 2 divisor, the only p ssible remainders are , and 3. The o same rule applies t all other divisors . 85

When the division is carried out beyond the decimal n o com point, the remai der may give a c ntinued or a

I f o f u i i o divisions n mbers, ind visible by the d vis rs ou used, are carried t so as to give decimals in the

t . quo ient , interesting results are obtained

2 h e e . . With as a divisor, t e only possible r maind r is 5 o With 3 as a divisor, there are two p ssible remainders, 666 t . d . bo h continue decimals, 333 and

o e 2 . . . With 4 as a divis r, the remaind rs are . 5, 5, and 75

2 6 . 8 . o n . . . With 5 as divis r, the remai ders are , 4, and 6 o or con With as a divis r, the remainder may be . 5, ’ ’ tinued r decimals ending in 3 s o in 6 s. 8 o the end a With as a divis r, remainders lways in 5,

two . 2 1 2 and are one, or three ﬁgure decimals, 5, . 5 5, 5 2 3 375, 5 and 75. o This leaves the numbers 7 and 9 to be c nsidered . n The remainders given by the number 7 are repete ds, e identical in all cases, except that the d cimals begin with different numbers ; the order o f the component digits is the same. The decimal remainders for the six possible n a re o : remainders of seve used as a divisor, the f llowing

2 . 857 14 .

4 28571 .

The ﬁrst o f these remainders is divisible by 7 with a 1 remainder of , reducing to the same decima l ; this is why e it is a rep tend. The quotient obtained by dividing it by o 7 is a rather curious successi n of numbers . u e The n mber 9 gives as r mainders continued decimals , of which each one is simply the repetition indeﬁnitely of 86

the rema inder ﬁ ur u o we ha ve to i i 2 g e. S pp se d v de 53

b . The uotient will 28 2nd 1 ver r i y 9 q be 0 , 0 a rema nder of 1 T t this n d m l w s m l . o pu i to eci a s e i p y repea t the

quotient is Su n h 2 0 i b ivid ppose owt a t 9 s to e d ed by 9 ; this gives a quotient of 32 and a remainder o f 2 ; in decimals the e 2 e i e remaind r is repeat d nd ﬁnitely, as a continued dec l ima , and the quotient is

This covers single digit divisors . The reader can try t r a d libitum t t o her diviso s , and will obtain qui e in eresting r esults .

A number divisible by 2 is called an em number ; one of — 2 6 8 such numbers end in the even numbers , 4, , 0 or else in a cipher, . A number is divisible by 3 when the sum of its n 1 2 digits is divisible by 3. Thus the umber 3 has as i t 6 d gi s As this sum, , is divisible by 3 l 1 2 2 2 it fol ows that 3 is divisible by 3. The digits of 5 add up to 9 ; As 9 is divisible by 3 it 2 2 two follows that 5 is also . The quotients of the a 1 8 examples given re 4 and 4. 6 Any even number divisible by 3 is divisible by . Thus the number 252 is an even number because it ends in 2 it is divisible by 3 ; therefore it is divisible by 6 ; divided 6 ot 2 by the qu ient is 4 . A number is divisible by 4 when its last two digits are

e . two i 82 2 divisibl by 4 The last d gits of 9 4 , namely 4, i o are d visible by 4, theref re the whole number is ; on t o 2 6 trying it the quo ient is f und to be 45 . DIVI SION 87

r i ending in 5 a e div sible by it . e 0 i All nding in are div sible by 5 . 2 All numbers ending in 5 a re divisible by it . A number ‘the sum o f whose even placed digits is equal 1 1 to the sum of the odd ones is divisible by . Thus the sum of the even digits of the number 1 538779 20 20 is , and the sum of the odd digits is also ; the number o e 1 1 theref r is divisible by . If the last three figures of a number are divisible by be 8 then the whole number is divisible by it . This is 1 i 8 so e the cause 000 is div sible by , whatever pr cedes last three figures it eventually come s to adding one o r ’ r u o a flect mo e thousands of them, which of co rse d es not 8 1 28 8 their divisibility by . Take ; this is divisible by ; 1 000 1 1 28 c if we add we have , which of ourse if divided 8 1 1 1 2 8 1 6 1 000 by gives 4 . 8 divided by gives and 1 6 1 2 = 1 1 N o divided by 8 adds 1 25 to it making 5 4 . matter how many figures are put before any three figures divisible by eight it only amounts to putting so many ’ 1 000 1 000 8 . s before the three, and each is divisible by i o t s Th s pr per y is practical as well as u eful , and is a parallel case to the similar property of 4. 2 8 u it— 79 is divisible by . Put any n mbers before say 2 8 o o 33 . This gives 3379 . Dividing we find that g es int 1 o e h 33 4 times and v r, so now we ave to divide our i e 1 000 for orig nal number increas d by , that is what the efiect o f 1 it 1 2 carrying is , gives 79 as the number to be It vi 8 divided . di des by without remainder . When the difference between the sum of the odd and v a b 1 1 o e en pl ced digits of a number is divisi le by , the wh le ' number is also divisible by it . 1 2 1 Take the number 549 ; its even digits are and 4, mmis r6 ; the difiermoe betv em s a nd 16 is 1 1 ; mere fore the v hole mnnber is divisible by n ; the quotient is 2 T k h mb 2 1 2 h r h v n di i h ve 499 . a e t e mi er 7 9 ; e e t e e e g ts a

divisibl b u n h h l n mb r is l e y a d t e w o e n e so a so .

I f a n m i in ivi i l b t a m in er n ber s d s b e y t without re a d , a nd we divide it b a nd ca rr it out in decima ls the y 4 y ,

. 2 h t e i . 5 , or dec mal 75 and no other. = Thus -s: 27 6-7s ’ Tb bov in vul a r fra ctions to . e a e reduces g fi, and 94 An a na logous rule obta ins for a ll such divisions ; the l rema inders for division by th ree will be fi a nd % ; for division by five the 96 and to The numera rs of the remainders, when expressed as vulgar fractions will begin with one and go to one less i e than the div sor, and the divisor will be the d nominator o f the fractions . Th t o o et e general sta ement of the above f ll ws, tog her with its application to the single ‘digits used as divisors o f numbers giving remainders . o i If the digits of any number are added t gether, the r sum or the sum o f the digits o f their sum is the remainder h which will be left on dividing t e original number by 9. Thus the sum o f the digits o f 875 is 20 ; the sum of the digits of 20 is 2 ; if we divide 875 by 9 the remainde r will 2 o f 20 be , the sum the digits of . The sum o f the digits o f 962 1 76 is 31 the sum o f the o f 1 62 1 6 digits 3 is 4 ; if we divide the original number, 9 7 the o be 1 06 08 by 9, qu tient will 9 with a remainder of 4, 1 the sum of the digits of 3 .

ifi d r in fi f n r The um fi me of the d end ca r y g ecessa y. remh of the subtra ction gives the next figure of the uo i n T r e is ca rried out ‘ th e end t he oo ss to . q e t. p iv 6 08 1 1 Put down 8 a s the i ht- D ide $ 4 by . r g hand fi u f n b 8 from he n x fi ure g re o the quotie t. Sutra ct t e t g , q o f the dividend ; this gives z the second figure of thé uti n w n t ca r Addin tbe 1 to be ca r ried q o e t ith o e o r y. g to the seeond fi ure o f the uo ient ives a nd sub g q t g 3, 3 tra cted fro m th next fi ure of the dividend 1 e g , 4, gives ,

th t i t o . e h rd figure of the quo ient, with n thing to carry Fina lly this third figure of the quotient is subtracted from 3 r f th 6 i the u the next figu e o e dividend, g ving 5, fo rth figure of the quofienn lea ving s of the quotient to be ' e o u e subtract d fr m the last fig re of the divid nd, 5 , leaving a s th i n 1 2 in h e uot e t 8 these be t e n . q , 5 , g figures fou d

ri To divide by 99, add the two ght hand figures of the dividend to the rest o f the number and put it down under the original number . Do the same with the smaller num K ber, and put it down under the other two . eep up this process until 99 is left or a quantity less than 99. Cut off the two right hand figures of everyone of the quan or e tities , either mentally by drawing a v rtical line ; add on o f n up what is left the left the line ; this is the quotie t . The number at the bottom of the right hand of the line e e 1 is the remaind r ; of course if the r mainder is 99, to has to be added the quotient . 86 2 Divide 943 by 99. The two right hand figures are 32 ; these are added to e : 86 i the r maining figures , thus 94 Th s 91

' is the ﬁr st qua ntity to be a dded to the original dividend . To get the next number we cut o ﬂfrom8726 its two right 26 a nd o f hand numbers, add them to what is left the 1 1 number ; we have thus : 3. This is the second number to be added to our original dividend . The same operati on is applied to 1 1 3 ; its two left hand digits = a re cut OE and added to wha t is left ; this gives 1 1 3 1 i o i o 4 . As th s is only a tw d git quantity n thing more is n now to be do e to it . We set down the quantities

I 4

82 m 1 . 87 with a re ainder, 4

of i 86 2 This is the result d viding 943 by 99. 2 661 Divide 3 by 99. Proceeding exactly as before we have a s our quantities

2 661 a nd - 3 , 99 ; these we treat as before

236 61 2 97 99

2 8 2 o ie . 3 with a remainder 99, giving 39 as the qu t nt

Properties of the Number Three in Division

' If we take any two numbers we will ﬁnd tha t either their sum or their difference or one or both of the num 1 1 bers is divisible by three . 9 and 7 seem rather hopeless , sum 6 6 but their is 3 , and 3 is divisible by 3 . 92 ARITHMETIC

Any number the sum o f whose digits is divisib le by 3 n divided b Melf A clue to the rea son for this ca be y 3 . lies in the fa ct tha t if we multiply a ll the single digits by three we sha ll ﬁnd the nine digits represented in the min l ﬁ ur s Thr im uns— ter a e . e t e r g e s 3, 2 1 2 2 — r the n , 4, 7 these are the p oducts of 3 by ine single numbers a nd the sa me nine numbers a re there in the la st

Now t 2281 1 — the sum places. ake a number, say 74 5 of i N ow r 0 . t its digits is 3 , this is divis ble by 3 y the num ber itself a nd its quotient when divided by 3 is 24742705 l with no remainder. The numbers 7 and 9 a so have the nine digi ts as the last figure s in their multiplications by n ue t e not o the ine first fig r s , but hat do s confer this pr p t but e e ty on 7, 9 poss sses it . S e the uppose a number is not divisible by 3, th n amount by which the sum of its digits exceeds the nearest lower multiple of 3 will be a number which if subtracted from the original one will leave it divisible by 3 without a n 8 o f i s remai der. Take the number 39 3 ; the sum its d git 2 2 1 2 is 3 ; the next lower multiple of 3 is , and therefore is the difierence to be subtracted from the original num h u t o 81 ber ; carrying out t e s b racti n we get 39 , which is e n b 1 2 t e divisibl by 3, the quotie t eing 3 7 and her is no remainder.

’ Lewis Ca rrol s Short Cut This is Lewis Carrol ’s short cut for dividing any mul

. o o f e tiple of 9 by 9 It is nly a matter int rest , and in o f great measure such , because its distinguished origin o f ator, the author Alice in Wonderland and of “ Alice in the Looking Glass the originator of the Won derla nd a l iﬁ arithmetic rules of Ambition , Distraction , Ugl

Vulp r h a ctions

A vulga r fra ction is one expressed a s a division ; the di vimr ch ssiﬁes the fmcﬁm a s being o f the ha lf d a m the

r n o f w o a r c . e t o e e eve A d fini e th e divis rs , which alled den s is en for n— the fi ure ominator , tak each fractio g s th r r expre sing e dividend is called the nume ato . To write a fraction the numerator is placed above a short horizonta l or dia ona l line or ba r a nd h g , t e denom ina tor is o h placed bel w t e same bar . ‘ a r the n 95, 360 e fractions ; first belo gs to the class of o thirds , as its denominat r tells us, the next one belongs i to the class of thirt eths as its denominator indicates . The numerator of the first fraction tells that only one of l t l its c ass is taken , the numerator of the nex one tel s that twelve of its class are taken .

Meaning of ﬁ e Fractional Ba r

One most important thing to be realized about fractions n is the true significance of the bar . It is a sig of division , just as truly as is the sign 943 may be expressed in words as five thirteenths or as five divided by thirteen ; o 1 it c uld correctly be written 5 3. The fracti onal bar may be and often is used as a sign 1 25_ o f — = division . 5 can be written S i both 25 n mean identically the same thi g. 94 95

The fractional bar is sometimes an oblique one ; this is

Although it is never used as the sign of division. a fra ction might be written pe r fectly correctly a s the in dica tion or setting down o f a calculation in division ; %0 t 1 6 for is migh be written ) 3, this the carrying out to its o o full meaning of the fracti n with its bar . If the divisi n is completed the result may be expressed in decimd o fracti n . Changi ng the Value of a Fracti on

There are two ways o f increasing the value of a frac on to tion ; e is increase its numerator, the other to m n n o . e diminish its de o i at r The r verse also is true, if the denominator is increased or the numerator diminished the value of the fraction will be diminished . 4 Thus is smaller than $ , and also smaller than it is larger than or than The reader will observe how the numerators and denominators are increased and i dim nished in these examples .

Reduci ng Fracti ons to the Same Cla ss or to a Common

If youwant to add a gallon to a pint you must reduce them to the same class either to gallon or to pint class ; the sum of the two is one and one eighth gallon or is of nine points . To add two fractions they must be the e o n same class , that is to say must have the same d n mi ator.

Addi ti on and Subtracti on of Fracti ons

1 To add to A we must ﬁrst multiply numerator and denominator of ea ch fraction by the denominator of the 96

the numerato rs are added because they tell how many there a re o f each — if there a r e two of them to be a dded

to three o f them the result is of course . The thin s , , 5 g r x m a dded a e si ths, a nd there a re ﬁve o f the . For subtraction the reverse is carried out ; the reduction to a common denominator is eﬁected and one numerator one o is subtracted from the other . subtracted fr m

$6 would give Ve The addition oi fractions o f different den ominators can be expressed in words without reduction to a co mmon denominator ; the above a ddition can be called a half and o o a third. This is c nstantly d ne in adding fractions to whole numbers— two added to a half is usually expressed as two and a half . It might be expressed as 96 or ﬁve halves .

Multiplication and Divisi on of Fra ctions

If we multiply a number by a fraction whose num I ca n crator is , we equally well divide the same number e o o 2 1 2 2 by the d n minat r . X 79 or give the same e r sult, namely , 36 or 1 . Any number may be regarded as a fraction ; as the denominator o f a fraction expresses what may be called o its class , and as a wh le number refers to the class of units , if a whole number is written as a fraction its de i o 1 nom nat r must be . Thus any whole number can be on written over the fracti al bar as a numerator, with the o o 1 1 2 ” den minat r, , under the bar . 5 can be written 91

never done , just as the decimal point is omitted in writing whole numbers . As a matter of correctness the number

The genera l rule for multiplication by a fraction having m r o l o multi l the denomina to r o f the m a nn e a t r , is t p y ul

' l n mm or t i n tip ica nd by the de o a t o f he fract o al multiplier. Now suppose the fra ction mnltiplier ha d a number gruter tha n unity for its numera tor ﬂt is clea r tha t it wmﬂd be tha t ma n times la r er so to multi l b it we y g , p y y ,

a to the l is numer r of fractional multip ier, or, what the sa e o e e o f m thing, we sh uld divid the d n minator o the n N mltipiica nd by it. ext we should divide the numera tor of the multiplica nd by the denomina tor o f the multiplier. The result is the sa me but it is simpler to multiply the

ti l two i mul p y fractions , mult ply the numerators together for a new numerator and the denominators for a new

A fraction can also be multiplied by a whole number by i n o t divid ng its de ominat r by the mul iplier. To multiply 2 t we o by 5 by this me hod , would btain the expression ,

1 — which is equal to 4 . 5g which is 5 . This is the

s t o n o o same re ul as that btai ed by the ther meth ds . i If a number is to be multiplied it mea ns that the num her is to be taken as many times as there are units in the o multiplier. If the multiplier is a fracti n, say it has one half a unit in it ; if a number is multiplied by therefore it ca n be ta ken one t time ; I O X If a fraction is multiplied by another fraction the same to be principle applies . 55 X 54 means that $4 is taken

one i o f o . to half time, wh ch gives c urse 34 If 56 is be multiplied by 34 it must ﬁrst be taken three times, because n on of the numerator, 3, and the e fourth time because of FRACTIONS 99

e the denominator, 4. If we take $6thr e times it gives if next we ta ke §§ one fourth time we ha ve which is one fourth of If the same operation of division or multiplication is effected on both numerator a nd denominator of a frac n id e tio , it is ev ent that the ratio of numerator to d nom

‘ e A, 34 , are all the sam in value ; the last two are

l and by 3 ; the ratios are the same in a l . If we divide the denominator o f a fraction by one tenth o f itself and the numerato r by the same the value o f the fraction will be unchanged but we will obtain a fraction o m i wh se deno inator w ll be a multiple o f ten. This is a clumsy way of reaching the desired result ; it is easier to divide the numerator by the denominator a o m oi But with due reg rd t the deci al p nt . as a matter of a i expl in ng it is well to try it the ﬁrst way .

Conversion of Vulgar Fractions into Decimal Fractions

Suppose we divide the two elements of the fraction , for 34 by one tenth o f the den ominator . We have the = u o I f n 2 . 2 . 2 or e n merat r 5, and the d nomi ator, = un 1 0 t o . , and the new frac i n is 71 0 The value is e e o n o changed and as it is a d cimal in its d n mi at r, it can o be written as a decimal fracti n, . 5 . o e to The m re dir ct way, and the way always used get o o the a decimal fraction fr m a vulgar fracti n , is direct n f o divisio o the numerat r by the den ominator. To reduce $6 to a decimal we divide as follows 2 ) the same result as obtained above in the mo re -5 rect way . a re divisible by a s; domg thls gtves the

”75 Or boﬂi mi bt h v been divided b i i i s g a e y s g v ng xo, a l o of the mme va l ue a s gs.

o e one un d llars and c nts. A cent is h dredth of a dollar d i 0 1 n a re the tenth a n is wr tten correctly $ . ; ten ce ts of a o ve of d llar, which howe r, as we never think this

t s 3 0. amount as a dime, but always as ten cen . is written $

The final 0 does not affect its value in any way whatever . Now suppose five dollars and fifteen cents are to be added to four dollars and twenty five cents . The additi on is carried out with dolla rs under dollars and cents under cents .

$94 0

Suppose five and fifteen hundredths gallons of turpen tine were to be mixed with four and twenty five hun dredths gallons o f linseed oil and that there was no

‘ shrinkage what would the total a mount o f the mixture be ? The numbers would be put down precisely as above o m Un except that the dollar mark w uld be o itted . its n n would go under u its , which brings the 4 u der the 5 ; then going to the right o f the decimal point we put the 2 I a re o under the , because they b th tenths ; 5 is put o o under 5 because they are b th hundredths . If an err r is made in putting the digits where they belong the result o will be inc rrect . Although the cipher is always used in writing such o ten am unts as cents or ﬁfty cents , the latter can just as o e 1 1 0 c rr ctly be written $ and 5 as $ or 50 . The sum of the last example could have been written Ciphers never need be pla ced after the last digit of a 103

d had its decimal point misplaced , say one to the right— it would then have read $94 example o f a frequent error in arithmetical

keeping units under units and r tenths and so for all integers and fo r all misplacing of the decimal point should occur

ever think o f adding dollars to cents in the same column ; precisely the same applies to decimals when added . Subtraction of Decima ls

o o f e To The same applies to subtracti n d cimals . subtract one o f the numbers o f the last example from the o other, they w uld be written down in exactly the same way ; hundredths would have been subtracted from

o u o a nd hundredths, tenths fr m tenths and nits fr m units fo r o . 0 or . r remainder w uld have been, 9 9, one and same ; it is only a matter

way youselect to write them . appears between decimals and ih as many ciphers as are convenient may right o f a decimal without changing its or one reason or another ciphers are often so it is not so o ften that ciphers are placed to e f an integral number, although it is perf ctly i to log cal do so . Multiplication of Decimals s a matter of carelessness to make an error in 1 point in additi on or subtraction ; in multi plica tion a nd division it is not a n impossibilty a nd it is n n not i freque t. To multiply decimals or mixed numbers there is no necessity of ‘ putting units under uni ts a nd tenths under e u o f ra c t nths , altho gh as a matter discipline and good p i m t ce it is to be recom ended . The decimals in a product are equal to the sum of the decimals in the numbers multiplied . If in the multipli 0 cation a ﬁnal appears in the product , it counts as a decimal and should be written because it has to be taken into account in ﬁxing the place of the decimal point .

Multiply by and also by 9-5

5 9 8 5 3 9 0 0

Both multiplications a re done regularly as in the case of integral numbers ; each of the o riginal numbers has one

e . decimal , therefor their product must have two decimals a r n u s In the case of the first both e sig ificant fig re , and have a value in addition to fixing the place o f the decimal point ; in the case of the second product there are two ciphers ; these a re without numerical value and can just om o as well be itted , but have t be put down to fix the o place of the decimal p int .

A decimal may be deﬁned as a multiple of ten or as some power o f ten or as a dividend of some power of ten . c If the latter it is alled a decimal fraction . by the a mexm’ g o f cipb ers on ﬁi e right ha nd o f the unt en a mmnns to a shiftimo f the decima l point to the right ; this m ld ha ve to be done were it not cus tmmry to m ft the decinn l point in putting down

Division od Decima ls

To divide a decima l fra ction by a decima l number the reverse opera tion ha s to be done ; the decima l point is mo ved one point to the left for every cipher in the i 1 2 i 1 2 1 0 t t . 0 u to i . divisor. Th s d v de 5 by is writ en 5 ; vi r o wri m 1 2 to di de it by o te it 5 . In the division of whole numbers by decimal divisors a m u l i 1 2 the s e r le is fol owed . To div de 5, a whole num her 1 0 s o one , by the unexpres ed decimal point is m ved to 1 2 — z place the left, and we have 5 If it wa s ‘ to be divided by 1 00 it would beco me A number may lie upo n both sides o f the decimal S point ; it is then a mixed number. uch are and with who le numbers on the left o f the decimal point and decimal fractions on the right. They may be ‘ 1 2 1 2 e i a writt n as mproper fr ctions, 940 and 9400, or as 2 {ﬁ lm m 1 2 3 3 d 1 W 1 2 nu bers, 940 9400” hich reduce to ” 1 and 34 . In division o f one number by another there is no need that the divisor shall be smaller than the dividend . The decimal point takes care o f the relati ons of one to the other and enables the division to be carried out as far as e o desir d if a division with ut remainder is impossible . A vulgar fraction is mer ely a statement of division to be o e d ne , the divisor b ing the denominator and usually

larger than the numerator, which is the dividend . _I I I I I V I 1 U 0 . lI . - ! 3 If such divisions are carried out we obtain decimal fracti ons and here is where the decimal point must be 1 2 watched . Dividing by gives 5 or ﬁve tenths ; dividing

1 2 0 or one . by 5 gives . 4 four hundredths

A mixed number is treated in exactly the same way . 6 to Take Dividing 3 by 5 gives . ; this is annexed 1 2 the whole number and we have or 940. Or we may write 1 2% as an improper fraction and carry out the division ; All is done exactly as in the division o f whole numbers but with clo se regard to the decimal point . The inserti on o f a cipher between a decimal fra ction or o o m a wh le number and the decimal p int, or the re oval of a cipher therefrom does a ﬁ ect its value as just described . It is in the division of and by decimal fractions that the greatest liability to error occurs . The rule is that the decimals i n the quotient are equal to h m i those in t e dividend di inished by those in the div sor. If the divisor has more ﬁgures to the right o f its decimal e ha s to point than the divid nd , ciphers are be annexed to m a the dividend to make the deci als equ l in number. 2 Divide by . 5 . Express it as a regular division o i and go thr ugh the operat on. 5 . As the decimals in both are equal there a re none to go into the

1 2 . 2 . H Divide . 5 by 5 We have ere there is one o m re decimal in the dividend than in the divisor, ere o one th f re there is decimal in the quotient . v d rz z W n h Di i e s by s. e a n ex two cipher s to t e

divisor ; this gives : As the decimals in the divisor are as many as those in the dividend there are no decimals to go into the quotient. Although the annexing of the ciphers could have been u dispensed with , and the decimal point co ld have been te o f t t e de rmined mentally, for the sake cer ain y it is b st to a dd ciphers until there are at least as many in the in a re dividend as the diviso r. It is immaterial if there

00 1 . u Divide . by 95 Add eno gh ciphers to make the dividend numerically equal to or greater than the w : i o f . e div sor, irrespective the decimals Thus have

or carrying it still further

no e In each case , as there are d cimals in the divi sor, all that is necessary is to make the decimals in the quotient

equal in number to those in the dividend .

a in th hird xa m le a fte shiftin h im l by z s e t e p , r g t e dec a

The principa l a s rega rds its integra l ﬁgures wa s sta ted in dollars ; the answer reads there fore In interest calculations the decimal point should be kept in mind and used correctly— guessing at how many do llars a nd how many cents a re in the result of a multiplication is frequently done ; if it be remembered that a per cent is properly written as a decimal and if the placing of the d n e ecimal poi t is perfectly understood, the ﬁrst m thod is the best way to work . Since a per cent expresses really a stated number of hundr edths it can be written as a vulgar fraction ; six l per cent can be written 9400. Interest ca culations can be done with this expression . Take as the principal and calculate interest at

c. 7 p . , using vulgar fractions

1 oo 6834 !

Short Ways of Calculati ng Interest

I f the 360 —day year and the 30-day month is used in on 6 i interest calculati s , the process for % is s mplicity itself . The interest for a year is 6 % the inte rest for a month is the inte rest for a day is %oth o f that for m n a o th . With this as a starting point other standard e rat s of interest are calculated nearly as simply . If the 365-day year is used then each interest may as ll l e we be calcu at d in the regular way . I l l

sum

at the rate o f we can easily derive the others from it. 2 For 3% divide the 6% amount by . 6 For 470 subtrac t from the % amount. For t sub ract 54 .

Fo 2 r divide the last by . 1 For 7% add 6 . For 75470 add For 8 % add 56.

For 0 97 add 55. There are other ways of doing interest calculations in o suﬁicient to a sh rt way . It will be give the methods, as n o f out s no ﬁ s the carryi g them pre ents dif cultie . F 2 o or 4 % divide the principal by 5. This is not a sh rt a way ; in any case it is only an lternative method , and

explains what 4% means . For 20 5% divide the principal by . F 2 0 or % divide the principal by 5 . Calculate the interest on at 4% and at 45696 1 0 60 for three months and days , taking the year at 3 days . ’ One month s interest at 6% is the half of and this is for three mo nths it is ten days is taken as the third o f a month so the interest at 670 for that period is and the sum o f the two last is For 470 b one o r 6 0 2 0 fo r , su tract third 3 this leaves $4 ; subtract one quarter ; this gives The a n swet s are ther efore and

Reduction of Interest Periods

One 00 one half of which is in decimals 5, is ’ month s interest at 6% on the 360 day basis for one mo nth . i reduced to other ra tes b he h i n Tlﬁs s y t met ods just g ve . It be reduced to other er iods of time b a dditi n ca n p y o ,

6 Thc ﬁrst column i re for two months a t % . g ves the duction of one month to other periods ; the second f on column the reductio n or two m ths .

To reduce one month to To reduce two months to 1 20 days multiply by 4 1 20 days multiply by 2 u a u w 3 90 136 45 1 36 45 2° 04 1 5 divide

c c t for one o To al ulate the interes day on any am unt, the product o f the amount by the interest is divided by f the number o days in the year . If we assume the year to consist o f 360 days the quotient obtained by dividing 360 by the interest rate divided by 1 00 will give a divisor by which if the principal be divided the quotent will be i e on o i for the nter st the am unt in quest on one day. ° Suppose the interest at 670 is to be determined on for o one day . To btain the general divisor 6 i 06 divide the interest rate, , by This g ves . . Di i f i in 60 06 6000 o 6 . v d . or 6 g 3 by gives , the general div s r 9 6000 0 1 666 or 1 Dividing by gives . 95 cents as

the required inter est . The reason for using the above general divisor is that one o o o it saves perati n , the multiplicati n by the interest T ‘ 6 60 o . 0 rate . multiply by and divide by 3 is the same

— thing as to divide by 6000 the operations are identical . The above method of calculating interest is applied to

1 14 ARITHKETIC

of a for the number d ys which the interest is to run . What is the interest on $ 1 5752 5 at 470 for 27 days day year ?

360 1 5753 5 31 5-05

. 04 3 7 27

The interest is on The interest is $4 73 on

6 e 60 e . the 3 5 day y ar basis . the 3 day y ar basis of for The rate interest paid money, when bills due are not d not discounte , is always realized by those failing to t o f o for n take advan age disc unt quick payme t, or by t o f n A ff 1 2 h se of eri g it. usual system is to o er % or % discount if the bill is paid in ten days . Suppose the bill is paid in any case in thirty days ; then for the extra twenty days the 1 % represents a rate o f about 1 8% per

n 2 0 f 6 n S a num, and the 7 is a rate o 3 % per a num . ixty 2 E days net , less % ten days, is per annum. ven if we take a mo st moderate case we will ﬁnd that thirty fo r in days net , less cash ten days , is at the rate of u 9% per ann m .

To calculate what per cent one number is o f another o o r divide the ﬁrst number by the sec nd, divide the per o c ntage number by the base . What per cent is 99 of 1 08 ? Proceeding as above we

: 1 : + 08 . 1 66 or e n ﬁnd 99 9 expr ssing it as a perce tage, 115

Wha t per cent o f 99 i3 1 08 ? Here we must di vide ot e fo r the h r way, the percentage amount must always be i 1 08 — d vided by the base . Then 1 or 1 09 0996

1 Suppose a number 8 increased by a certain per cent . The question may be asked what per cent must be sub tracted from the increased number to get the original il i f number again . It w l be a d f erent per cent . 1 0 If % is added to 50 it will give 55. This is because 1 0 0 S et 0 5 is 70 of 5 . o to g 5 back again we must sub

our not 1 0 0 of . tract 5 from 55 . But 5 is 7 55 To ﬁnd

- = what per cent it is we must divide 5 by 55. 5 55 0 0 . 9 9 which is

A h 0 0 city a s inhabitants . Another city is 5 7 larger ; how much smaller is the ﬁrst than the last city ? The larger city has inhabitants ; this is because is 50 % of but as is of the smaller city is at one and the same time

smaller than the inhabitant city . If the above operations are followed out with regard to o l - be the decimal p int , it wi l seen that the percentage

0 or o to te sign , 7 , the w rd per cent , applied an in gral or ff of 1 00 be mixed number, has the e ect dividing it by , cause any stated percentage is as many hundredths o f the base number as ther e are signiﬁca nt ﬁgures in the 6 06 e 6 or 0 or . perc ntage . Thus per cent 7 all indicate the same thing .

Approxi mate Calculations by Percenta ges

By using percentage corrections or fractional ones r o o approximate esults , near en ugh c rrectn ess to be used i E in practice, may often be obta ned . xamples are very 116 ARITﬁ METIC

a mil T uce kilo A kilometer is a bout ﬁ of a e. o red

l ﬁ a Thi is a meters to miles we should multip y by . s

o s lti .6 two ﬁgure multiplicati n . It is ea ier to mu ply by and add $90 or even Reduce 23 kil ometers to miles by the three methods = 2 62 x . 26 a . 3 X . 4 miles. = . 6 .6 1 8 c. 2 b . 23 x 3 3 x

e . miles . mil s

In exa mple b béo is a dded ﬁ n exa mple c ﬁ b is added for a ll ordina ry purposes one method is a s good a s the other . S A mile is equal to about kilometers . uppose we t had to reduce 23 miles to kilome ers . We may multiply 23 by The ea sy way to do this l 8 to is to mu tiply by , giving and then multiply this 2 s by , giving kilometer .

to 6 of 2 . 6 Or else add the miles . their number ; 3 X gives which added to 23 gi ves kilometers as o bef re . A meter is equa l to a little more tha n 39 inches or a e little more than f et . To reduce meters to feet u m ltiply by 3 and add X2. 1 1 00 meters are equal approximately to ( 1 1 00 X 3) ’ (3300 X Az) which is 3575 feet It may be done by percentage ; we may multiply by 3 and add The result will be nearly the same ; for the 1 1 00 meters we should ﬁnd h not T is is so close, the percentage should be

Powers a nd Roots

The square of any number above 3 has two or more digits : the square of any number above 9 has three or more digits ; there is no rule for determining when a o in series of values , with one m re digit in each case than e thos of the preceding series begins .

Powers a nd Roots of Decima l a nd of Mixed Numbers

The square of a decimal fra ction must have at la st two o f decimal places , and the number decimal places u must be an even n mber . f 2 0 1 6 Thus the square o . is . 4 ; the square of .4 is . ;

1 1 . 0 1 2 1 the square of . is . It follows from the above that if we have to extract

oo of . or . the square r t a single ﬁgure decimal , such as 4 9, we must add a cipher and use the number with cipher e ann xed as the ﬁrst period for the extraction. e o f n Thus the squar r ot o .4 has to be take as the square f o . 0 f 0 000 on . 000 o . 0 root 4 , or of 4 , or 4 , and so , carry out to e ing it as many places as desired , always ann xing two fo ciphers , in accordance with the rule r extraction f o o . oo f the square ro t The square r t o .4 is o f that . 9 is the fact that on their face they appear to be single digit squares has nothing to do with o f the value their true square roots . 1 18 IWDVVE3CS

h bu n fl. b n t lAmmflo mui ha wu: for ld nr xnwens cxmd e n gm ufi p gg h, the above is sufficient to illustrate the laws affecting the powers of decimal numbers . l o f e It wi l be seen also that the powers d cimals, when a nd such powers are greater than the first power, the e l the sam applies to fractions, are smal er in value than o For o o riginal quantities . c rresp nding negative powers the reverse is the case ; the powers for decimals and i r i o fract ons are larger than the o iginal quantit es, for wh le ruun r t i r n r be s q uwe txnvens a e sn a ue . Rela ti ons of Numbers a nd Square Roots of their Powers The square root of any power of a number is the t f identical power of the square roo o such number . fra kxz the 4 i 1 6 1 6 e e its square root s , and is the id ntical pow r of the oo o f a of 2 square r t 4, because the squ re root 4, which is , o e o th o e o 1 6 raised t the id ntical p wer, the 4 p wer, is qual t . two o o o In the d uble c lumns bel w, the numbers 4 and r e f 9 a e used as the bas s or the illustration of this law . The left hand column o f each pair o f colunms con o o f tains the consecutive p wers of 4 and 9, the series n to th o bei g carried in each case up the 7 p wer . The right hand columns of each pair contain the square roots h of t e powers in question . It will be seen that these roots a re the identical powers of the square roots of 4

and of 9 respectively .

4 2 1 6 4 64 8 256 1 6 I 024 32 64 I 28 120

In the le ft ha nd column ta ke the cube o f mwhich is ﬁt. I ua 8 h in ts sq re root is , w ich is placed in the adjoin g N s a s 6 is cube 0f column in a line with 64. ow jut 4 the 30 is its s ua re root 8 the cube o f the s ua re root of 4, q , , q oo 2 and the original number, 4. The square r t of 4 is 2 8 is the cube of . of 8 Turning to the left hand column of the powers , we ﬁnd the 7th power of 9 to be 4782969 ; its square root 2 18 2 1 8 o the th o u is 7, and 7 is als 7 p wer of the sq are root

f . o the original number, 9 ; it is the square root of 3 The same will be found to apply to a ll the po wers in the s to two column , and the reader may try other numbers

N o e 2 r 8. A e xact square ends in , 3, 7 o number nding in one o f the other ﬁve digits may be a perfect square ; if it end in n f four i ca nnot be An n s o e o these t . eve end one o f 1 6 square may in any the other digits , , 4, 5, o o or 9 f ll wed by an even number of ciphers . Thus 02 00 e r 0 i 3 5 is a p rfect squa e ; its root is 55 . Th s gives a way of telling that a number has no perfect square root ; it is a pity tha t it does not go a little further a nd tell tha t

a number is a true square, but it does not . If a square o f a number ends in 4 the figure immedi ' one must b ately preceding 4 , the last figure but , e an even r f o . o 8 60 number a cipher Thus the square 9 is 9 4, the a r o f 62 8 squ e is 3 44 . If a square ends in an odd figure the figure preceding it must be a n n The even o e. s ua re q s of 5, 7 and 9 are 2 81 e examples , 5, 49 and resp ctively . If a s ua re ends in a n even number xce h q y e pt 4, t e la st fi ure ne will be a n odd Thus h g but o one . t e squa re of 1 6 2 6 is 5 .

The number s a nd 6 a re ca lled circula r numb rs s e . The prowty which is a ppea led to a s a ba si s fo r the na me or title is the fa c tha t their s ur s nd h t q a e , cubes a a ll ot er owers end res ectivel in he numbers n 6 t a . p p y , 5 d All - o rs o f end in a nd a ll owers o f 6 end in 6 p we s s p .

Properties oq ua res

E a s or h 1 very number squared, it is, else w en is

s r t r m . ubt ac ed f o it , is divisible by 3 and by 4 Try such e a s 8 1 1 to s c squar s , 49, for cases where has be ubtra ted ; 6 1 4, 44, and many others are divisible without the sub

if n e b 1 if this do not ma ke it divi sible it is i cr a sed y , 0 r es 1 T it n the then it will be when diminished by . ry o numbers above and on others . S a odd o 8 1 ome square numbers re nes, such as , 49 and o 1 the like . If fr m any odd square we subtract the 2 remainder will be divisible by 8 . Thus 7 9 is an odd 2 su c 1 square number, it is the square of 7 ; btra t and we 2 1 8 8 . have 7 , which is 9 X If the sum of the squares o f two numbers is a perfect e o o f squar , then the pr duct the two numbers is divisible r e o f by 6. This would be a very useful p op rty squares s — if it only worked the other or both way but it does not. f 6 6 Thus the product o and 9 is divisible by , but the of 1 1 sum their squares is 7, which is not a perfect square . H ere is where it does not apply . If the difference of the squares of two numbers is a perfect square the sum and diﬁerence of the original a re or o f numbers squares the d uble o squares . Try it POWERS OF NUMBERS 123

’ i 1 1 2 n 1 1 ’ — 6 6 0 . o w th and , and then with a d 3 Thus = 6 1 0 6 is 4, which is a perfect square ; the sum of and 1 6 diﬁer nce t o f c and the e is 4, bo h which are perfe t

S i . squares . uch numbers are not part cularly numerous u the Write in a line the sq ares of natural numbers , 1 2 n U w eri 1 , , 3 and so o . nder them rite the s es , 3, 5,

7 . Then the squares will be obtained by adding the series numbers to the squares one by one .

a 1 2 . Natural numbers , , 3, 4, 1 1 6 b. Their squares , 4, 9, ,

c. The 1 series , 3, 5, 7, 9, 1 1 6 d. Sumo fb a nd c , 4, 9, ,

It will be seen that line d and line b are identical .

Property of the Squa re of Two

The number 2 is the only integral number whose square is equal to its double ; this amounts to saying tha t 2 2 2 2 one e X X , taking multiplication as the xpress of ion squaring, and the other as the expression of simple multiplication .

’ Fermat s La st Theorem

’ What is known as Fermat s last theorem may be ex pressed as follows : There are no three integral numbers r to m o so elated that, if all are raised the sa e p wer, and the o f two power is higher than the square, the sum any will be equal to the third . We know and have seen that there are any number of quantities the sum o f the two squares of two o f which are equal to the square of the third , but if we go above the squares o f numbers no similar relation can be found to exist . ve on oessi numbers is analogous to the relati of the squares, o The pr perty refers to the numbers 3, 4, 6 5, and , and is expressed as below

The volume of a cube is equal to the cube of one of e To o a o the l its sid s . s lve the cl ssic pr blem of dup ication sixn le o of the cube by p arithmetic, which of c urse was r eel not contemplated in the o iginal problem, we must ed of he second cube cula te the length of the ge t , such tha t its cube will be twice the cube of the edge o f the le t smal r cube. The relation of the edges mus be as 4 1 N in e r number b f v is to . o t g a l s ca n e ound gi ing this

Orders of Diﬁerences of Powers

S row n on e uppose a of numbers is set dow a lin , and that on the line below them the successive diﬁerenees ff ua are placed, each di erence nat r lly below the space that

f n is called the first order of dif ere ces , and there will be one d f th N less if erence than e original numbers . ext if the rows of differences are subtracted from ea ch other o f s difi erences the new row figures is the econd order of , and it will again be one less in number than its parent r w o . N ow try this with the squares of numbers . The of 1 1 2 square is , that of is 4 and so on.

I 2 4 9 6 5 36 49 64 are. First order 5 7 9 1 1 1 3 1 5 Second order 2 2 2 2 2

4 t +s +s= 9

x ° u= 6 +3 +. 5 +7 +9 + 3 l +3 +5 +7 +9 +u+1 3 =49 9 +u+

ro es All the sums are perfect squares. The same p c s The the can be ea rried out as far as desired . number of quantities added tells the square root . Thus the addition of seven quantities gives a number whose square root is a nd 7 , so for all other additions by this method. Taking the number 3 as a starting point if we add t two or ogether the ﬁrst uneven odd numbers , namely 3 and 5 we have the cube o f the number of digits used . We need the digits 3 and s; adding themtogether gives 8 ; the e n h f 2 y a re two in numb r a d 8 is t e cube o . Now r a nd take the next th ee they will give the cube of 3, 2 : N ow the which is 7, thus 7 take next a o f i 6 : four, and we sh ll get the cube four, wh ch is 4, thus I 3 1 5 4 I 7 This relationship of the odd numbers to the success ive cubes holds good as far as we wish to go . H o o ere is another curi us pr gression . If the cubes of the natural numbers in their regular order are written the d o f e out, a diti n o the succ ssive cubes will give a series o f e o f 6 1 0 1 squar s the series 3, , , 5,

1 2 Natural numbers, , , 3, 4, 1 8 2 Their cubes , , 7, 64 , o a ll 6 1 00 22 Additi ns, squares , 9, 3 , , 5 . Roo of o 6 1 ts the additi ns, 3, , 0, Thus ta lring the second line and a dding consecutively we find tha t l a nd 8 give 9 0 f the line below; then 1 and 8 and 27 of the second line give 36 o f the third line and a nd so fo r the rest or as far as it may be carried , the l roots of the squares of the third line give the fourth ine, e which is the series referr d to in the text.

Relati ons of Squares of Two Numbers

Take any odd number and divide it into two parts such l i f l that they wi l d f er by unity . These two numbers wi l be the roots o f two squares which will diﬁer in amount 1 by the original odd number. Thus take the number 3 . 6 nd S a . It can be divided into two parts, 7 quaring e 6 o fe n th se we get 3 and 49, wh se dif rence is the origi al 1 odd number, namely 3. The same applies to any num o 2 ff ber whatever ; 5 divides int and 3, and the di erence

r or . of their squa es 4 and 9 is 5, the iginal number

Many other curious relations can be traced out. Any series o f even numbers beginning with 1 2 and going e o f 1 2 1 6 20 2 a nd so ou— on with incr ments 4, , , , 4 is t 2 o be divided by . The number thus obtained is to be iﬁerin 2 e divided into two parts d g by . These numb rs squared will give two numbers whose difference will give ri 1 6 for 2 the o ginal number . Take example ; divide by 8 u f 2 . 8 two giving From we obtain n mbers dif ering by ,

o 8 . S e namely 5 and 3, wh se sum is quaring these we g t 2 ff 1 6 o 9 and 5, whose di erence is , the riginal number. This cannot be done with even numbers indivisible by e 2 4 , b cause such numbers when divided by give odd o numbers , and these cann t give the two aliquot parts diﬁe in 2 r o . for g from each ther by Try it yourself . A well known problem is the following : Find a num ber s-eh th t if ra a nd a be mocessiv l a dded it s e y to .

' the dfi eruee bet v een the two squa res will be zs— u cc 1 d m n mber ifi n 3 . M u difi e to two u s d eri g by

T v m w v he t o m b ers into hich we di ide 1 3 a re 6 a nd 7, their s ua res a re 6 n a nd number in q 3 a d 49, the a sked for the roblem is — 1 2 — 2 e t e s p f or 49 5, i h r ubtraction ivin wh h e n w r g g u ic is th a s e . ‘ ' l he rooes a he i i p s e n be ea rried to ot r d flerences . Take ' the sa i es of nnmber difierin b multi l f s g y p es o s. What squa res difier by 35 ma y be a sked? Divide by s which ives divide in two a rts differin b na mel 1 a nd g 7 ; p g y 5, y 6 s the i 1 n 6 : t se g m a d 3 , two squa re numbers W ing le ss “re number 3s oould ha ve been ma de to give a nother a nswer b rn tin it a s a n odd number sim l y t g p y. Thus divide it into two pa rts difler ing by 1 ; 1 7 a nd 1 8 ; square these which gives fi g a nd m both squa res a nd difi ering by ss a lsa A genera l rule is when youha ve the choiee o f two mm m ivi w vi the sma ller hu bus d de hh di de by . T s we ot a n a nswer b dividin b the sma ller number g y g 35 y , 5 ; ha d we i i s d v ded by 7 , which is the other divi or, and the h r er n we should ha ve fa il d to et a n a nswer g o e e g . Also divide even numbers b even ones a nd odd ona b dd y s y o .

The results will be erfect s ua re a bel p q s s ow.

180 M C

md the wmhers of the third pa ir diﬁ er by mThe render ca n try the sa me ca lcula tio ns with so a nd with 75

m em oi r-o m lf the pro duct o f 4 by a number is such tha t the

such pri me number a nd a ny power o f it is the sum of

Thns ta lee the pro duct of 4 X 3 which is 1 z a nd a dd 1 This ive 1 rime number formed b et . a ther o g s 3, p , y du f Thir n is th moi a dding unity to a pro ct o 4. tee e su s r n mb r N ow ta ke a ower of 1 9 +4 both qua e u e s. p 3;

again the sum of two squares . k 1 o n a d 1 this ive 1 Next ta e 4 X a d d thereto , g s 4 , a ’ rime number a nd a a in two p g the sum of squares , 5 and N s i 1 681 o f ext quare this : it g ves , the sum two 0 ’ squares , 4 and we may cube it next ; ’ a nd this is the sum o f e 2 6 two squar s , 3 and If two numbers a re such tha t the sum of their squa res is a lso a s ua re then the roduct o f the two numbers q , p is divi ible b 6 s y . The numbers 3 and 4 have as squares 9 and 1 6 ; the 2 l sum of these squares is 5, which is a so a square ; there

6 d = 1 2 6 by , this pro uct is 3 X 4 , into which goes t wice without rema inder. To ﬁnd two such numbers, the sum of whose squares shall be a square take any two numbers and multiply therv e tog ther. Twice their p roduct will be one o f the nut! f o f t bers , and the dif erence heir square s will be the other Ta ke a nd a s the numbers to sta rt 4 7 with . Their 131

28 6 i on product is , and twice this is 5 ; th s is e of the ff o f numbers to be found . The di erence their squares 6 = 1 is o ou . is 49 33, which the ther number s ght The square of 56 is 31 36 ; the square of 33 is 1 089 ; the sum of i 22 the two ﬁgures s 4 5, which is a perfect square. 6 whose root is 5 .

The Square of the Hypotheneuse

There is one right angle triangle which is very well o on e 03 kn wn , and which has l g b en used for laying o o o o f i e square w rk, such as f undati ns bu ldings , f nce o e i n c rners and th like . Th s triangle is o e who se sides for are in the proporti on o f 3 to 4 to 5 . Thus the corner f h ni Then if we of a field we ma y ta ke 1 0 eet a s t e u t. ff 0 n o measure o 3 feet o a side kn wn to be correct , and then with a 40 foot string scratch an arc o f a circle from the corner end of the field as a centre, and next with the outer end o f the 40 foot line as a center and a 50 foot string strike a n a rc of a circle intersecting the a re o o o f two other , a line fr m the intersecti n the arcs to f lo e the corner o the t will be a right angle . A squar can be extempo rized from three laths on the same prin i l in c e . On p e lath is cut a little over three feet length , the others a little over four and five feet respectively . H o o f on oles are b red near the ends the laths, exactly r the axis , and exactly th ee, four and five feet apart . Jo ining them by tightly fitting wire nails youwill have

an accurate square . The following is an elegant rule for la ying out any ﬁ n d of right angle triangle . Take any two numbers . For one o f the sides of the triangle take twice the product f of the two ; for the other side, the dif erence of the 132 squa m of the numbm ; a nd for the hypotheneuse the umof h ua s t eir sq res. Su oe we ta lse z a nd a s the n mb T he pp e 7 u ers . wice t product o f the two i3 2 X 7 X 2 = 28 This is one side The diﬁerence of thei r squa res is 49 This is si e Then h h the other d . for the ypot eneuse we ta ke the sum of the squa res— 4

triangles can be calculated . It is obvio us that the num ' e k Th b rs ta en must be diﬁerent oncs. us try 3 and 7 as the numbers and the resulting triangle will be almost equilateral. 4 and 7 give a triangle whose short side is n - h o hen almost exactly o e half the yp t euse. The formation of the following series o f mixed num i n 1 2 bers hardly needs explanat o . The series is 55, 36,

394, 496 and so on as far as desired . The integral numbers and the numerators of the fractions are success ively increased by the addition of 1 ; the denominators of the fractions are successively increased by the addition 2 N v o f . owif these numbers are con erted into improper 1 2 4 o o : . fracti ns they will run as f llows 56, 96, 56, 96 These improper fractions give the sides of perfect right f angle triangles , that is to say o right angle triangles i w th integral number sides . 2 Taking the fraction 59; this gives one side of the o D h otheneuse triangle as 7, the ther side as 4 and the yp as the square root of the sum of the squares o f 7 and of 0 1 68 1 4 . The sum of these squares is , and the square o o f 1 681 1 th h otheneu r ot is 4 , which is e yp se of the triangle in question . If we try the same with any fraction o f the series the sum o f the squares of its numerator and denominator will give a perfect square .

134 ARITHMETIC

o do is to suocessivel e1ttra ct the roots indica ted b t , y y

lf the sixth root of a number is to be extra cted a t one But i n c s n o n ta . operat on it is a lo g pro e s , i v lvi g ten tion if the squa re root is ﬁrst extr a cted a nd the cube root of the square root extracted this gives the sixth root without n oot the tedious extractio of a high r . For of e the extraction roots , whose indic s are prime numbers this method is inapplicable .

The follo wing method of calculating the squares of 1 o numbers is available Up to 00. Bey nd that it could s o o a l till be used by a good c mputat r, but it is especi ly recommended for numbers under the century mark . It a o n o r is b sed the principle pr ved in algeb a, that the product of the sum and difference of two numbers is e 2 1 qual to the difference o f their squares . Thus 9 + 2 — 1 multiplied by 9 1 gives . If therefore we i 0 28 for 2 1 mult ply 3 by , these are the values of 9 and 2 1 t 1 9 , and if o their product is added the result will 2 28 0 be the square of 9. It is easier to multiply by 3 t 2 than o directly square 9. To square a number proceed as follows Add to the number or subtract from it a quantity o 20 which will pr duce a decimal number, such as or the 1 nearest product o f units by 0 . This gives a single digit to n t multiply by, which is very easy . The sub racting the o o m same quantity fr m the riginal nu ber, multiply the two new numbers together and to their product add the

o f r square the number added o subtracted . S quare 79. POWERS OF NUMBERS 135

subtract 1 ; this gives 78 a nd80 ; 78 X 80 f m adding the square o the incre ent , which in 62 1 1 r of . s , the squa e 79 is the result , 4

o e add and subtract 3, obtaining as our w rking 6 i 6 0 80 4 and 70 . Mult plying 4 by 7 gives 44 , 3 o f must be added the square the increment, 3 ; re is 9 and which is the s 6 quare of 7. The clue to the pr ocess is by additi on or subtraction to et for f g a decimal number one o the multiplie rs . The next exa mme shows the use o f subtra ction to get the desired decimal number .

Subtracting and adding 4 gives the two multipliers as

a 0 8. o we may c ll them , 5 and 5 It will be bserved that e the d cimal number this time is obtained by subtraction . We have therefore 50 X and adding the o f re 1 6 o f square the inc ment, , we have the square 54, 2 1 6 which is 9 . Subtraction is used to get the two working numbers when the smaller increment is requi red to give the r o f o decimal number . It is only a matte c nvenience i whether addit on or subtraction is employed .

Short Wa y of Calculati ng Squares of La rge Numbers

The square o f the sum o f two numbers is proved in 0 be equal to the sum o f the square o f the ﬁrst the square o f the sec ond and to twice the product he rst by t second . The law is best illustrated

o f examples . =4oo + 576

By a little pra ctice this method becomes very ea sy of it n e to a s execution , and can ofte be us d advantage a method o f ca lcula ting squa res o r a s a wa y of proving e r sults. e ers 6 When applied to mix d numb , such as 556, % the e e u s : to and like , a g n ral rule may be th s tated Add h on i t e integral number double the fracti , and multiply th s by the integral number ; add to the sum the square of the fraction. To calculate the square of 9% by the above rule e is io Add to th integral number, which 9, twice the fract n , giving 936: multiply this by 9 ; this gives 9 X d s ad ing the quare of the fraction , 96, we have (954? 8756 This squaring is particularly easy because 9 is divisible by 3. If 8% is raised to the square by this process it will be uite diflicult not q as simple , although there is no y about it ; thus : 8 X and adding the square of the i s 6 fract on , which is 36, give 9% which is the square in question . 62 Aliquot parts apply often here . 5 may be taken as 6 54. The method is applied with due regard to the = decimal point ; first we get 6 X 6% 39 ; to this is added o f i = 62 .o the the square wh ch is He 5, and final re 062 for i sult is 39 5, the decimal point d sappears as the original number contained none .

Ap ia the prodnct of 20 by 80 is t 6oo ; ha lf the diﬁ 0 if ua i a dded to r6oo it i vcs umoe is 3 ; its sq re s , g 1600 00 = 2 oo i o f +9 s , which s the square the inter n b o r t ot o mdia te um er . he squa re s . To sqna re a number ending in 54 or wha t is the sa me thin for this method one endin in rooeed a s g , g 5, p follows : In the ﬁrst ca se multiply the integra l po rtion of the number by the next higher integra l ; a dd to the product md the resnlt is the s ua r o f th ori n ix % , q e e gi a l m ed number. Thus the squa re o f 8% is

If the nnmber end in the is ta ken a s re r es tin s 5, 5 p en g the yﬁ of the ﬁrst ca se ; thus the squa re 0 f 65 is (60 x 70 ) +2s= 4225

re e 2 2 To squa any number betwe n 5 and 75, subtract 5 the e l i 1 from numb r, mu t ply the remainder by 00 and add f the square of the dif erence between the number and 50 . Thus to square 46 proceed as follows :

46 50 — 46= « a nd 4’ 1 6

’ s To square any number made up entirely of 9 , proceed thus : o 1 - its ut Put d wn for the right hand ﬁgure , on left p ’ ciphers one less than the 9 s in the o riginal number ; next ’ ’ 8 s one less s put an , and ﬁnally 9 , in number than the 9 o in the riginal number . By this rule we can write out the square o f any such number at once ; thus the square of 8000 1 800001 9999 is 999 ; the square of 99999 is 9999 . POWERS OF NUMBERS 139

The following metho d o f squaring numbers between 0 60 to ro McGifi ert o f the 4 and is due P f . James Rennsel a er Polytechnic Institue . The difference between the number and 50 is called the o n 0 o c mpleme t . If the number exceeds 5 then the c mple ment is added to 2 5 and the square o f the complement is

not d to t . S annexed , adde , the resul uppose 57 is to be

f r 0 squared . The dif e ence between 57 and 5 is 7, this is o r ut : 2 the c mplement . Car ying o the rule we have 5 = 7 32 ; annexing the square o f the complement or 49 2 to 32 we ha ve a s the squa re o f 57 the number 3 49. I f the number 15 less tha n 50 the complement is sub o tracted and the square is annexed as ab ve . Thus the o e 2 — 6 = 1 f c mplem nt o f 44 is 6. 5 9 ; the square o the e n 6 to 1 1 6 compl me t is 3 ; annexing this 9 we have 93 , the f square o 44 . Except fo r the one variation (addition or subtra ction) the processes are identical and are based on the rule for the square of the sum and diﬁerence o f numbers— for

o f o i to . the squaring bin m als , use an algebraic term The rule will Operate on any numbers but it is most a dva n o ta geous within the range given . The reason it c mes so on o f 0 nicely is because e the numbers is always 5 , 2 00 whose square is 5 and the second number, namely the is complement , is a monomial which a number less than 1 0 o o o , wh se square is in the multiplicati n table , i . e . wh se o o square is kn wn to every ne . Let us hope so . If a number is doubled the square o f the new number will be four ti mes that o f the original . By multiplying 2 1 2 2 any number between and 9 by , we can apply the 1 6 n 2 1 6 Then with a nexed 1s 9 , and this divided 2 e ur f 2 b ives th a e o . y 4 g 7 9, sq 7 If we divide any even number from 82 to 98 by 2 and l one apply the rule, it wi l give us quarter of the square o of the riginal number . This process can be extended to apply to all numbers 1 00 of under , if the calculator will only learn the squares 1 2 d the numbers from 3 to 4, if the oubling process be 2 used to supplement the method for numbers under 5. h i u 60 and t e halv ng process for n mbers above . Pro e n fessor McGiﬂert says th se squares are easily lear ed .

The reader is advised to try it . F r o f t 1 00 1 1 0 o the squa re a number be ween and , add the complement to the number and annex the square of o e O f u ou the c mpl ment. co rse if y have learned the squares as just recommended this rule will apply to any 100 number above . 1 Take 09 ; the complement is 9. Add this to the number— 1 09 9= 1 1 8 ; to this annex the square o f the ’ — — 1 1 88 1 the complement 9 and we have , as square 1 0 of 9, which is correct . If the square of the comple ment has only one digit 3 cipher must go before it, of o between it and the sum the number and its c mplement, " : i 0 a nnued thus 3 w th 3 preceded by , 1 060 1 0 giving 9 as the square of 3 . Between 90 a nd 1 00 the same rule applies except that ou y are to subtract the complement from the number. Thus to square 95 subtract the complement which is 5 e 2 02 and ann x the square of the complement, 5, giving 9 5, the square of 95. by ha lf of the exponent or if the exponent is a n odd mb r ra ise it t0 the ower o f wha t ma be ca lled the nu e , p y “ x onent r t larger half o f the e p . For a cube oo , where

root it would be 3. Divide the original number by this quantity ; the quo tient will be an a pproximation to the cube root sought . Add to it twice the or iginal approximate root and divide by 3 ; this gives an avera ge which is a still closer a pproxi mation to the correct root . With this new approximate root repeat the operation

a d libitum o e o o . and so , until a cl se n ugh result is btained Two operatio ns are close enough for a ll ordinary pur

but r o e The rule seems long, the Ope ati n is v ry short l n a m i e as wi l be seen i the ex ples g ven h re . e o o f 2 0 To xtra ct the cube r ot 5 , proceed as follows A mental trial shows that 6 is the nearest integral root o f 250 ; it is raised to the second power in accordance with the third paragraph of the rule, and the number is o : divided by it . We have theref re

2 1 6

r o The quotient , is an app oximati n to the root sought i for ; it is added to tw ce the ﬁrst approximate root, 143

h 6 the e of th r w ich was , and averag e three is squa ed and o used as a new divis r, because it is a closer approximation e o f the to the true root than ither others .

6X 2 = 1 2 694

1 8 94 average is and the next

X 2 : 238 92

1 8 898 average is and is the next divisor and the quotient of 2 50 divided by is Using this to get an aver t in the age exac ly as preceding cases , we have

X 2

average is 6 2993

If required we can go on as far as we desire . But compa ring the th ree approximations so far obtained they are so close that it is not wo rth while to go further for o any ordinary calculati n . The three approximations with their cubes a re the following

whose cube is a u

249-96

e We now will xtract the cube root o f 377. The cube 6 2 1 6 of 8 1 2 of is , the cube of 7 is 343, the cube is 5 ;

Extra ctions of the ﬁfth root by regula r a rithmﬂica l method ca n be ca rried out to a ny de sired degree of

Th is method ca n be uscd for the s ua re root a lthouh q , g

e he a r f 2 81 b c o . L t t s ur e oot o e re uir fa t ry q 9 q ed. t r iv r a lo h a r xim t re As he fi st d iso we t e s, t e pp o a e squa ro 2 -fi ure ot of 9, the first two g period of the number in

In arra nging the addition for the avera ge we must i o e e take nto account the fact that the divis r, 5 , was r ferr d to h the ﬁrst period , so in addition we put t e two ﬁgure ' 5 s under each other exactly as if the rest o f the quotient i was a dec mal fraction .

We now do the second division .

average of the divi sor and quotient ( 548 544) which is the squa re root a s fa r a s the figures co but u are ncerned , as there are four fig res in the original u number the decimal point must go after the second fig re, so that the root is POWE RS OF NUMBERS 147

The regula r ca lcula tion ca rried out to four pla ces of e i a s i es a i a h u in d c m l g v pr ct c lly t e same. By c tt g oﬁ no decima ls a nd carrying the operation through some more divisions the root could ha ve been brought to a ny e ee of a ua es e d gr cc r cy d ir d . The a bove methods a re interesting a nd instructive a nd a i a for the e a nd i oo s e e pr ct c l cub h gh r t , but wh r a higher ro ot with a prime number for the exponent is to be ex e it is e t d i a s tract d b s to o t by log rithm . CHAPTER X

w a rm

H a m her gm ter tha n unity is multiplied once by M it is sa id to be ra ised to the second f , power ; i m lfipfied by imdf twieg it is sa id to be ra ised to third r “ na me nomencla r is ca rried f ll powe . e tue out or a ‘ ' wr lhis s stemis not in on i n i m m po e s . y c s ste t a s t a y see ' o fi umerica l va lue o f the o ex o rst s ht. The n w g p er , re b does not e p ssed y the exponent, dir ctly indicate the m ber of multiplica tions involved in ra ising the number to the given pov er fi t indica tes the number of timec the i e n i number ra s d e ters into the mult plication . To raise a nnmber to the second wer a s it is multi li n po , p ed o ce b itself it is ob h i il n e y , vious t a t t w l e t r twiee into the multipliea tion ; if raised to the third power it will enter three ime n n t s a d so o .

An o r f number such ower b in x y p we o a , p e g e pressed b n itive conta ins the o i na l y a integra l pos number, r gi number a s fa ctors a s ma ny times a s the exponent of the

The exponent of a power may be a positive or a nega tive integral or fractional number ; if it is an im proper fra ction it ma y be written a s a mixed number e if d sirable. S 2 uppose that the quantity , , is to be raised to the ﬁfth n I power, whose expo ent is 5 . t must be multiplied timw l : 2 2 2 2 four by itse f, thus X x X X 148

An expoo ent nny eonsist o f the smn o f sever a l nnm

the n e the diﬁerent indica ted owers nn i r r to p , multiply oe b the other a nd the comunued roduct o f a ll will o y , p

m a is h or i ina l number directl o the a n d r e t e g , 4 , y t

Retnrnmg to the consideration o f a mixed number W a mixed number is the sum o f a n integra l h f re re nm er md a ra ction. The fore if we a to reduce a mmher v ith a nﬁmed number exponenuwe ma y sepa ra te the x on n w a r s a n int r l a nd f l e p ent i to t o p t , eg a a ra ctiona a rt l l t ea s a p . ca cu a e ch epar tely, and multiply them to

“ Whet is the va lne o f g ?

1 “ 9

ra ise to the eube ivin z a nd extra ct t e s ua re g g g y g, h q f i ives z bef T l he root o m wh ch g y a s ore. he resut is t n me iuboth ca ses ; there a re two wa ys of doing the

now come n l on t We to a other ru e c cerning exponen s .

one number from h In such ea se h n l a not er . t e or igi a number ma y be ra ised to the power represented by the m edca l diﬁerenee between the ex on nts p e , with the m+or of the la rger exponent preﬁxed to the EXPONENTS 15 1

e e fwe i be xpress d as But may, if we wish , ra se the e to on one numb r each partial exp ent, and divide the with o n the p sitive exponent ( the mi uend) by the other. M ’ ‘ Taking 4 as before we may divide 4 by 4 o r 64 by n 1 6 or 4, givi g ; we may say and the

ot . answer, b h results being identical Next take following the rule we divide 4 by 4 4 I

as before. And we may say 4' 64 1 6 I or as bef e . 1 6

oo In the present b k algebra has been avoided , as our obj ect is to treat the subj ects on a strictly arithmetical l a . u e b sis A s ight depart re is inevitable here, as a negativ exponent falls within the scope rather of algebra than

of arithmetic . Zero as a n Exponent

o o one Assume any number t have tw equal exponents, S o subtracted from the other. uch w uld be a t e and the like . We have seen th wh n a number e has two xponents separated by the minus sign , as in o f o e these cases, the value the expressi n is obtain d by dividing the number raised to the ﬁrst exponent by the n o same number raised to the second o e . But here b th exponents are the same ; therefore the number raised to the po wer represented by one exponent will be the same M 2 = = L as for the second exponent . Thus 4 +4 to 2 — 5— 2 = I The same applies which gives 7 7 , and — 1 = t No w o o f 8 . we must n te the value the exponents of the numbers ; the value of 2 — 2 is o — we ﬁnd numbers tha t of 3 3 is a lso o. So tha t these 0 ower a nd ha a n f thc a re of the zero or p , t t y number o

O power is equa l to unity.

x onen o is e ua l to t . e p t , q nver h uni i lw s This is in line with the co se, t a t ty s a a y e or equal to unity , whether an xponent is given it not ;

buted to a Chinese source . If 2 i to o is ra sed any power, wh se exponent is a prime n 2 ra so umber, and if is subt cted from it when raised, the result will be divisible by the exponent used without re th a mainder ; e quotient will be an integral number . e o f 2 Take 3, a prime number, as the xponent ; this s 8 o 8 2 6 give ; fr m subtract and is left, which is divisible by 3, the exponent used . The same may be tried with i e on . 2 5, also a pr me number as the xp ent 3 , and 2 — 2 = o 3 3 , which is divisible by the exponent used, which was 5 .

A negative exponent indicates division o f unity by the o number with the negative exp nent, after such number to n n has been raised the power indicated by the expo e t . " ' 2 o f 2 8 Thus indicates division unity by or by , and I putting it in fractional form we have 2

in is the same th g, $6.

The pra a iea l unit o f reei sta nce ﬂl e ohm is equa l to n f i I . S o uits res sta nce . f the rp oomq ooo C G .

N u mis l i lied b self 8 im3 it will ter a s a m t p y it t , en fm b mme opa a fim nine times ; hence it will be ra ised t the ninth ov er so tha t l o with a n x nent o p , e po . g v ill upresent the one thousa nd millions expressed a b v ’ o e h fignres fi t is ro . The exponent o f a ny power o f ten indica tes the m her of ciphers which come a fter the l in the numeri a l re n a i n f Th bov prese t t o o that power. e a e number conta ins nine ciphers to the right o f the l ; therefore the nnmher is the ninth po wer of ten a nd is expressed by the

The volt is the pra ctica l tmit of potentia l ; it is equa l t one n r n me ni . n o hud e d nnllio s ﬁ s the C G S . u t of pote tia l ; this fa ctor is expressed by t o with a n exponent 8 ; ' it is ro .

4 ex onmt o f t 1 0 e p , thus , m aning 340 The unit of a pa city is one one thousand millionth o f h t . G. S i s C . unit ; th s give the multiplier a negative ” n of B expo ent 9 ; it is w . ut this pra ctica l unit is still inconveniently la rge ; it is ma de one millionth this va lue by ra ising the nega tive exponeot o f ten to the— rsth EXP ONENTS 155

o f to o I o power . But instead having write d wn a foll wed i r to n by ﬁ fteen c phe s , all that has be do e is to change the o so t exp nent , hat the multiplier will be When two or more numbers have identical exponents they can be multiplied by simply adding the exponents together and putting down the o riginal number with t e o f the new exponent. This is ra h r an indication multi n l o i plicatio than a real multip icati n . Applying th s rule to powers of ten we have such results as the following :

1 ° ' = " 3 4 = t " t “ 1 0 X ro t o ; 10 X to o x o .

If the exponents have different signs the diﬁerence for th o e between them must be taken e new exp n nt, and it is to have the sign of the larger numerical exponent i e o : aff xed to it . Some exampl s f llow

“ " “ ° 1 4 " t t 1 t 10 4 t o“ m 1 0 =x o x o 03; o x ; X o .

T u f o divide n mbers with dif erent exponents, but nu merica ll l y identical, subtract the exponents a gebraically, as in the following examples .

“ 3 ’ = I O ; I O’ = 1 0 ; IO

The subtraction is to be done algebraically ; accord n l t o o the of i g y in the ﬁrs and last example , f ll wing laws a e lgebra , the sign of the subtrah nd was changed and the two exponents were added ; this constitutes algebraic i subt ract on . In the other cases the operation was done arithmetically . CHAPTER XI

Squa ring the Circle

A long historica l treatise could be written on the subject of the rela tion between the dia meter a nd circum n n S fen ee of me circle . The ancie t term quaring the ” ’ Circle expre sses one of the world s most famous prob n f lems, the finding of the le gth o the side of a square e ua l in a rea o a circl f a iven dia meter To deter q t e o g . nfine k the one thing necessa ry to be known is the ra tio of the diameter to the circumference of a circle. If n r So that is k own the rest is simple eve yday arithmetic . a e the term squaring the circle is rather antiqu t d . One of the first things ma ny of us were taught in the line of pra ctical mathematics is that once around a circle u b t is three times th rough or across it . This wo ld e mos e e but u conveni nt and is easy to rem mber, has one tro ble , no s o o the it is t so . It is the mere t appr ximati n to truth v it is so fa r from the truth that it should ne er be used .

Approximations of For mer Times In ancient times the Babylonians and the Jews are o to o supp sed have used this inc rrect ﬁgure, 3, as the e r multiple to produc the circumfe ence from the diameter . The ancient Egyptians a re credited with doing better o on than this, with the fracti nal expressi which reduces to a very fair attempt at the value. 156

M o i the cirele one ﬁﬁ h of the side o f the squa te he edded it v ill gi ve the circumference o f the circle circle is equa l to the dia gona l of the squa re inscribed kn i i n x within it uhe side o f the squa re ta i g ts d a go a l a s ,

mr mot o f one ha lf which is . o 1 a nd this is the sq e , 7 7 ue i 1 whieh i .ooz l ha n th tr iv v s . t s ess t e d ided by s g e 4 4, deciml ca rried to the number o f ia ces indica ted p .

s is the Greek letter,

a a nd is supposed to st nd for peri mete r .

Ai d to l emorizing Pi

for the factor or r se rried to the fourth decimal l mbe f he p a ce. The nu r o letters in each word gives t i d gits of the factor .

3 1 4 1 6

Y es l ha ve a number .

n he of r A other rhyme by which to remember t value , carried out to the twelfth decimal place is the following ; e o f e r as b fore, the number letters in ach wo d gives the o i c rrespo nd ng digit .

3 I 4 1 5 9 See I ha ve n rhyme assisting 2 6 6 3 5 9 9 o i My feeble brains its tasks s met mes resisting.

t The mathematical value is bet er than the grammatical. SQUARING THE CI RCLE 159

Cmi ous Determina tion of wby Proba biliti es

Buﬁ on and Laplace determined that if a stick o f a length less than the distance apart o f a series of parallel lines was dropped upon the surface marked with such

o i o o one the lines , the pr bab lity that it w uld fall acr ss of

o u 2L 1rA L lines is expressed by the f rm la / , in which is

o f i the length the st ck, which must be less than the

o f or L A A distance apart the lines , < , and is the distance

f o o ﬂ apart o the lines . S metimes a b ard oor with boards o f equal width is cited as the ruled area . The experiment of throwing a stick on such a surface d u has been trie , with results curio sly near the true value

r c ro of . In one a se the stick was th wn times and

u for w t o f 600 gave the val e, ; ano her trial throws th gave and in e third case, throws gave f x or .

’ e o The r ader is referred to A . de M rgan s Budget of o o 1 8 2 Parad xes , Lond n 7 and in the Messenger of 1 8 V l 1 1 1 1 E . o ii . Mathematics , Cambridge , ng 73, . , pp . 3, 4

Squa ring the Circle Li terally H aving the multiplier, . with which to obtain the length o f the ci rcumference fro m that o f the

diameter, we can square the ci rcle ; this is to ﬁnd the side o f the square equal in a rea to the circle of any t given diame er . The area o f a circle is equal to the

o o f u o f 1 Fo r pr duct the sq are the radius by . a circle o f t 1 8 . 82 the diame er, , this area is 7 539 The side of a square o f thi s area is equal to the square root of e 8862 the same numb r ; this root is . . This number is the 100 m r nmmc length o f the side o f a squa re of the area of a circle of

1 . It will to give the following a s the value of r ; it is far more a ccura te tha n will ever be required r r a er by ou e d s. It erro re is u on -th t i ion The r he abo t e ir y b ll th, not far from a n error of sixteen feet in the dista nce of the sun

t l n Write out di e odd in egra umbers, 3, n s Cro s ou ever o a s fa r a s de ired. s t y ime ft for non of these a re r . a er 3, e p in addition to these every fifth number a every seventh number after seven and so a n excessively laborious after high numbers a re we t s If write out the odd in egral , 3, 5, 7 , 9, we will find that the numbers eliminated by e h fr re ev ry t ird one om3, a 9, elimina ted by counting every fifth one fmmS a re I s 2 ( a lra dy eliminated by 5, This leaves as

e 1 1 2 2 1 . prime numb rs 3, 5 , 7, 7, 9, 3, 9, 3 If the seven elimination be applied it will only touch up to this i 2 1 po nt the numbers and 35, both of which have been m ne o 2 1 eli inated , o by the three eliminati n ( ) and one by the ﬁve elimination

The aliquot parts of a number are factors or integral numbers which multiplied by other integers give the i . 1 is he or ginal number The number included . Thus t e 1 2 nd number six has as its int gral parts or numbers, , a

3. These three numbers multiplied respectively by 6, 2 6 e by 3 and by give as the product in ach case. A number the sum of whose aliquot parts gives the o number in questi n is called a perfect number. If the l o f 6 d t 1 2 a iquot parts are adde toge her, we have +3 =6 o 6 r ; theref re is a pe fect number .

The next perfect number in the numerical order is 28. 163

1 4. Each of thos e 28 when multiplied

number. Another way to put it is to say that an aliquot part of a number is a factor of the number ; it is any number, which can divide it without fi a remainder . N ow if these ve numbers are added t t o e 28 oge her we will btain the original numb r, ; therefore 28 is a perfect number . To find all the perfect numbers take the geometrical r 2 8 1 6 2 c p ogression , , 4, , , 3 , arrying it as far as de i . o s red Fr m this series select those numbers, which 8 . S diminished by unity give prime numbers uch are 4 , , 2 1 28 8 1 2 ut o 3 , and 9 , if the series is carried o far en ugh u u e to give it . Diminished by nity these n mbers becom 1 1 2 8 1 1 E 3, 7, 3 , 7, 9 . ach is now to be multiplied by the geometrical progression which preceded the one from i which it is der ved. Thus 3 is derived from 4 of the o 2 so riginal series , and 4 is preceded by in the series , 3 2 i 6 is multiplied by , g ving , the first of the perfect num Foilowin to l bers . g out the rule 7 is be multip ied by 4 ,

3 1 is to be multiplied by 1 6 and so on as far as desired . The result is very meager for there are very few per 1 1 fect numbers ; up to 1 0 Hutton gives only eight of them . 28 All perfect number s o r nearly all end in 6 or in .

Amicable Numbers Amicable numbers are numbers so related to each other that the sum of the aliquot parts o f one of them gives the other number. we 220 o 1 2 I f take the number , its aliqu t parts are , , 10 1 1 20 22 1 10 4, 5, , , , , 44 , 55 and ; the sum of these is 28 Th o 2 1 2 1 1 2 4 e aliqu t parts of 84 are , , 4, 7 and 4 . 10‘ ARITHHE TIC

md ﬁxe snm of these is a zo nherefore the two numbers , i l um s m a nd ﬂa re a ma b e n ber . Other a mia bie nnmhers a re 1 7296 2nd 1 84 1 6; a fter

‘ 8 0 6 It ill be seen tha t ths e come 93635 4 a nd 9437 5 . w m s ve they a re e ces i ly ra re. To ca lenla te them ﬁ a rt with the sa me geometrica l

progression a s in the determination of perfect numbers . seri Multi l ea ch term b a nd write it Ca ll it es A. p y y 3 n fr which it wa s derived Ca ll this under the o e om .

ch o f B series B . Multiply m number series hy the num - ber preceding it in the sa me series a nd dimini sh it by W rite hes down ea ch under the one fromwhich unity. t e v Th foll n a re s l this senes C . e owi it wa deri ed. Ca l g

2 8 1 6 2 Series A. 4 3 64 r 2 Se ice B . 6 1 24 48 96 1 92

S . 1 28 1 1 1 6 eries C 7 7 5 4 07 1 843 1 . i S s . 1 1 2 1 er e D 5 3 47 95 91 .

Subtra ct x fromea ch ﬁ ure o f nd r i g series B , a w te it under the number from which it was derived . Cail this series

Take any prime number of series C, subject to this con i : e d tion the number under it in seri s D, and the number n s u precedi g this one in serie D, must be prime n mbers . the two e Multiply numbers of series D togeth r, and mul tiply their product by the number in series A directly

e one. 1 over the larg r Thus in series C, 7 is a prime num so 1 1 ber ; are and 5 of series D . Following the rule we 1 1 = 220 h have 5 x X 4 , one of the numbers . T e other is obtained by multiplying 7 1 by the same number in ' = series A ; 71 X 4 284 ; this is the corresponding ami

s This i S I is squa res of the dia meters ; or a 4 is t0 9. s a Th ro l f rc o f inertia is in the ra tio or to m. o pe ling o e ro ortion fli e cubes r a s 8 is t0 2 this is th e ra tio p p ot , o 7 ; la s 2 i h of x to 3% . Tbe rgcr hell with only 54 t mes t e resista nce o f skin friction a nd cross-section of tha t o f the la rger shell ha s 3fi times the propelling force o f its inertia which a s we knowis in ro ortion to its wei ht Tha t is p p g . s f e a why a h rge projectile goe urth r tha n sma ll one. This rule has many applica tions other than the one o o f a on of pr jectiles . The force gravity cts all bodies in proportion to their weight ; in a vacuum all bodies s r r la rge or sma flfa ll with the a me speed. In the a i thei fall is retarde d by their skin friction and cross- sectional esis a c r t n e a s it ma y be ea lled. Again the rule of the square and the cube comes into play and the larger body th o a l falls e faster . Dust falls with extreme sl wness though it may be o f the same material as a great rock

which would fa ll with tremendous velocity .

o n m a r The clock faces , which Ro an numerals e used , ’ r the it will be obse ved, have four indicated by four I s, IV o f instead of by . It is said that the reason this goes as fa r ba ck a s Cha rles the Fifth or a s it is wr itten Charles

V. He is said to have declared that nothing should go the V l before of his title, so he thus e iminated the usual a nd IV, there was nothing to do but to use four letter PS .

The Number 1 08 in Our Solar a nd Lunar Systems

The diameter of the sun is approximately 1 08 times that o f the earth ; the distance of the earth from the sun 167

i m l t oB mes ﬂre dia meter o f the snn Th e is a pprox r te y ti . distance from the moon to the ca rth is approximately

108 times the dia meter of the moon.

il wheel there is a s eciﬁc tire For a ny size a utomob e p ,

This is a tire o f one ha lf inch la rger dia meter o f ca liber h t o r z Th i e ha s to fit file sa me t a n tha f the regula si e. e t r rim ; the problem is what will be the increa se in full diameter for the half inch increase in ca lib er . As it ha s to fit the sa me rim the inner circle o f the

r r s sa he sm r la ge tire mut be the me a s tha t of t a lle one. The increa sed ea liber a dds a ha lf inch a ll a round the

Ther f s tire. e ore a there is a ha lf inch extra on every h i side of t e tire it results in giv ng a tire one inch larger. r the ersiz f r a r Therefo e ov e o 34 X 4 inch ti e is 35X 4% . Add an inch to the cross diameter of a ny given tire and a ha lf inch to the ca liber or a s it ma y be termed to the t small diame er and youwill have the oversize. If an automobile tire is priced at a certain amount the the one of next larger caliber, as a 4% and a 4 inch tire , to o on seems always be priced dispr porti ately high . But suppose we take a three inch tire to be compared with a one three and a half inch . If both were of identical thickness and of equally heavy construction the larger -u one would have 5/30ths extra material in its make p. This is almost so on this basis alone it will be seen t that here is a reaso n for a large increase in price . But a n inch on the cross diameter although it sounds like more is much less ; fo r two ti res o f 30 and 31 inch size Tw cleﬂrs mrt work in the sa me oﬁ ce a t the sa me

M Sso eve ry six months ; the sa la ry o f the othcr a r The l r a i a ch ha lf is ra ia ed tzoo every ye . a a ies a re p d e

' ch clerk fa res th best ? ye ar. t e At ﬁrst sight it would seem tha t the clerk whose sa i a ry m ra ised ﬁ oo ea ch yea r would get the mosnbm it will be found on ca lcula ting the a nnua l returns in ea ch a se dxa t the clerlr receiving the $ 50 ra isq every six

months kee s consta ntl hea d o f the h . A e nd , p y a ot er t th e of the first yea r the first clerk ha s received $ 1 p 50 a nd the other one at the end o f the second the first clerk will receive for his ea r o f 1 2 0 th othcr y work $ , 5 ; e one It kee s on in the sa me wa he i rk , p y ; t first c e is a lwa ys $ 5o a hmd o f the second 0ne

Let a tumbler A l of , , be ha f full water, and a second tumbier B be ha lf full f A s , , o wine. tea poonful of wine is oured from B into A a nd a te s f p , then a poonful o the mixed wa ter a nd wine is ta ken out o a nd is put B into . The uestion is a fter a ll this ha s q ; , the tumbler, B l m r , ost o e wine tha n the wa ter, which ha s been lost by the tumbler A ? M ost people will sa y a t once tha t more wine wa s lost b B tha n the a ter ios s y , w t by A ; the correct a nswer i tha t the come out even one losin iess tha n a ms oon y , g p symme m inder will be the tlue e nnmbus in thdr

s r f 2 o o e . Supp e the numbe s thought wer , 3 and 4 We then pmeeed a s follows : =ss: 58“ a nd ﬁna ll 8 — 0 = 2 which a re ﬂae mree nnm y. 5 4 35 34 , bm thought of in the order in which they were entend

A Pm dox of the Time Ca rd

The following is a good illustra tion o f confusion be

A derk a pplied for a job in a business house a nd mid he m r n H s oid be mrth um. e t $ , 500 per a n wa not worth it .

Y ousleet ours a da y (li of a da y)

There a re 52 $ unda ys in the yea r

Days left Y ouhave one ha lf day on Saturdays

Days left 17 1

’ Y ouhave two weeks vacation

Days left 1

th o so udo no wo . This is July 4 and we cl se, that yo rk at all

It is generally unsafe to trust to the memo ry for too a many telephone numbers . But there are in many c ses e et i n peculiariti s about them , which give a basis for r ain g t in mo Technim or hem mind . Me ria Artificial Memory i o s s the term describing such meth ds a we illustrate here. o 2 : Take the teleph ne number 579. It is made up thus two i s the sum of the first numbers , 5, g ve the third u n mber, 7 . The sum of the first and third number,

e . giv s the last number, 9 Try the number 1 1 40 ; the sum of the first two numbers gives half the third one. Tr 2 2 e y 5 9 ; the sum of the first, second and last numb r e giv s the third one . 1 1 2 1 the o i e Try . The sum of first tw numbers g v s the third number. 1 0 1 8 1 1 0 Try 9 and 7 . and 9 make , this gives the final cipher for 1 90 ; the sum of the first and last number 1 8 gives the middle number of 7.

’ 6 two di n b e Try 447. The sum of the mid e um ers giv s 8 ; two greater than the ﬁ rst number and one greater than the last one. e Many numbers involve sequences or inverted s quences . n 6 8 Inversio may be partial or complete . Such are 4 7 68 and 76. l a t ing tbe l ine b iglts in Proper Order Add np to One

It is required to combine the nine digits keeping them mo i r e order so tha t th sha ll a dd to . in die r p op r , ey up The triclr f it is little bctter is done b combinin , or , y g s wo 8 l , ca the t . mu tipli tion of la t , i e 9 and , with the a ddition i h ﬁrst seven di i s o t e g t , thus

8 — (9 X ) . l

Th e sa me ﬁ gits mn be ma de to give the sa me sum in severa l other wa s n im orta nce o : y , no e of any p , a s bel w

xs+36 2 = roo ( 98

The next series of operations brings out the nine digits t u a nd a lso ert in heir nat ral in their inv ed order. The

( 1 8) t =9 ( 12 X 8)

( 1 234 X 8)

( 1 23456 x8)

( 1 2345678 X 8) ( 1 23456789 8) +9= 98765432 1

’ ’ 1 1eft ha nd fi ures of the suooessive 98765432 . 1he g ill the di its in re ula r order the la st pro ducts w be g g , ’ r mse will be a nd there will be 8 s e n figue in ea ch g, b twee ’ fi s n s di its the numbcr o f such 8 s b in n the r t a d la t g , e g o e es or le ha nd fi ure o f li i s tha n the first ft g the multip er . Th uc e i : 1 8 288 e s cessiv products thus w ll be 9, 9, (as

Sometimes to ma ke it mo re stii king a subtra ction of 1 from mch o f the products is pr escri bed ; this gives ’ ’ 8 s n tea s ﬁnal i s d of 9 . The successive multiplications may be arranged in a i sort of pyramidal form, similar to the sets of multipl i cat ons above . I f 8 the digits, omitting , are written down in their e m e r gular order, we shall have a nu b r, which, i f multi ’ ll 1 s plied by 9, wi give a succession of units or . The multiplica tion follows

I I I I I I I I I

One curious thing about this is the cﬁect of ’ 8 o i 1 the ; this m ssion gives the succession of s . If the 8 not e is omitted, a cipher mak s its appearance in the pla ce be l c n low it. The mu tipli atio follows : MI SCEIJA NE OUS 175

1 1 1 1 1 1 1 1 01

The rea der may try placing the nine digits e order and multiplying th m by 9. The following is a curious set of multiplications or of squarings : 1 1 X 1 1 1 2 1 = 1 1 1 X 1 1 1 1232 1 1 1 1 1 = 1 23432 1 1 1 1 1 1 X 1 1 1 1 1 1 2345432 1

The same operations ca n be carried o ut as i e t the s r d , but the symmetry is impaired af er ’ 1 s exceeds nine .

Properties of Numbers “ It is not easy to define the term Properties of Num ” rs o o o f s e be , but the c ncepti n ju t what the term m ans one is easily reached by exa mples. Thus of the proper oi nd ties the number one is that its square, cube a all t n u o her powers a re equal each o e to unity . Any n mber whose last two digits a re divisible by four is divisible i r by four throughout . Th s may be taken as a prope ty i f of the number four . A number is a perfect number e the the sum of all its factors is qual to itself . Thus 6 1 2 o f t h factors of are , and 3, and the sum hese t ree es the n 6 One figur is origi al number . of the properties 6 t t of hen is that it is a perfect number . The fac ors of 28 1 2 1 i l are , , 4, 7, and 4 ; add these together and you w l 2 e have 8 . Th refore one of the properties of 28 is that it is a perfect number. ' Th e number nine exceeds in mtere sting a nd practica l ro erti l i i s It is a onder t ha s not p p ee a l other d g t . w tha it heur more popula r with fiie trihe o f a strologers a nd uh r o f flmt da s a s a hi ck or unluck nmnber in e s s y y ,

If g is used a s the multipiier o f a ny numb er from 1 to zo the sum o f the digits or ﬁguros o f the product will be a was and r +r +7 =a Or 18 = 162 X 9 , a nd as before Nowwrite down the series o f products o f 9 by a il the numb ers from 1 to zo a nd see if there is not something c l e 2 = 1 8 pe u iar in this seri s of properties . Thus 9 X and 9 X 9 ==8 1 ; each of these products is composed of the same digits but in revers e order. Try 9 multiplied 1 n 1 e 1 1 1 1 by 3 a d by 9 ; the pro ducts are resp ctively 7 and 7 , the same digits but the last two are reversed. The num e s 10 1 1 n 1 6 not b r , a d are the only multipliers which do r t o a o the pai wi h any thers as b ve , and if cipher, o, is e o n e l ft out of acc u t, the refractory numbers r duce to 1 1 and 1 6 ; their products by 9 are respectively 99 and ' 1 e o inca 44. A mom nt s inspecti n shows that they are p able o i reversing in the way all the other products are ; hence they fall out by the wayside . Coming back to our paired numbers we will ﬁnd that the sum o f the mul ti liers fo r d 1 0 i p each pair un er t mes 9 will be the same , 1 1 2 = 1 8 off = 8 1 namely . Thus X 9 pairs with 9 X 9 ; ==6 off =6 two 4 X 9 3 pairs with 7 X 9 3. The pairs o f e re 2 1 1 multipli rs a and 9 whose sum is , and 4 and 7 1 1 i mul whose sum is also . This appl es to single digit tipliers ; now if we try the same with multipliers from

178 difierence 18 by 9 without a rema inder indica tes fiie fact tha t two figures were tra nsposed a nd ma kes me loca tion ier it wouid be without this o f the er ror ea s tha n clue. The rule is not infallible ; a difference divisible by 9 h ist a nd be due to a n inversion but it is a mig t ex not , case o f probability ; and inversion or transposition probably was the cause o f the trouble . In the second pair the error is in the middle of the number but the same n o f e rule holds . It also holds for the tra sposition sev ral

ﬁgures . A Mystery in Money

Take any sum o f money less than a n dollars transpose the digits and take the diﬁerence between the to ff two sums thus o btained . Then the di erence a dd its own tra nsposition and you will always get the same H see . amount, namely Try it and ere are a few examples .

sum

O f course the dollars have nothing to do with it ; they me rely make it more picturesque and if it is used as a i the doila r l sort of trick or mag c sign obscures it a litt e, i a des rable feature in parlor magic and the like . Thus a person is told to write down a sum o f money o a nd less than ten dollars , and wh se ﬁrst last ﬁgures f must be dif erent. Then he is to write down the trans MI SCEIL ANE OUS 179

i e a the posed figures d r ctly under it. Ask him wh t s o r fir t last figure is . As the sum of the first and last figures is always nine and the central figure is always tell i nine youat once him what the number s. Then he s e i is told to tran pos this last number, writ ng it under h h t two . t e t e o her, and to add the Then he is told what t T resul is , namely o make it a little more mys terious it is well to tell him to put any number youname below the lower pair o f numbers and then to add them . All you ha ve to do is to add this to and tell him the result The object o f this is to prevent the repeated production of The examples below illustrate

. ou 2 this In one case y can tell him to add 5, in the other 1 case to add 3 .

A pro fessional magician as a rule avoids performing the too e r same trick oft n , so it is well to va y this one by not i o e but sometimes giv ng y ur fri nd the third number, to let him finish the operation befo re telli ng the number e a difierent found . It is w ll lways to add a number as s ju t described , perhaps just once to omit the addition and telling him that he has ten dollars and eighty- nine n n ce ts as the result of his figuri g . After that always f add in some number and always a dif erent one. M a n us so u to b e much more puzzling.

' nu nced 12 : nd the ounds a nd eno must diﬁer “ a , p p e v lwa b i m l 8s 1 1 d. E m i le . The m h ina ys e . We do it vith fu mﬁ a nd with £3 7s zd

Divinh g a Sumof DEgits

difier s s s r a nd ua r it. T a ns o it nd u , q e r p e a q a e the result. Then on subu'a cting one from the other the tesult will be a nnmber without a ny excess o f nim Thus a person is toid bo thinko f a number nhm to squa re the number ; tlm to tra nsposc a nd squa re the tra nsposed one no sub tra ct one fr om the other a nd to a dd the digits o f the diflumoe a nd to tell youthe first or the la st digit o f tlfis A r sum. t onoe youtell him the numbe which the a ddi tion o f th di its s i n All ouha ve to d e g ha g ve . y o is to ' a nnu to the number given a number which wili ma ke it a multi le o f nine If he sa s the first number is x p . y , the wh l sum f t i 1 z o e o he dig ts i3 8 ; if 2 it is 7.

If thh numb er is snbtn cted from fli e nm highest mifl wh of ninq the difierwoe between it a nd tb e next bigh the number ss t a t multiple of nine will be cro ed ou. 1 6 mm 4 97 .

146 641

r u 6 228 C ro out 8 C oss o t ; ss , ing 1 0 ; this s 1 8 — xc = highe t multiple 8.

' If any of the difierences had been multiplied it would ' ha ve ma de no difierence ; it ma y be a ssumed tha t they a re f t 1 . en mul iplied by Thus multiply the first dif er ce, ’ by 3 ; this gives 7884 ; cross out one of the 8 s lea ving 784 ; the next highest multiple o f 9 is 9 X and

o crossed ut . Take any two numbers less than 10 ; multiply one of lt 2 t them by 5, add 7, mu iply by , add the o her number, subtract 14 ; the number resulting will be made up of the digits o f the two numbers . h e 6 Let t e two numb rs be 3 and . The successive = = opera tions are these : 3 x 5 1 5 ; 1 5 +7 zz ; 183

44 ; 50 which are the two i dig ts we started with . Next try the same operation with numbers o f two 1 28 i o digits , say 7 and . Carry ng out the operati ns ex t o i e a s 1 8 h ac ly as bef re g v s the ﬁnal result , 9 . T is s n seem to be wro g . But if the middle digit is split up it ca n give We therefore read 1 98 as 1 7 and

28 . , which are the two numbers we started with This can be done with all two digit numbers . The dividing the middle digit into two gives the original numbers . The same sort o f operation can be applied to three lt 10 single numbers . Mu iply the first number by and add the second ; multiply the sum by 1 0 and a dd the third number ; the three numbers will then appear in their o r pr pe or der. 2 Take the numbers , 4 and 7 . Carrying out the opera tio n gi ves (2 X 1 0 ) (24 X 1 0 ) ro the three numbers we started with in their p per order. o o out e If this operation be f ll wed , it will be se n that nothing has been done except to multiply the first of the e 1 00 1 0 thr e numbers by , the second by and the last is i t ca o r not multiplied by anyth ng. These mul ipli ti ns th ow n n the three numbers i to the hundreds , tens and u its places in the final number . The above principle ca n be applied to the investiga r tion of the inversion ope ation just given . It gives a f general formula fo r a ll cases . Call the three digits o u (I c i h to e b . t e n mber be tr ated , and Writ ng them as a s the numbers, on the b is of multiplications as above , ‘ 1 000 1o h o gives +b +e. Writing t is ut again and e o placing b neath it its inversi n , remembering that units d o now become hun reds and hundreds bec me units, and subtracting we have : tuo c +1o b +o

— 99 c ==99 ( ft — C )

Suppooe our number is 795 ; thm a — c is 7 a nd 99 x z ==198 is the number which will be obta ined

mvmg nnupmor rm by mm

If 20 is divided by 1 1 the quotient is If 30 is to be divided simply increase the ﬁrst number of the la st quotient by 1 a nd decrea se the second number ir a s lon a s o u is a lso by 1 a nd repeet the pa g y w h. The sa me rul i l the line a nd the r ults of e a ppl es a l down , es d e f 1 0 1 1 l e ividing multipl s o and are tabu at d here .

6o 70

Inspection will show perfectly howthe rule o f a dding 1 and subtracting is carried out. Ano ther thing to be observed is that each pair or couple of digits in the quotients above is divisible by 9 ; it makes no difference what pair o f numbers are taken as long as they are next r to one another in any of the quo tients . F om the ﬁrst o f o the 1 8 8 1 quotient the tabulati n we get pairs and , t bo h divisible by 9, and the same applies to all the other quotients .

Fractional Bar

Fractions Addition a nd Subtraction of

Multiplication a nd Division o f 96-99

Group Additi on

H otheneuse The S of the 1 1 1 2 yp , quare 3 , 3

Index Addition Interest and Discount s c o Intere t, Can ellati n in Interest for One Day r How Inte est, Calculated t P o c o o In erest eri ds, Redu ti n f s o in Intere t Rate, Divis rs n How I terest Rate, Expressed t e Sho In er st, rt Cuts

’ wi Cs rrol s S o Cut 2 Le s h rt 9 , 93

’ Metius Value o f Pi ( t ) c o o Multipli ati n, Aliqu t Part An Addition By a ny Number Between Twelve a nd Twenty By Nine By One Hundred and Eleven

By Two Numbers at Once Complement M 0 0 0 0 o o o o c e e e o o e o o o o e o o o “ M e o o o o q o e e o o o o o o o o o o o k

h 0 0 0 0 0 0 0 w 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m !m se-ss inverted 56 M i les of -80 01 Deeh nls 103, 104 Of Fra ctions 9699 Of Nu bers Ending in s 66

M oL b Exeeuo h les

163 . 163

Orders of Diﬂerencee of Powers

1 14. 1 15

1 18, no

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