The "Greatest European Mathematician of the Middle Ages"

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The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci).

His names

Fibonacci

Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. There are a couple of explanations for the meaning of Fibonacci: 1. Fibonacci is a shortening of the Latin "filius Bonacci", used in the title of his book Libar Abaci (of which mmore later), which means "the son of Bonaccio". His father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English names of Robin-son and John-son. But (in Italian) Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as his family name (as in "the Smiths" for the family of John Smith). Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"; the uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian and written Latin, common at that time. However he did not use the word "Fibonacci". This seems to have been a nickname probably originating in the works of Guillaume Libri in 1838, accordigng to L E Sigler's in his Introduction to Leonardo Pisano's Book of Squares (see Fibonacci's Mathematical Books below). 2. Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune").

Statue of Fibonacci

Other names Other names He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his Hnea mise p, eLrehoanpsa rmdor Pe icosarnreoct. Olycca calsileodn Laelloy nhaer dalos oo wf Proistea Loer,o unsianrdgo a B laigtinoilsalo tsiionnc oef, in hTisu scanamney,, bLiegoonlloa rmdoea Pniss an tora. veOccaller.si onally he also wrote Leonardo Bigollo siWnece sh, ianl lT justsc acany,ll h bimig oFlliob omneaaccins a as tdraove mlolestr. modern authors, but if you are looking him Wupe ishn oalldl ejurst b ocaoks,ll h ibme Fpirbeopnaareccid taos s deoe manoyst o mf tohdee arnb oaveuth voarrs,ia btiount sif oyof uh isa rnea me. lo[Wokiithn gth hainmks u pto i nP oroldf.e Cr lbaouodksio ,G bieo mprinepi oafr eRdo tmo ese foer ahneylp o of nth teh ea bLoavetin vaanrdia Ittioanliasn o nf ames hinis tnhaism see.ct ion.] [With thanks to Prof. Claudio Giomini of Rome for help on the Latin and Italian n ames in this section.] Fibonacci's Mathematical Contributions

Fibonacci's Mathematical Contributions Introducing the Decimal Number system into Europe

Introducing the Decimal Number system into Europe He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point Haen dw aa ssym oneb oolf ftohre z feirrsto: people to introduce the Hindu-Arabic number system into E1u 2ro 3p e4 -5 th6e 7 p 8o 9si t0io nal system we use today - based on ten digits with its decimal pHoiisn tb aonodk ao nsym howb otol f odro ze arrioth:m etic in the decimal system, called Liber abbaci (meaning 1B 2o o3k 4o f5 t6h e7 A8b 9a cu0 s or Book of Calculating) completed in 1202 persuaded many HEius rboopoeka no nm haothwe mtoa dtioci arnitsh mofe htics dina tyh eto d uesecim thaisl syst"newem" syst, calelemd. Liber abbaci (Tmheea bnoinogk Bdoeoscrk oibf eths e( inA bLaacutins) tohre B rouoleks o wf eC aallcl unloawtin lge)a rcon matp eleletemde innt a1r2y0 sch2 ool for paedrdsuinagd neudm mbaenrsy, Esuurbotrpaectanin gm, amthueltmipalyiticinga nasn do fd hiviisd dinagy, ttoo guesteh ethr isw i"tnhe mwa" nsysty preomb.le ms to Tilhluest braootek tdhescr meibtheosd (s:in Latin) the rules we all now learn at elementary school for a d d 1in g7 n4u +m b e 1r s7, su4 -b t r a ct1 i7n g4, xm u l t1ip 7lyi 4n g÷ a2n8d dividing, together with many p r o b l e2m 8s t o i l l u st2 r8a t e t h e m2 e8t h o d s: is 1- -7-- -4 + - - 1-- -7 4 ------1- 7 4 x 1 7 4 ÷ 28 2 20 82 1 24 86 3 42 8 0 + 6 isre mainder 6 ------1- 3 9 2 2 0 2 1 4 6 3 -4- -8-- -0- + 6 remainder 6 ------14 38 97 22 ------L e t ' s f i r s t o f a ll l o o k a t 4th 8e 7R o2m an number system still in use in Europe at that time ( 1 2 0 0 ) a n d s e e h o w a w --kw---a--r d it was for arithmetic. L et's first of all look at the Roman number system still in use in Europe at that tRimoem (a1n20 N0u) manedr aselse how awkward it was for arithmetic. The Numerals are letters Roman Numerals The Numerals are letters The method in use in Europe until then used the Roman numerals: I = 1, T hVe = m 5e,t h od in use in Europe until then used the Roman numerals: XI = = 1 1, 0 , V = 5, X = 10, L L = = 5 500, , CC == 110000,, DD == 550000 aanndd MM == 11000000 YYoouu cacann ststilil l seseee tthheemm uusesedd oonn ffoouunnddaattioionn ststoonneess ooff ooldld bbuuiildldiinnggss aanndd oonn ssoommee clcloocks.cks. TThhee AAddddiittiivvee rruullee TThhee sisimmpplelesstt systsysteemm wwoouuldld bbee mmeerreelyly ttoo uusese tthhee leletttteerrss ffoorr tthhee vavalluueess aass inin tthhee ttaabblele aabbooveve,, aanndd aadddd tthhee vavalluueess ffoorr eeaacchh leletttteerr uusesedd.. FFoorr ininststaanncece,, 1133 cocouuldld bbee wwrritittteenn aass XXIIIIII oorr ppeerrhhaappss IIIIIIXX oorr eevevenn IIIIXXII.. TThhisis ooccuccurrss in inth teh eR oRmomana nla nlagnugaugaeg eo fo Lf aLtaint inw hwehrer e2 32 3is isps sopkeoken na sa tsr etrse est e vit gviingtini wti hwichhich tr anslates tarasn tslhraetes a ansd tthwrenet ya.n Ydo twu emnatyy. rYeomue mabye rr ethmee nmubrseer rtyh erh nymursee Sryin rgh yma Seo Snign go fa SSoixnpge onfc Se iwxpheichnc bee wghinisch begins SSiningg aa sosonngg ooff sisixxppeenncece AA ppoockeckett ffuulll ooff rryeye FFoouurr aanndd ttwweennttyy bblalackbckbirirddss BBaakekedd inin aa ppieie...... AAbbooveve 110000,, tthhee LLaattinin wwoorrddss uussee tthhee ssaammee oorrddeerr aass wwee ddoo inin EEnnggllishish,, soso tthhaatt wwhheerreeaass 3355 isis qquuiinnqquuee eett ttrrigigininttaa ((55 aanndd 3300)),, 223355 isis dduucecenntti i ttrrigigininttaa qquuininqquuee ((ttwwoo hhuunnddrreedd tthhirirttyy ffiveive)).. IInn tthhisis sisimmpplele systsysteemm,, uusisinngg aaddddititioionn oonnly,ly, 9999 wwoouuldld bbee 9900++99 oorr,, uusisinngg oonnllyy tthhee nnuummbbeerrss aabbooveve,, 5500++1100++1100++1100 ++ 55++11++11++11++11 wwhhichich ttrraannslslaatteess ttoo LLXXXXXXXXVVIIIIIIII aanndd bbyy tthhee sasammee mmeetthhoodd 11999988 wwoouuldld bbee wwrritittteenn bbyy tthhee RRoommaannss aass MMDDCCCCCCCCLLXXXXXXXXVVIIIIII.. BBuutt sosommee nnuummbbeerrss aarree lolonngg aanndd itit isis tthhisis isis wwhheerree,, ifif wwee aaggrreeee ttoo lelett tthhee oorrddeerr ooff leletttteerrss mmaatttteerr wwee cacann aalsolso uusese susubbttrraactctioionn.. ThTeh es usbutbrtarcatcivtiev eru rluel e

TThhee RRoommaann lalanngguuaaggee ((LLaattinin)) aalsolso uusesess aa susubbttrraactctiioonn pprriinncicipplele soso tthhaatt wwhheerreeaass 2200 is isvi gviignitni t1i 91 9is i"s1 " 1fr ofrmom 2 02"0 o" ro ur nudnedviegviignitni.t iW. We eh ahvaeve it int inE nEgnligshlish w hwehne nw ew esa say tyh teh etim e tiism "e1 0is t"o1 07" t ow h7i"ch w hisic nho its thnoe t sathme esa ams e" 7a s1 0""7. 1T0h"e. Tfirhste mfiresta mnse 1a0n sm 1in0u mteisn ubtefso re ( or bsuefbotrrea ct( oerd sfurobmtra) ct7 e0d'cl forock,m) w7h 0e'clreoack,s th weh seerceoansd t hmee seancos 1n0d meinauntes s1 a0d mdeindu teos ( or aadftdeerd) 7to o ('cloro ackft.e Tr)h 7is o is'cl aolck.so rTehfliesct ise da lsoin Rreofmleacnte ndu inm Reroamls.a nT hnius maebbrarels.vi Tathioisn makes athbeb roervidaetri oonf lmetatekerss i mthpeo ortradnetr. oSfo l eift tae rsms imalpleorr tvaanlut.e S cao imf ae smbefaolrleer tvahel unee xtca lmareg er one, bite wfoarse sutheb tnraectxte ladr agnedr oifn iet ,ca it mwea sa fsuterb,t rita wctaesd aadndde idf .i t came after, it was added. FFoorr eexaxammpplele,, XXII mmeeaannss 1100++11==1111 ((sisinnccee tthhee ssmmaallelerr oonnee cocommeess aafftteerr tthhee lalarrggeerr ten) tbeunt) IbXu mt IeXa mnse a1n les ss1 ltehssan t h1a0n o 1r 09 .o r 9. BBuutt 88 isis ststilil l wwrritittteenn aass VVIIIIII ((nnoott IIIIXX)).. TThhee susubbttrraactctioionn iinn nnuummbbeerrss wwaass oonnllyy ooff aa uunniitt ((11,, 1100 oorr 110000)) ttaakekenn aawwaayy ffrroomm 55 ooff tthhoosese uunnititss ((55,, 5500 oorr 550000 oorr ffrroomm tthhee nneextxt larger lamrugletirp mleu oltfi p1l0e (o1f0 1, 01 0(100 o, r1 1000 o0r) .1 000). UUsisinngg tthhisis mmeetthhoodd,, 11999988 wwoouuldld bbee wwrritittteenn mmuucchh mmoorree ccoommppaaccttlyly aass MMCCMMXXCCVVIIIIII but bthuits t htaiske taske a sli tatl eli tmtleo rmeo trime eti mtoe i ntote irnptererpt:r 1e0t:0 100 +0 0( 1+0 (01 l0e0ss le tssha nth 1a0n0 100)0 +0 )( 1+0 ( l1e0ss than le1ss00 )t h+a 5n +1 010 +) +1 5+ +1 .1 N+ o1t e+ t1h.a tN ino teh eth UaKt i nw teh eu seUK a wsiem uilsear systa simemila rfo systr timeem w fohre n t6im:5e0 w ish eonft e6n:5 sa0 iids oafst e"nte san tiod 7a"s a"ste wn etoll a7s" a"6s fwiftey"ll ,a sis m"6il afirfltyy" f,o sir m"ai lqaurlayr ftoer "toa 4" qmueaartneirn tgo 34:"4 5m. eIna nthineg U3S:4A5,. 6In:5 t0h eis U soSmA,e 6ti:m50e sis sp soomkeenti mase s"1 s0p okef 7n". as "10 of 7". LLooookk oouutt ffoorr RRoommaann nnuummeerraallss uusesedd aass tthhee ddaattee aa ffilimlm wwaass mmaaddee,, oofftteenn rreecocorrddeedd on othne t hscre screene ewnh wichhi chgive gisve itss ictse ncesonrs ocer rcetifircatifitcaionti oonr opre rphearphsa pthse t hvee rvey lrays lta istm aimgea ge ooff tthhee mmooviviee ggiviivinngg ccrreeddititss oorr cocoppyryrigighhtt ininffoorrmmaattioionn.. Arithmetic with Roman Numerals

AArritithhmmeetticic wwaass nnoott eeaasysy iinn tthhee RRoommaann systsysteemm::

CCLLXXXXIIIIIIII aaddddeedd ttoo XXXXVVIIIIII isis CCCCIIII CCLLXXXXIIIIIIII lelessss XXXXVVIIIIII isis CCXXXXXXXXVVII FFoorr mmoorree oonn RRoommaann NNuummeerraals,ls, seseee tthhee eexcxceellelenntt FFrreeqquueennttlyly AAskeskedd QQuueeststioionnss oonn RRoommaann NNuummeerraalsls aatt MMaatthh FFoorruumm..

Stems from Fibonacci Decimal Positioning Stems from Fibonacci Decimal Positioning

TThhee DDeecciimmaall PPoossiittiioonnaall SSyysstteemm

TThhee systsysteemm tthhaatt FFibiboonnaaccicci iinnttrroodduucecedd ininttoo EEuurrooppee cacammee ffrroomm IInnddiaia aanndd AArraabbiaia aanndd uusesedd tthhee AArraabbicic sysymmbboolsls 11,, 22,, 33,, 44,, 55,, 66,, 77,, 88,, 99 wwitithh,, mmoostst imimppoorrttaannttlly,y, aa symsymbbool l ffoorr zzeerroo 00.. WWitithh RRoommaann nnuummbbeerrs,s, 22000033 cocouulldd bbee wwrriitttteenn aass MMMMIIIIII oorr,, jujustst aass clcleeaarrlly,y, itit cocouulldd bbee wwrritittteenn aass IIIIIIMMMM -- tthhee oorrddeerr ddooeess nnoott mmaatttteerr sisinncece tthhee vvaalluueess ooff tthhee leletttteerrss are aadrdee add tdoe md atoke m tahkee n thume bnuemr inb etrh ein o trhigei noarilg (inunaal (bubnraebvibarteevida) tsysted) esystm. Wemit.h W thiteh the aabbbbrreeviviaatteedd systsysteemm ooff IIXX mmeeaanniningg 99,, tthheenn tthhee oorrddeerr ddiidd mmaattteerr bbuutt itit seseeemmss tthhisis sytsyteemm wwaass nnoott oofftteenn uusesedd inin RRoommaann ttimimees.s. IInn tthhee ""nneeww systsysteemm"",, tthhee oorrddeerr ddooeess mmaatttteerr aalwlwaaysys sisinncece 2233 isis qquuititee aa ddififffeerreenntt nnuummbbeerr ttoo 3322.. AAlsolso,, sisinncece tthhee ppoosisittioionn ooff eeaachch ddigigitit isis imimppoorrttaanntt,, tthheenn wwee mmaayy nneeeedd aa zezerroo ttoo ggeett tthhee ddigigititss ininttoo tthheeirir ccoorrrreectct pplalacecess ((cocolulummnns)s) eegg 22000033 wwhhichich hhaass nnoo tteennss aanndd nnoo hhuunnddrreedds.s. ((TThhee RRoommaann sysyststeemm wwoouuldld hhaaveve jujustst oommitittteedd tthhee vavalulueess nnoott uusseedd soso hhaadd nnoo nneeeedd ooff ""zezerroo""..)) TThhisis ddeecicimmaal l ppoosisittioionnaal l systsysteemm,, aass wwee cacall l itit,, uusesess tthhee tteenn symsymbboolsls ooff AArraabbicic oorrigiginin aanndd tthhee ""mmeetthhoodds"s" uusesedd bbyy IInnddiiaann HHinindduu mmaatthheemmaattiiciciaannss mmaannyy yeyeaarrss bbeeffoorree tthheeyy wweerree imimppoorrtteedd ininttoo EEuurrooppee.. IItt hhaass bbeeeenn cocommmmeenntteedd tthhaatt inin IInnddiaia,, tthhee coconncecepptt ooff nnootthhiningg isis imimppoorrttaanntt inin ititss eeaarrlyly rreelilgigioionn aanndd pphhiliolososopphhyy aanndd soso itit wwaass mmuuchch mmoorree nnaattuurraal l ttoo hhaaveve aa symsymbbool l ffoorr itit tthhaann ffoorr tthhee LLaattiinn ((RRoommaann)) aanndd GGrreeeekk systsysteemms.s. "Algorithm"

EEaarrlileierr tthhee PPeerrsisiaann aauutthhoorr AAbbuu ‘A‘Abbdd AAllalahh,, MMoohhaammmmeedd iibbnn MMuussaa aall--KKhhwwaarriizmzmi i ((uususuaallyly aabbbbrreeviviaatteedd ttoo AAl-l-KKhhwwaarrizmizmi i hhaadd wwrrititteenn aa bbooookk wwhhichich ininclcluuddeedd tthhee rules rules of arithmetic for the decimal number system we now use, called Kitab al jabr wa‘l- muoqfa abraitlham (Retuicle fso ro ft hre stdoerciinmga al nndu meqbuear tsysting) edmat iwneg nfrowm uasebo,u cta 8ll2e5d AKDita. bD aEl jKanbur th (in thew ear‘rla-mtau fqoar bthaela se (Rcoulneds oedf riteiostno arinndg tahnirdd eeqduitaiotinn go)f dhaist in"Fgu fnrodmam aebnotuatl 8A2lg5o AritDh.m Ds "E) giveKsn uththe (fiunl lt nhea merer aatbao fveor tahned se sacoysn dit caednit ibone atrnadn slthairtedd e adsit ioFna tohfe hr iosf "AFbudnudlalamhe, ntal MoAhlagmormithemd,s" so) ng iovef sM tohsee fs,u lln naatimvee oaf bKohvwe aarnizdm s.a Hyse iwt caasn a bne a sttraronmslaotmede ra tso Fthaeth caer liopfh at BagAhbddaudll a(hn,o Mw oinh aIrmaqm).e d, son of Moses, native of Khwarizm. He was an astromomer to the caliph at Baghdad (now in Iraq). Al-Khowârizmî is the region south and to the east of the Al-KhAorwaâl Srizmea îa irso tuhned r tehgeio tno wson untohw a ncad ltleod t hKeh eivaast ( or Urgoef nthche )A orna lt hSee aA maruo uDnadr ythae r itveowr.n I tn wowas c paallertd o Kf thhiev aS ilk R(ooru Uter,g ae nmchaj)o or ntr athdein Agm pua tDhwarayya breivetwre. eItn w tahse pEaarstt o af nd the SEilku rRoopuet.e I,n a 1 m20a0jo irt wtrads ining P peartsihwa abyu tb teotdwaeye nis in thUezb Eeakistst aannd, Epaurrto opfe t.h Ien f1o2rm00e rit UwSaSs Rin, nPoerrtsiha o bf uIrta n, today is in Uwzbhicheki gstaaine, dp aitrst oinfd theep efonrdmenecer U iSnS 1R9,9 1. north of Iran, which gained its independence in 1991. Prof Don Knuth has a picture of a postage stamp issued by the USSR in 1983 to commemorate al-Khowârizmî 1200 year anniversary of his probable birth date. Fro Pmro tfh De otintl eK onfu tthi sh abso oak p Kictituarbe a ol fj ab pro wst'alg-me ustqaambpa laissu wee de bryive th oeu Ur SmSoRd einrn 1 w98o3rd t o algcoebmram. e morate al-Khowârizmî 1200 year anniversary of his probable birth date. Th eF rPoemrsi thaen taituleth oofr t'sh insa bmoeo kis K citoamb male jmabora wte'adl -imn uthqea bwaolard w aelg doerriitvehm o.u Irt mhaosd echrna nged ovewr othrde aylegaerbsr faro. m an original European pronunciation and latinisation of algorism. Alg oTrhiteh mPes rwsiearne aknutohworn's o nf abmefeo ries cAol-mKmhoewmâorirzamteîd's inw rtihtien gwso, r(dfo ar legxaormithpmle., IEt uhclasid 's Elechmeantgse ids ofuvlel ro ft haelg yeoraithrsm fsro fmor agne oomrigeitnray,l Einucrluodpienagn o pnreo ntou nficinda ttihoen garneda ltaetstin icsaomtiomno n diviosof ar lgoof rtwismo n. uAmlgboerirtsh mcasl lwede rEeu kncliodw'sn a olgf obreitfhomre t oAdl-aKy)ho. w ârizmî's writings, (for Theexa UmSApl eL,i bEruarclyi do'fs CEolenmgreenssts hisa sfu all olistf a olgf ocirittahtmiosn sfo orf gAelo-Kmheotwryâ, riinzclmuî daindg hoinse w to rks. Oufri nmdo tdheer ng rweaotredst "a colgomrmithomn" d divioesos rn ooft tjwusto naupmplbye tros tcahel lreudle Esu oclf iadr'sit hamlgeotriicth bmu tt omdeaay)n.s any T phree UciSsAe sLeibtr oarfy i nosf tCruocntgiorensss fhoars p ae rlifsto romf icintga taio cnos mofp Ault-aKthioonw âwrhizemthîe ar ntdh ish ibs e a mewthoordks. fo llowed by humans, for example: Our modern word "algorithm" does not just apply to the rules of arithmetic but a comeoakinnsg arneyc ipree; c ise set of instructions for performing a computation whether a knthiistt ibneg p aa tmteertnh;o d followed by humans, for example: travel instructions; a ca a rco moakinnuga lr epcaigpee ;f o r example, on how to remove the gear-box; a m ae kndicaittiln pgr poactetdeurnre; such as removing your appendix; a ca tralcveulal tiinosnt rbuyc htiuomnsa;n computors : two examples are: Wi lalia camr S mhanuksa lw phaog coe fmorp euxatedm tphlee ,va onlu heo owf ptoi troe m70o7ve d ethciem gael aprl-abceox;s by hand last cen atu mrye odvicaer la pbrooucte 2d0u yere asursch up a tso r1e8m7o3vi -n bgu yot huer waapsp ewnrdoinx;g at the 526-th place when it wa sa ch caelcckeuladti obny bdye skhu camalcun lcaotomrsp iunt o19rs4 4: !t w o examples are: Ea rWlieilrli aJomh Sanhna nDksase w hhoa dco commpuptuetde dth pei vacolrure ctolfy p tio t o2 0750 7d edciecimmala pl lpalceaces sin b 1y8 h4a4n wd hen ageladst 2 0ce bnutut rthy iso vewar sa bdounte 2 c0o myepalerste ulyp itno h1i8s 7h3e a- db ujut sthe w wriatins gw trhoen gn uamt tbheer 5d2o6w-nth after woprkilanceg ownh iet nfo irt twwaos m choenckeths!d! by desk calculators in 1944! or mEaerclihear nJoichaalnlyn bDya msea chhaidn ecos (msupuchte ads p pi lcoacirrnegct chly itpos 2 a0n5d dcoecimmpaoln pelnatces ast inco 1rr8e4ct4 plawcehse no na gae cidr 2cu0i tb buot athrdis two agso dinosined eco yomuprl eTtVel)y in his head just writing the number down after working on it for two months!! or mechanically by machines (such as placing chips and components at correct places on a circuit board to go inside your TV) or automatiicalllly by ellectroniic computters which store tthe iinstructiions as wellll as ddaattaa ttoo wwoorrkk oonn.. SSeeee DD EE KKnnuutthh,, TThhee AArrtt ooff CCoommppuutteerr PPrrooggrraammmminingg VVoolulummee 11:: FFuunnddaammeennttaal l AAlglgoorritithhmmss ((nnooww inin ititss TThhirirdd EEddiittioionn,, 11999977))ppaaggeess 11--22.. TThheerree iiss aann EEnnggllishish ttrraannslslaattioionn ooff tthhee "".... aal l jajabbrr ...."" bbooook:k: LL CC KKaarrppininskiski RRoobbeerrtt ooff CChheesstteerr''ss LLaattiinn TTrraannssllaattiioonn ...... ooff aall--KKhhoowwaarriizmzmii ppuubbllishisheedd inin NNeeww YYoorrkk inin 11991155.. [[NNoottee tthhee vavarriaiattioionn inin tthhee spspeellilningg ooff ""AAll--KKhhoowwâârrizmizmîî"" hheerree -- tthhisis isis nnoott uunnuususuaall!! OOtthheerr spspeellilninggss ininclcluuddee aal-l-KKhhoorreezzmmi.i.]] IIaann SStteewwaarrtt's's TThhee PPrroobblleemmss ooff MMaatthheemmaattiiccss ((OOxfxfoorrdd)) 11999922,, IISSBBNN:: 00--1199--228866114488--44 hhaass aa chchaapptteerr oonn aalglgoorriitthhmmss aanndd tthhee hhististoorryy of othf eth nea nmaem: ech: chapateprte 2r1 2: 1D: iDxiitxi At lgAolgroizmrizim. i.

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DiDdi dF iFbiobnoancaccic iin ivnevnetn tth tihsi sS eSreierise?s ?

FFibiboonnaaccicci sasaysys hhisis bbooookk LLiibbeerr AAbbaacici ((tthhee ffirirstst eeddititioionn wwaass ddaatteedd 11220022)) tthhaatt hhee hhaadd ststuuddieiedd tthhee ""nnininee IInnddiaiann ffigiguurrees"s" aanndd tthheeirir aarritithhmmeetticic aass uusesedd inin vavarrioiouuss cocouunnttrrieiess aarroouunndd tthhee MMeeddititeerrrraanneeaann aanndd wwrroottee aabboouutt tthheemm ttoo mmaakeke tthheeirir uusese mmoorree commonly counmdmerostnolyo du nind ehrisst noaotdive in Ihtaisly. n aStoive h eIt aplry.ob Saob lhye m peroreblay binlycl mudeeredl yth ine cl"ruadbebdit tphreo blem" "frraobmb iot npero obfl ehmis" cforonmta ctonse a onfd h disid c noontt ainctves natn edi tdhiedr nthoet inpverobnlte emit hoer rt hthee se proiebsl eomf or tnhuem sebreierss wohf inchum nboewr sb ewahri chhis n noawm bee.a r his name. DD EE KKnnuutthh aaddddss tthhee ffoollolowwiningg inin hhisis mmoonnuummeennttaal l wwoorrkk TThhee AArrtt ooff CCoommppuutteerr PPrrooggrraammmminingg:: VVoolulummee 11:: FFuunnddaammeennttaal l AAlglgoorritithhmmss eerrrraattaa ttoo sesecoconndd eeddititioionn:: BBeeffoorree FFibiboonnaaccicci wwrroottee hhisis wwoorrkk,, tthhee seseqquueennccee FF((nn)) hhaadd aalrlreeaaddyy bbeeeenn ddiscuiscussessedd by bIny dIniadnia schn scholaors,lar ws,h woh hoa hd alodn lgo nbge benee int einrteesrteesdte ind rinh ytrhhytmhicm pica tptaerttnesr nths atht art ea rfoer med fforromme odn fero-bme aotn aen-bde tawt oa-nbde atwt noo-btees.at Tnhoete ns.u mThbee rn oufm subechr o rfh suythcmh srh hytahvimngs hna bvienagts n baeltaotgse athltoerg eist hFe(rn i+s1 F);( nth+e1r)e; ftohreer ebfootrhe G bospth aGlao s(pbaelfao r(eb e1f1o3re5 )1 a1n3d5 )H aenmda chandra (c. H1e1m50a)ch maenndtiroan (ec.d 1th1e5 0n)u mebentriso n1e, d2 ,t h3e, 5n,u 8m, b1e3r,s 2 1,, 2..,. 3e,x 5p,li ci8,t l1y.3 , 21, ... explicitly. KKnnuutthh rreeffeerrss ttoo aann aarrtticliclee bbyy PP SSiningghh iinn HHististoorriaia MMaatthheemmaatticaica vovoll 1122 ((11998855)) pages p2a2g9e-s2 4242.9 -244. NaNmaminign gth teh eS eSreierise s

IItt wwaass tthhee FFrreenncchh mmaatthheemmaatticiiciaann EEddoouuaarrdd LLuuccaass ((11884422--11889911)) wwhhoo ggaaveve tthhee name nFaimboen Faicbcoin naucmcib neursm tboe trhsis t os ethriiess se anride sfo aunndd fmouanndy motahneyr oimthpeorr timanpt oarptapnlitca tions as awpeplli caast ihoanvis nags twhe llse asri ehsa viofn ngu tmheb esersr itehsa ot fa rneu mclobseersly t hrealta atered tcloo tsehel yF ribeolanteadcci to tnhuem Fbibeorns a-cci the n Luumcabesr sN u- mthbee Lrsu:ca 2,s 1N, u3m, 4b,e 7rs, :1 21,, 11,8 3, ,2 49,, 74,7 1, 1..,. 18, 29, 47, ... named after him. named after him. Fibonacci memorials to see in Pisa Fibonacci memorials to see in Pisa

H e died in the 1240's and there is now a statue commemorating him located at th e Leaning Tower end of the cemetery next to the Cathedral in Pisa. [With spHeeci daiel tdh ainn tkshe t o1 2N4i0ch'so alansd F thaerhrei, iasn n eoxw-p au pstila otuf eW coinchmmestemero Craotilnlegg eh,im fo lro tcahet ed at ptictheu rLee oafn tinhge Tstoawtueer. ]e nd of the cemetery next to the Cathedral in Pisa. [With special thanks to Nicholas Farhi, an ex-pupil of Winchester College, for the picture of the statue.] References

References D E Smith's History of Mathematics Volume 1, (Dover, 1958 - a reprint of the orignal version from 1923) gives a complete list of other books that he wrote and is Da Efu lSlemr irtehf'es rHenistceo royn o Ff Miboanthaeccima'st ilcsife Vaonldu mweo r1ks., ( Dover, 1958 - a reprint of the Tohriegrnea ils ve anrsiotohne rf rbormie f1 b9i2o3g)r agpivehys o af Fcoibmopnlaeccite lwisth iochf o itsh epra rbto oofks Ka trheant hHeu nwgreorte and Piassh a afulll'lse r( Vreirfgeirneian ceUn oivne Frsiibty)on Taccihe 'sa rlti foef aAnldg ewborarks. fro m from al-Khwarizmi to Viéte: A TShtuedrey iisn athneo tNheartu brraiel Sf beiloegctriaopnh oyf oIdf eFaibso inf ayocciu wahnicht t ois r epadrt mofo Krea raebno Hutu tnhgee r hPistaoshrya ollf's m (Vatirhgeimnia tUics.niv ersity) The art of Algebra from from al-Khwarizmi to Viéte: EAi gShtut dHyu innd trheed N Yaetuarrasl YSoeulenctgio bny oAf IFd eHaosr aifd yoamu (wUannivet torsi retya do fm Noerwe aEbnogulat nthde) in Thhiest Aoruyst orfa mliaant hMeamthaetimcs.a t ics Teacher Vol 31, 1985, pages 123-134, is an in tEeirgehstti nHgu annddre rde aYdeaabrlse Yarotiuclneg o bny F Aib Fo nHaoccira,d haims n(Uamnieves rasintdy ofr iNgienws aEsn wglealnl da)s inh is mTahthee Amuasttircaalila wno Mrks.ath Heme aretifcser sT teoa achnedr e Vxpola 3n1d,s 1 u9p8o5n, pthaeg efosl l1o2w3in-1g3 a4r,t iicls ea.n.. Tinhte rAestutiongb iaongdr arpeahdya obfl eL aerotinclaer doon PFibsoanaocci R ,E h iGs rnimame, sin a Fnidb oonriagcciins Qaus awrteelrl lay svo l 1h1i,s 1 m97a3th, epmagaetisca 9l9 w-1o0rks4. . He refers to and expands upon the following article... L Tehoen aArudt ofb Pioigsara apnhdy tohfe L Neeowna Mrdaoth Peimsaantioc sR o Ef Gthreim Mmid, dinl eF Aibgoensa cciby QJ uaanrdte Frl y Gvoies,l 1 T1,h 1o9m7a3s, Ypa Cgreosw 9e9ll- 1p0u4b.l ish ers, 1969, 127 pages, is another book with much o nL tehoen baardckg ofr oPuinsda taon Fdi bthoen aNcciew's Mlifaet hanedm watoicrk.s o f the Middle Ages by J and F DGeiells,a vTihtao me adse lYle C orpoewreel l dpiu Lbelioshnearrs,d o1 9P6is9a, n1o27 B paaldgaessas, isrr ea nBoothnecor mbopoakg wnii,t hR momuceh, 1o8n5 4th ise tbhaeckg onrlyo ucondm tpol eFteib opnrianccited's v elifresi aond o wf Foirbk.o n acci's 1228 edition of Liber A bDaecill.a vita e delle opere di Leonardo Pisano Baldassarre Boncompagni, TRhoem thee, 1M8a5t4h iFs othruem o'ns lya rcochmivpelse toef pthrein tHeidst veoryrsi oof nM oaft hFeibmoantaicscci d'si sc12u2ssi8 eodni tgioronu opf coLnibtearin A ab aucise. f ul discussion on some of the controversial topics of Fibonacci's n aTmhes t haen dM laifteh (FFoerburmua'sr ya 1rch99iv9e).s U osef th iets Hniestxot>ry> o lifn Mk atoth feomlloawti csth ed itschruessiad oonf tghreo up dcoiscuntssiaino an uthsreofugl hd iscuits 6ssi emona iolend so comnetr iobfu tthioen cos. nItt rtaolveksr siabaol utot pthicse uonf cFeibrtoaninalccity o'sf his bnirathm aensd a dneda ltihfe d (aFteebsr aunadry h 1is9 9n9a)m. Ues.se I ti tses neemxst> t>h alitn Fk ibtoo nfoalccilow n tehvee trh refaedr roefd t htoe hdimiscuselssif aso n"F tihbroonuagccih it"s b 6u te tmhiasi lweda sco an ntrickibu-ntioanms.e Igt itvealksn t oa bhoimut bthye la utenrc ewrrtiateinrls.ty of his birth and death dates and his names. It seems that Fibonacci never referred to himself as "Fibonacci" but this was a nick-name given to him by later writers.