Braille Mathematics Notation

Home , Plus and minus signs, Roman numerals

BRAILLE AUTHORITY OF THE UNITED KINGDOM

BRAILLE MATHEMATICS NOTATION

Royal National Institute of the Blind

Bakewell Road Orton Southgate

Peterb orough Cambridgeshire

PE XU

Braille Authority of the United Kingdom

Registered Charity No

Printed by RNIB Peterb orough

CONTENTS

Memb ers of the Mathematics Committee ii

Memb ers of the JointTechnical Committee ii

Intro duction to edition iii

Intro duction to edition iv

Table of braille mathematical signs

Mo dication of signs

Braille Mathematics Notation

General remarks

Numeral and letter signs

The spacing of braille signs

The oblique stroke and fraction line

Brackets

Indices

Sp ecial functions and other words and abbreviations

Delimiting the argument of a function

Co ordinates and sets

Matrices determinants and other arrays

Worked calculations logical gures

Miscellaneous notation

Layout of mathematical text

Units

Alphab ets used in mathematics

Index i

MEMBERS OF THE MATHEMATICS COMMITTEE

A W Chatters Chairman Bristol University

J A Allnut Central Electricity Research Lab oratory

T K Devonald Royal National College for the Blind

J Fullwo o d Marconi Systems

T D Maley Braille House RNIB

Miss D M Nicholl Braille House RNIB

S J Phipp en Braille House RNIB

W B L Po ole Chairman Braille Authority of the United Kingdom

Miss J Short Chorleywo o d College

D Spyb eyWorcester College

PJTalb ot Principal Queen Alexandra College

MEMBERS OF THE JOINT TECHNICAL COMMITTEE

D Bo den Braille Computer Asso ciation of the Blind

N Brown New College Worcester

S A Clamp Visual Impairment and Sp ecial Needs Advice VISpA

S J Phipp en RNIB Peterb orough

W B L Po ole Chairman Braille Authority of the United Kingdom

D Spyb ey New College Worcester

P Southall Dorton House Scho ol

C Stonehouse New College Worcester

PTo oze Pia

M E Townsend TorchTrust for the Blind

R West Braille Computer Asso ciation of the Blind ii

INTRODUCTION TO EDITION

This edition of the Brail le Mathematics Notation has b een prepared by the Mathemat

ics Committee of the Braille Authority of the United Kingdom It arose from the need to

extend and amend the edition in print the edition with Addendum and

this also seemed to b e a go o d o ccasion to rewrite the co de b o ok in a more convenient form

It should b e emphasized that no changes have b een made in such basic notations as those

for addition subtraction multiplication division brackets equality unions intersections

inequalities and so on

Two of the most imp ortantchanges made in this edition are concerned with unit ab

breviations and with letter fount conventions The abbreviations for such units as metre

second and gram will now follow the numb er of units rather than precede it so that

braille will follow the standard print convention The rules ab out fount indicators have

b een changed so that letters will now b e assumed to b e small Latin unless shown otherwise

Another imp ortantchange is the extension of the use of the sign to co de any sort

of sp ecial function eg lim exp det etc as well as the trigonometric and hyp erb olic

functions At the same time some of the old word abbreviations suchas for

rectangle have b een deleted

A few alterations have b een made in order to simplify the rules and increase conceptual

clarityFor example the numerator of a fraction will always b e shown even when it is

so that Similarly the exp onent rather than will now b e written

will always b e shown so that x will b e written rather than It will

no longer b e necessary to decide whether an upp er symbolinprint should b e treated as

a sup erscript or as an exp onent the same sup erscript sign will nowserve in b oth

cases The sp ecial metho d for dealing with limits of integrals has b een deleted Some new

symb ols have b een added to the co de to deal with certain printsymb ols whichwere not

catered for b efore

There was a strong feeling in the committee that the time had come to rewrite and

restructure the co de b o ok The two part structure of the old co de has b een abandoned

and it is hop ed that the way the new edition is organized will makeitmuch easier to use

Iwould like to thank all the memb ers of the committee for b eing so generous with

their time and exp ertise and for making our discussions lively but go o d humoured I am

sure that all the other memb ers will join me in expressing our appreciation of the work

done by the Braille House sta in pro ducing all the do cuments so eciently in b oth braille

and print

A W Chatters

Bristol November iii

INTRODUCTION TO EDITION

The current edition of Brail le Mathematics Notation contains some minor amendments

to the edition prompted by the publication of the edition of British Brail leThe

main issues relate to the new rules on capital indicators For conformity with British

Brail le the double capital indicator is now used for a sequence of two or more capital

letters rather than for sequences of three or more This change has b een carried through

to other fount indicators in this co de as well as to unit abbreviations Also single capital

letters standing alone or starting a mathematical expression in ordinary text now require a

letter sign b efore the capital sign as in British Brail lethus removing a p ossible ambiguity

with single letter wordsigns

in line with British The oblique stroke has b een changed to the cell sign

Brail le The force of the numeral sign no longer carries over the hyphen again in line with

British Brail le

Unit abbreviations starting with a lower case letter followed by a capital letter such

as mW now require a letter sign b efore the m whichwas previously not the case

June iv

TABLE OF BRAILLE MATHEMATICAL SIGNS

The braille signs in the following table should generally b e used to representthe

corresp onding print signs whatever their meaning In a few cases however more than one

braille sign has the same print equivalent eg the signs for for these the correct

braille sign should b e used according to the meaning or use stated in brackets

Sign Meaning Sign Meaning

plus

minus

or approximately

equal to

multiplied by cross

divided by

varies as prop ortional to

plus or minus

is dened as

Def

minus or plus

is pro jective with

oblique stroke fraction

line sign

less than

continued fraction symbol

less than or equal to

multiplication dot and

greater than

in logic

greater than or equal to

etc an op eration sign

eg x y

much less than

etc a second

much greater than

op eration sign

greater than or less than

ro ot

less than or

equals

approximately equal to

greater than or

equivalent to

approximately equal to

swung dash

numeral sign

Table of Brail le Mathematical Signs

Sign Meaning Sign Meaning

decimal p oint intersection

comma or space marking

union

thousands etc

and meet vector pro duct

recurring decimal sign

or join

bar over negative

numb er eg

n set dierence

contained in

contains

containedinor

equal to

contains or equal to

is an elementof

reverse of

angle bracket

not

angle bracket

amp ersand and

j vertical bar

mo dulus determinant

for all

such that divides

restricted to

there exists

evaluated at etc

empty set

k double vertical bar

norm

is a theorem

matrix bracket

a reverse of

therefore

jisvalid

since

reverse of j

Table of Brail le Mathematical Signs

Sign Meaning Sign Meaning

sup erscript or subscript

arrow indicating vector

eg AB

sin

sup erscript or subscript

arrow indicating vector

sin arcsin

eg AB

cos

arrow with long shaft

cos arccos

arrow with long shaft

tan

tan arctan

sec

sec arcsec

cosec

cosec arccosec

cot

cot arccot

sinh

sinh arcsinh

cosh

cosh arccosh

tanh

Table of Brail le Mathematical Signs

Sign Meaning Sign Meaning

is a normal subgroup of tanh arctanh

or equal to

sech

reverse of

sech arcsech

k parallel symbol

cosech

p erp endicular symb ol

cosech arccosech

integral

coth

closed line integral

coth arccoth

innity

log

t etc end of pro of

log antilog

hash

colog

factorial

grad gradient

point notation of

curl or rot

partial correlation

or regression in

div divergence

statistics eg x

r nabla del

copro duct

angle

curly d partial

derivative d

circle symb ol

dash prime eg x

triangle symbol

symmetric dierence

star eg x

t square symb ol vector

bar egx

or wave op erator

hat egx

is a normal subgroup of

inverted hat egx

reverse of

Table of Brail le Mathematical Signs

Sign Meaning Sign Meaning

index separation sign tilde egx

y y

dierential op erator

dagger eg x

separation sign

actuarial symb ol

sup erscript sign

years eg x

subscript sign

dierentiation dot

egx

top index sign

double dierentiation

b ottom index sign

dots egx

index termination sign

triple dierentiation

dots eg x

matrix line indicator

degrees

word indicator

minutes

small Latin letter sign

seconds

capital Latin letter sign

radians

small b old Latin letter sign

Aangstrom

capital b old Latin letter sign

p ound sterling

small Greek letter sign

dollar

capital Greek letter sign

c cent

double capital Greek

letter sign

euro

small letter sign

percent

capital letter sign

unit separation sign

small letter sign

mathematical hyphen

capital letter sign

separation sign

Table of Brail le Mathematical Signs

Sign Meaning

small letter sign

capital letter sign

small letter sign

capital letter sign

crossed symb ol sign

Mo dication of Signs

a When it is required to show that a nonalphab etic symbol is printed in b old or large

or typ e as distinct from the usual print sign a dot is inserted after the

for comp ound signs with these prexes

This indication is not made when it is just a style of printing and of no mathematical

signs in braille signicance It is not usually shown for b old r or t

b A symb ol consisting of a struck out letter or sign is co ded by inserting a dot this

follows the for signs prexed by replaces the for signs prexed by

and precedes all other signs ie precedes a letter fount sign if present If a dot

is already present at this p oint it must b e retained

Ex R

AxB

a a

Feynman slash

Analogously is co ded as and as etc

c Symb ols printed in a circle or a square may b e co ded by placing the sign in round or

square brackets resp ectively with the comp ound sign ob eying the usual rules for the

enclosed sign

Ex x y

a b

BRAILLE MATHEMATICS NOTATION

GENERAL REMARKS

Set out mathematical expressions are generally brailled b eginning in cell with

runovers in cell whatever the setting in print See x for further details concern

ing layout of mathematical text

Dot precedes any punctuation sign except a hyphen or dash used within or imme

diately following a mathematical expression See for example x and but

note the examples in x and x for unit abbreviations

Dot is used as a mathematical hyphen when it is necessary to divide a mathematical

expression at the end of a braille line whether or not a division is made at that p ointin

the print

The following rules for dividing mathematical expressions should b e observed

a Do not separate letters from their letter signs or numb ers from their numeral signs

b Do not divide an expression so as to separate indices or dashes stars hats etc from

their terms or functions from their arguments unless the function is unusually long

c Do not divide immediately after a sign which is normally spaced b efore but not after

for for eg etc see x or after the signs

or an op ening bracket

d Do not divide immediately b efore a sign or b efore a closing bracket

e Do not divide an expression at a p oint where a space is sp ecially omitted to unite it

eg in indices x or co ordinates x

It is generally not go o d practice to divide a short expression eg x whichcould

b e conveniently brailled complete on a new line

If the print text contains symb ols which are not listed or otherwise covered in this

co de sp ecial braille signs may b e devised to represent them or else the meaning of existing

signs maybechanged for the purp ose Anysuch nonstandard notation should b e stated

b eforehand eg at the b eginning of the work in which it app ears or at the p ointofuse

See in particular x

NUMERAL AND LETTER SIGNS

The numeral sign indicates that immediately following letters aj representnu

merals and until the sequence is ended by a space a letter kzorany other sign

except those given in x below

Numeral and Letter Signs

The decimal p oint is co ded as dot

The print comma or space separating groups of digits such as thousands or thou

sandths is co ded as dot

Recurring decimals are co ded by inserting a dot b efore the recurring sequence

that sequence b eing indicated in printby dots placed ab ove its rst and last digits or its

single digit

A bar indicating a negative part of a numb er a notation used for logarithms is

co ded by inserting the sign b etween the numeral sign and the numb er

Anumb er should only b e divided using the mathematical hyphen dot if it is to o

long for a whole line In such cases the force of the numeral sign carries over the dot and

is not reasserted at the b eginning of the new line

The force of the numeral sign do es not carry over the literary hyphen

Ex May

Numeral and Letter Signs

Numeral signs may b e omitted altogether from tables or worked calculations to save

space or to allow an uncluttered presentation in such cases advance notice should b e

given

Simple numerical fractions are co ded by stating the denominator as a lower number

following the numerator in the upp er p osition

If either the numerator or denominator has a comma or space in the print indicating

thousands etc the comma may b e omitted in the braille Alternatively if the comma is

and a second numeral sign should b e inserted to b e retained an oblique stroke

with the denominator written in the upp er p osition This latter metho d retaining the

oblique strokemay also b e used for fractions without the separating dot s when it is

considered desirable to emphasize the fraction line eg in calculations

In mixed numb ers the numeral sign is rep eated for the fractional part

A letter fount sign gives the fount of the immediately following letter only

Small Capital

Latin

Greek

Bold or underlined Latin

See x

Bold Greek

Bold numeral two kinds rather

than small and capital

Numeral and Letter Signs

Small Capital

Other founts See x

is generally omitted all letters b eing assumed to b e The small Latin letter sign

small Latin unless shown otherwise It must however b e stated when the letter

a Stands alone or b egins a mathematical expression in ordinary text

b Is an aj immediately following a numb er but not a lower numb er eg the denomi

nator of a simple fraction even at the b eginning of a line

c Immediately follows a sequence of letters brailled with a double letter fountsign

d Immediately follows a vertical bar j

e Is an o within brace brackets f g

f Is an a immediately after an op ening square bracket in ordinary text

g Is the rst letter of a small Roman numeral or a capital Roman numeral co ded as

suchx

h Immediately follows a prexed function or expression see x

See x for unit abbreviations

Ex x

F xy

fxjxng

fa e i o ug

Numeral and Letter Signs

hpjH jr i

The interval a b

Ex A ladder of length l width w and mass f makes an angle

with a wall and an angle with the ground

The sign is also used b efore a single capital Latin letter p ossibly followed bya

simple numerical subscript when the letter

a Stands alone within ordinary text

b Begins a mathematical expression in ordinary text and is followed by a space

Ex In the equation A R S the total area is AandR is the radius

of the base

Let X equal Abc

Note that in accordance with the ab ove a capital Latin letter acting as a lab el in a

mathematical diagram and not adjacent to ordinary words do es not require a letter sign

If two or more letters of the same fount other than small Latin or Roman numbers

treated as suchx o ccur in sequence they are preceded by the appropriate double

letter fount sign which has force until the sequence is ended by a space or any other sign

except a dash star or numerical subscript Toavoid misreading as bar the Greek letter

small or capital should always b e preceded by the single letter fount sign and therefore

terminates a double Greek letter fount sign sequence

Ex AB

Numeral and Letter Signs

In triangle AB C sides AB and AC are equal so angles at B and at C

are also equal

AB C de

B B B B

AB C D E F GH I J

Exceptionally is used as the double capital Greek letter fount sign to distinguish

it from double bars

kf xk

Undesignated letter fount signs are used for other founts according to the particular

requirements of the text When they are used their meaning should b e stated If the fount

is particularly predominant and frequent it may in some cases b e advantageous to adopt

a single cell letter fount sign for the purp ose to avoid over use of a two cell fount sign

Such nonstandard use should b e explained

Letters printed or written with an underline or undertilde used as a fount sign eg

may generally b e co ded with the dot or dots letter fount signs without further a

Numeral and Letter Signs

comment unless b old letters also o ccur in the text or if the underline or tilde needs sp ecial

or individual indication

Dotisalsousedtoshow the b old fount of other signs see paragraph a P and

is the standard fount sign used to indicate curly d in partial derivatives etc without

further comment

A sp ecial script l is sometimes used in print to distinguish the letter from number

This is unnecessary in braille co de the letter as an ordinary letter l

P Q

and indicating summation and pro duct are conventionally normally co ded as

and resp ectively whether or not written as b old letters in print

aleph and an unusual letter denoting the Weierstrassian elliptic function

are co ded as letters a and p resp ectively with suitable letter fount signs This notation

should b e explained when rst used

The Dirac or Planck constantsymbol h used in physics should b e co ded as h unless

this notation would b e ambiguous in the particular context Otherwise it may b e sucient

to co de it simply as h if that letter is not otherwise used or else a particular letter fount

sign should b e adopted for the purp ose in b oth these latter cases the notation should b e

explained when rst used

Roman numerals are normally written as in literary braille if small or brailled as such

ie in noncapitalized braille they are preceded by the dots letter sign If however

they are indistinguishable in the print from ordinary algebraic letters they may b e co ded

as such according to the rules in the rest of this section

Note that even in noncapitalized braille it may b e necessary to show that Roman

numerals are capital

Ex V

noncapitalized braille

iii

The Spacing of Brail le Signs

noncapitalized braille

aIII

noncapitalized braille

III

noncapitalized braille

f x

Sp ecial letter fount rules exist for brailling chemical formulae for this reference should

b e made to Brail le Science NotationHowever it is p ossible to represent such formulae

unambiguously using the conventions of mathematics braille In such cases it is advisable

to avoid using the dots sign b etween the letters of a twoletter elementsymbol byinstead

using a dot sign b efore the rst letter in the symbol

Ex NaOH

HCl

H O

CH CH Br

Refer also to British Brail le where examples are given of such formulae brailled

using literary conventions onlyinwhich the subscript sign is retained

THE SPACING OF BRAILLE SIGNS

The following gives general rules for the spacing of braille signs within a mathematical

expression Spaces are however omitted when these signs are used in indices or co ordi

nates etc these cases are dealt with in the particular sections see for example x

An initial space is also omitted when the sign immediately follows an op ening bracket

and similarly a following space is omitted when the sign immediately precedes a closing

bracket

Op eration and relation signs etc prexed by or except for

or k and for all arrows the signs for

parallel to and the negation of these signs are spaced b efore but not after

The Spacing of Brail le Signs

Ex x y z

A B

x X

A B

xuu x

xy x y

X a b Y

The sign for not may b e immediately preceded by a sign whichwould

normally b e unspaced after eg etc In such cases should b e brailled

unspaced The same also applies to the signs for etc however when two signs

b eginning with o ccur in succession the is omitted b efore the second sign

Exceptionally brackets should b e used for the juxtap osition of and signs as

means etc See x

Ex p q r

sin x

Arrows b etween ordinary words eg implies should remain spaced

and the negation of these signs The signs for j a

are spaced on b oth sides

Ex mn

p q q p

The Spacing of Brail le Signs

Punctuation is generally spaced according to print though it should not usually b e

left spaced b oth b efore and after in braille

Ex f X Y or

f X Y

x x h

G H

f g

Other signs are freely brailled unspaced sub ject to satisfying the constraints on ad

jacent signs given by the ab ove rules x See however x for the treatmentof

adjacentword segments representing sp ecial functions etc

Ex a b c

pq r

P E jF

fxjxg

kxk

log sin x

t sin

Refer also to the various examples throughout this co de

The Oblique Stroke and Fraction Line

THE OBLIQUE STROKE AND FRACTION LINE

is used in braille to represent b oth the oblique stroke and the horizontal The sign

fraction line in print except where it is omitted for simple numerical fractions x

When it represents the horizontal fraction line brackets must b e inserted in braille

in the following cases

a To unite the numerator or the denominator of a fraction consisting of two or more

spaced terms or expressions in braille or which is itself a fraction brailled with the

sign

a b

b c

a b

a c

c ad bc a

b d bd

x y z

a b a b

If the numerator or denominator consists only of terms multiplied together and not

separated by spaces extra brackets are not required

a bb c

xy z

x y y z z x

b To separate a fraction brailled with an oblique strokefromafollowing unspaced mul

sign see tiplying term unless the fraction is an index and thus terminated bya

Ex c

Indices

x Y

x y

BRACKETS

Rules concerning brackets o ccur in various sections of the co de eg x The Oblique

Stroke and Fraction Line x Indices x Co ordinates and Sets x Matrices The follow

ing should also b e observed

All brackets in print should b e represented bybrackets of the same typ e in braille

Mo dulus bars serve as brackets

ja bj

c d

Brackets should b e inserted to enclose a signed numb er or term when preceded bya

or sign

Ex or

Insertion of brackets may b e desirable in the braille to add clarityeven when not

sp ecically required by the rules

INDICES

The sup erscript sign indicates that the expression which follows is printed in the

sup erscript p osition Unsigned or negative simple whole numb er sup erscripts are written

in the lower part of the cell without a numeral sign except for this case sup erscript

expressions must b e terminated bya sign if not by a space or a closing bracket

enclosing the whole term

Arrows suchas or immediately following the sup erscript sign should b e preceded

by the dot separator to avoid misreading of the cell

Indices

x y

x y x y xy

x y

A B

a c

a d

p p

x a

q q

y b

d x

d y

X Y

Indices

indicates that the expression which follows is printed in the The subscript sign

subscript p osition Unsigned or negativesimplewholenumb er subscripts are written in

the lower part of the cell without a numeral sign and when such indices follow letters

closing brackets or the integral sign the subscript sign is also omitted except for these

sign if not by a space or a closing cases subscript expressions must b e terminated bya

bracket enclosing the whole term As in print x may represent xoneone or xeleven

etc

Arrows suchas or immediately following the subscript sign should b e preceded

cell by the dot separator to avoid misreading of the

Ex x

x y

x b

T S

Indices

x y

f xdx

R R

b d

g x y dxdy

a c

Ro ot exp onents are co ded as lower righthand indices

x Ex

Lower numb ers are used with the sign representing the p oint in partial regression

co ecients and partial correlations etc in statistics but are not used within indices in

conjunction with any other sign except the minus sign as in x and x ab ove

Ex x

Expressions with several levels of indexation are co ded by asserting sup erscript or

subscript signs as required rememb ering that a sup erscript or subscript sign do es not

cancel the force of a preceding such sign but indicates a deep er level

The index terminator only cancels level of indexation ie the last uncancelled

sup erscript or subscript sign lower numb ers act so as to cancel immediately preceding

Indices

If more than one level needs to b e cancelled sup erscript or subscript signs without a

signs should b e used at a p oint in an expression the appropriate number of

Ex x

x y

a b

y x

Indices consisting of two or more terms etc whichwould normally b e spaced in braille

see x should b e united by omitting the spaces b etween those terms if not already united

by b eing enclosed in brackets In such cases a dot separator should b e inserted b efore

signs suchas for whichambiguities could otherwise arise As exceptions

the sign for and the signs which are normally spaced eg should retain their

normal spacing so that brackets enclosing the index are necessary in braille Spaces are also

retained in sequences of word segments denoting functions etc see x so brackets are

again required Subscript co ordinates or sets enclosed in brackets are co ded as in x

Ex x

a x

Indices

n a

x a

When omitting spaces it will b e necessary in some cases to insert the sign to

terminate an index within the sup erscript or subscript

Ex x

f E

E E

Brackets should b e used to enclose an index consisting of two or more separate

expressions

Ex x

lim f x y

lim f z

z z I

Indices

As shown in the last example when brackets are used to enclose the index and the

individual expressions are brailled unspaced a separating comma maybeomittedasis

done with co ordinates etc x

Brackets rather than the omission of spaces should b e used to unite more complex

indices and those in which the spacing needs to b e retained as well as to unite indices

which are to b e divided at the end of a braille line

Ex a

n m

max x

i n

Indices on the left are preceded by the appropriate or signs and followed if

necessaryby signs

Ex A

n r

Additional brackets may b e required in braille to make clear the prop er attachment

of such indices in an unspaced sequence

Ex C

n r

a P

Indices

s p P

The signs and are used to indicate indices directly ab ove or b elowa

term resp ectively when the distinction b etween these and standard indices is necessary

index signs if a combination of such and The same rules are employed as for the

signs o ccurs in an expression eachmust b e cancelled separately

When a sup erscript or subscript arrow extends over several signs that extentmay

generally b e indicated in braille by using brackets Lower numb ers are not regarded as

full signs here but as b eing part of the sign to which they are an index so that cases such

and a may b e co ded without brackets Sp ecial functions see x are also regarded as a

as single entities in this connection

Ex abcd

abcd

a brackets unnecessary

In the particular case of vectors or directed line segments determined bytwo p oints

b oth AB and AB may b e compactly co ded as andbothAB and AB

as etc

Ex AB B C AC

AX AY

If a dash star hat tilde dagger etc is in its usual sup erscript p osition in print it is

brailled without an index sign

Ex x

Indices

The index terminator is not required for such signs

Ex fdx

x y

f x

If such a sign is printed in the subscript p osition the subscript sign should b e used

unless this degree of explicitness is not required in which case it may b e co ded without

index terminator should b e inserted as ab ove When the subscript sign is used the

b efore following unspaced expressions not in the same subscript p osition as usual See

also the note in x concerning subscript bars and tildes used as fount signs

Ex f x

x y

fdx

In geometry the print notation AB C or AB C the angle AB C is exceptionally

more conveniently co ded as AB C rather than using the hat sign

Indices

Bars hats etc in braille are to b e understo o d as referring only to the immediately

preceding symbol or bracketed group brackets should thus b e inserted in braille to show

that such a bar hat etc extends over several signs in print As in xlower numb ers

are not regarded as full signs here but as b eing part of the sign to which they are an index

a is co ded a as so that and etc Sp ecial functions see x are

also regarded as single entities in this connection

Ex xy

x y

An uninterrupted string of dashes or stars is co ded by giving only the initial dot

Ex x

f x

For other suchcombinations each dash star etc should haveitsown dot or dots

Ex x

The signs etc are used to indicate resp ectively single double

triple dots etc ab ove a letter or symbol A terminator is not required after such

signs

Ex x

Indices

Aor sign in an index p osition may b e co ded without the sup erscript or subscript

sign should b e used to separate sign by placing the or close up to the term The

such a term from an immediately following expression

Ex e

The sup erscript or subscript sign should however b e used for such expressions if

ambiguities arising from nonindex or signs could o ccur eg to distinguish from the

expression x or if greater explicitness is required

Ex L

When several indices are attached to a term they should b e brailled in the most nat

ural order according to their meaning ie as they would normally b e read In particular

P Q S T

for the integral sign summation sign pro duct sign union sign intersection

sign etc subscripts should b e brailled b efore sup erscripts See the various examples in

x When such considerations do not clearly apply indices should then simply b e stated

in the most convenient order for co ding In particular it is generally neatest to braille

subscript numb ers attached to letters b efore dashes bars etc as the subscript sign can

thereby b e disp ensed with eg braille x rather than as

Special Functions and Other Words and Abbreviations

The order of indices should not otherwise b e regarded as critical though once a conven

tional order for a typ e of expression has b een established in a particular piece of work it

should b e adhered to throughout for consistency

SPECIAL FUNCTIONS AND OTHER

WORDS AND ABBREVIATIONS

These are words parts of words or sets of initials used to denote functions etc and

are generally printed so as to b e distinguishable from algebraic letters ie usually printed

in ordinary typ e rather than italic typ e for algebraic letters Such functions etc are

indicated in braille by placing the sign b efore the letter or letters which denote it

Some of the more common functions eg sin cos log have sp ecial braille abbreviations

as given in the table of signs in other cases the word segment is brailled in full without

literary contractions

A prexed function or expression may b e immediately followed byany mathe

matical sign an immediately following small Latin letter thus requiring its dots letter

sign This latter rule applying also to the signs for and for t

Ex sin

sin x tan A

cosh x

sech x

sina sina cos cosa sin

log x log x

logsin x

grad r

lim x

slim A

ln x

Special Functions and Other Words and Abbreviations

exp x

exp x dx

PrE

z Rez i Im z

n dim E dimE

n w ord s

Sp ec A

Cardn

trexp B

tjtr Aj

gcd

gcda bd

A function with a sp ecial braille abbreviation may b e used to representa variety

of print forms with the given meaning

Thus

represents sin arcsin arc sin

sinh arcsinh arsinh

log antilog etc

A sequence of two or more spaced word segments whichmay denote separate functions

or together a single function not interrupted byany other mathematical sign is generally

co ded with a single initial sign and with the spaces b etween word segments repro duced

A function within the sequence which has a sp ecial braille abbreviation should however

b e prexed by its own sign with the space separating it from the preceding word

segment omitted

Ex log sin

Special Functions and Other Words and Abbreviations

exp sin

t sin ln

r tan arg tan arg

tr exp A

log cos Re z

im f ker coker f

f esssupf

H st graph lim H

tr ln M

det M e

brackets are required according to x

lim sup x

lim inf y

brackets are required according to x a

Re tr a r

det a

Indices are generally co ded in the usual way with the sup erscript or subscript signs

the subscript sign should not b e omitted b efore lower numb ers here to avoidthembeing

misread as punctuation marks The inverse logarithmic trigonometric and hyp erb olic

functions however have sp ecial designated braille signs eg represents sin

etc

Ex sin

sin

Special Functions and Other Words and Abbreviations

log x

lim x

lim f f

sin

cosh x

co dimY X inf co dimZ X

Z Y

Functions b eginning with an initial capital in printmay b e brailled by inserting a

dot b efore the initial letter after the sign when this use of capitals in printis

mathematically signicant

Ex Log z

Sin x

Arg z

The sign may b e used generally to indicate and distinguish ordinary words or ab

breviations within mathematical expressions though a certain degree of discretion should

b e exercised here since in many cases particularly with indices the expression is more

simply co ded and read without the sign byjustleaving the expression uncontracted

as distinct from A Ex A

mercury

out

Delimiting the Argument of a Function

f x constant

not A or C and B and not A

DELIMITING THE ARGUMENT OF A FUNCTION

If the argument of a function integral sign summation sign etc is a fraction brailled

sign the argumentmust b e enclosed in brackets in braille whether or not with a

this is done so in the print Brackets should not b e inserted where the printisambiguous

in such cases

Ex sin

sin a

a b

sin

Gal

x y n

i i

C sin m xa sin nya

t mn

Delimiting the Argument of a Function

If spacing is used in print to indicate that the argument consists of two or more

terms which are spaced in braille that indication of extent should b e shown in braille by

enclosing the whole argument in brackets Brackets should not b e inserted where the print

is ambiguous in such cases

Ex sin cos

f g

E n A

Cases such as log cos Re z are brailled without inserting brackets in accordance with

It is not necessary to insert additional brackets for arguments consisting of unspaced

terms in braille The conventional print use or nonuse of brackets should b e sucientin

such cases

Ex sin t

log cos

cos t

sin a cos b

sina cos b

log x log y

The ro ot sign is generally to b e understo o d as applying only to the immediately

following numb er letter function or bracketed group ie that following the ro ot exp onent

if present Thus if a large ro ot sign is used in printtoshow that it applies to a fraction

whichwould b e brailled with the sign or if a horizontal bar is used in printtoshow

that the argument extends b eyond the rst letter or numb er brackets should b e placed

around the whole argument in braille if they are not already present in the print Indices

mayhowever b e attached to the argument without the use of brackets when the meaning

is clear in particular when brackets or the horizontal bar is not used in print

Ex xy

Coordinates and Sets

sin x

x x or

COORDINATES AND SETS

Commas shown in print to separate elements or arguments of sets co ordinates

functions etc displayed in brackets are generally omitted in braille the spacing b etween

those elements b eing sucient A comma b efore an element preceded by an op eration sign

usually or should however b e retained to avoid ambiguity

f x y z

f g

Coordinates and Sets

a b

ptjH jp t

When the arguments of a function generally printed with commas are separated into

groups by a dierent punctuation mark eg a semicolon this latter punctuation mark

should b e retained in the braille

Ex f x y z s t

Elements consisting of twoormoretermswhichwould normally b e spaced in braille

may b e united in such expressions by omitting the spaces b etween terms if not already

united by b eing enclosed in brackets In omitting spaces it will b e necessary in some cases

to insert a sign to terminate an index

Ex x y

x t y z

a c

x y

b d

f x x x

y z t x

x x

In particularly complex cases it may b e preferable to preserve the spaces b etween

terms by retaining the commas b etween elements This metho d may also b e necessary

when an element contains spaced word segments x since these spaces maynotbe

omitted

st st

Ex A B sin cos x A B sin cos x

Matrices Determinants and Other Arrays

An element with spaces omitted to unite it according to x should not b e split

between terms at the end of a braille line the element should rather either b e brailled

with spaces by enclosing it in brackets so that it can b e split b etween terms or else it

should b e taken down complete onto the next line

When sets or co ordinates are divided b etween elements at the end of a braille line

a dot mathematical hyphen should not b e used as it would imply that the element

continued nor should a comma b e used if commas are generally omitted in the braille

Ex Consider the set E fa b c d e f g g

MATRICES DETERMINANTS AND OTHER ARRAYS

Round or square matrix brackets are brailled as columns of signs of the appropri

ate length Long vertical bars eg for determinants or double vertical bars are brailled

as single or double columns of signs resp ectively Columns of elements are spaced

from each other by clear cell running down b etween the columns Elements within a

column should b e brailled with their lefthand cells aligned unless prexed byaor

sign which is normally brailled to stand out Elements b eginning with a small Latin letter

do not need to b e prexed by a letter sign unless they immediately followa vertical

bar sign

a b

c d

a b c

Matrices Determinants and Other Arrays

x x

Elements consisting of two or more terms whichwould normally b e spaced in braille

are united by omitting the spaces as is done with co ordinates or sets x

a b a

a a b

A site in a matrix marked with a dot in print should b e marked with a dot in braille

Ellipses are brailled as ordinary literary ellipses which should b e aligned in an appropriate

matrix column and a line of dots is brailled as a line of dot s

b b b

s s ss

signs for horizontal Partition lines in matrices maybeshown by a sequence of

lines and by signs aligned vertically for vertical lines

Matrices Determinants and Other Arrays

x x

C B

Matrices or determinants set out on a separate line in print are normally indented to

b egin in cell or cell for runovers to an equation as usual for mathematics A cell

start may b e used for wide matrices if it allows the array to b e brailled without runovers

when this would not b e p ossible with the normal indentation

In a matrix equation or for matrices within ordinary text all the matrices whicht

across the page should b e brailled with their top rows and any other intervening single

line expressions or text on the same braille line Single line expressions or text following a

matrix should b e brailled on the b ottom line of the matrix unless another matrix app ears

on that same line in which case the expression or text is placed on the top line b etween

the matrices

Ex AB C

Ex The matrices and are linearly indep endent

Matrices should not b e brailled directly b eneath one another with one or b oth of the

long brackets aligned Toavoid this one of the matrices can b e moved on or cells or

else a blank line left b etween the matrices

Matrices Determinants and Other Arrays

A matrix which is to o wide for the braille page may b e split b etween columns with

the remaining part of the matrix placed b eneath and indented cells from the start of the

matrix Facing pages may also b e used for wide matrices if convenient

sin sin cos sin sin cos

cos cos cos cos sin cos

sin cos cos cos sin sin Ex

cos cos sin cos sin sin

sin cos sin sin cos

A linear metho d may alternativelybeusedtorepresent matrices and other such

arrays This metho d is favoured for binomial co ecients and for Christoel symbols

whereby the particular typ e of bracket used in print can b e represented in braille but it is

not the primary metho d used for representing matrices in general transcription work In

this metho d the rows are brailled in order from the topmost downwards with an unspaced

sign used to indicate the start of a new row The whole sequence is enclosed in the

appropriate standard mathematical brackets and may b e divided at any convenient p oint

at the end of a braille line a dot mathematical hyphen not b eing used if the division is

between spaced elements

Ex a binomial co ecient

Ex a Christoel symbol

a c a a a

a a c a a

C B

a a a c a

a a a a c

Worked Calculations Logical Figures

WORKED CALCULATIONS LOGICAL FIGURES

Horizontal lines used in worked calculations are brailled as a line of signs of the

appropriate length Digits tens and hundreds etc should generally b e vertically aligned

and op eration signs placed as in print It maybeconvenienttoomitnumeral signs so as

to leave the braille less cluttered but this convention should b e explained b eforehand in

general transcription work

Ex Numeral signs are omitted

etc

Worked calculations with algebraic expressions should generally b e arranged so that

the and signs and terms of the same degree are vertically aligned This maybe

achieved compactly by omitting spaces as well as p ossibly leaving or inserting spaces

where necessary or less compactly by just inserting the necessary spaces

Worked Calculations Logical Figures

Ex x x

x x x x

x x

Horizontal lines in logical gures may also b e represented by a sequence of signs

Ex D D

D K

Miscel laneous Notation

D D

IIc

DN N DNN

IIb

DN DN N

by denition of K

D K

MISCELLANEOUS NOTATION

The signs and are used to representa variety of op erator symb ols as

required

Ex x y

f g function comp osition

Continued fractions are co ded using as a mo died fraction line sign indicating

that all the terms following it are part of the denominator This notation should not b e

used in other contexts

Ex ab

The dierential op erators etc should b e separated from their follow

dx x xy

ing op erands by a dot if they the op erators are not enclosed in brackets

Ex f x

Miscel laneous Notation

i j x y

x y

df f

etc are co ded as ordinary fractions

dx x

A dot or blank in print indicating a missing digit or an unsp ecied argument etc

may b e shown in braille as a single dot

Dashes are also freely used for similar purp oses However it is usually inadvisable

for the dash to immediately followa minus sign if in this case the dash represents a

number a numeral sign can b e inserted b efore it

Ex A B A B

a c

A general alternative in cases like the last two examples is to substitute a symbol

which is unambiguous in the context for example a literary dagger An explanation of the

metho d would then b e needed for the reader

Ex p

Ellipses representing omitted or unsp ecied terms in an expression are spaced in the

same way as the terms that they representwould b e An ellipsis is co ded as an unspaced

sequence of three dot s

Miscel laneous Notation

Ex sin x x x x

z z z z

f x x x

The Landau notation O y and oy used in analysis etc should b e co ded with

letter os ie not zeros

Ex sin x x x O x

sin x x x ox

for small x

In expressions such as mo d the bracketed remark should b e spaced from

the preceding term and literary brackets used

Ex mod

In the notation b elow for cyclic p ermutations the numb ers or other symb ols used

may b e brailled spaced or unspaced according to print Eachnumb er should have its own

numeral sign whether spaced or unspaced

a a a

The notation for a general p ermutation may b e brailled as a matrix

cos

The notation x meaning cos x or sin x etc may b e co ded on one line by using

sin

brackets

Layout of Mathematical Text

cos

Ex X px

sin

rd and ths should generally b e omitted The terminal letters in expressions suchas

in braille after fractions but not after whole numb ers

rd Ex

LAYOUT OF MATHEMATICAL TEXT

The following remarks are additional to those in x

Centred headings should not b egin b efore cell to avoid confusion with set out

mathematics

Equation numb ers should b e placed in literary brackets whether or not brackets are

used in the print starting in cell of the rst line of the equation or equations to which

they refer If any further line of an equation has a reference numb er it should b e placed

in brackets at the b eginning of the line to which it refers starting in cell and followed

bytwo blank spaces b efore the equation continues

If the reference symb ol is not a simple numb er but wholly or partly some other sign it

is treated in the same way with a dot separation sign inserted b efore the closing bracket

if necessary

The following are cases where it may b e necessary or advantageous to dier from the

usual cell with runovers in cell mathematics layout

Wide arrays x

Worked calculations x

To enable simultaneous equations to all start in the same cell if the rst equation

is preceded by an equation numb er or otherwise do es not b egin in cell

Ex x y

x y

Units

To allow cell or cell indented text indicators eg question numb ers or letters

to stand out in parts of text consisting principally of such indicators and set out equations

a The layout used in x ab ovemay b e used where applicable or

b The equations may each b e started in cell or

c Blank lines may b e inserted to separate the numb ered or lettered sections of text

retaining the usual cell and layout of equations and those numb ers or letters

Note Set out equations not b eginning in cell by virtue of x or x ab ove

should not b e allowed runovers If equations are thereby to o long ie any equation of

a set of simultaneous equations a standard cell and format should b e adhered to

instead Equations simply preceded by cell equation numb ers are normally allowed cell

runovers

If the righthand side of an equation has several options sub ject to dierent conditions

often group ed with a large bracket in print the large bracket is omitted in braille and

one of the following alternativelayouts may b e used

a The lefthand side and rst option may b e brailled as a complete equation and the

second and further options placed b eneath with their equals signs in the same cell as

that of the rst option if the options are all short enough to t without runovers or

b The lefthand side of the equation if it is reasonably short may b e rep eated for each

option so that the options are each brailled as complete equations or

c The lefthand side may b e brailled once followed by the rst option with the subse

quent options each b egun on a separate line with their equals signs in cell and with

all runovers in cell

x x

Ex jxj

x x

If conditions or words of comment are asso ciated with and interrupt a mathematical

expression they may b e inserted as in print at the appropriate place but should not b e

preceded or followed bya dothyphen The mathematical expression should continue in

cell of a new line

UNITS

Tables of standard unit abbreviations are given at the end of this section for refer

ence

Note The examples in this section have b een chosen to illustrate a varietyofprint

forms as found this choice is not intended to indicate recommended practice

Units are placed b efore or after the numb er to which they refer according to print

Units should b e spaced in braille apart from the signs in x denoting units of angle and

length in feet and inches monetary unit abbreviations preceding the numb er and single

letter monetary unit abbreviations or symb ols following the numb er

Units

Unit abbreviations are generally co ded using the usual conventions of literary and

mathematics braille notation eg as regards the use of the letter sign and capital sign

Note that the letter sign is not required b efore unit abbreviations consisting of two or more

lower case letters b elonging to one word eg cm for centimetre but it is required where a

lower case letter is followed by an upp er case letter at the b eginning of a unit abbreviation

eg mW for milliwatts

Capitals should normally b e indicated even in noncapitalized braille However

conventional informal abbreviations such as MPH MPH MPG etc can b e treated as

lower case in noncapitalized braille

mmHg should b e co ded with a dot b efore the abbreviation Hg for mercury unless

the sp ecial braille co de for chemistry is b eing used see Brail le Science Notation

ohm co ded as Aangstrom co ded as is co ded as

p ercent co ded as p ounds sterling co ded as dollars co ded as

euro as See however x c cent co ded as and

Ex metres

metres

p ounds sterling

x p ounds sterling

p ounds p ence

million p ounds

Units

euros cents

p ercent

seconds

sec

seconds

mol

mole

kilometres

ftins

feet inches

teslas

milliamp eres

hertz

pascals

GeV

gigaelectronvolts

megawatts

angstroms

Units

microseconds

millilitres

cubic centimetres

mph

miles p er hour

MPH

miles p er hour

years

Unit abbreviations should generally b e co ded in the same way whether or not ac

companying a number

and spaced co ded as spaced should however b e co ded as

These signs should also b e used when the symb ols o ccur in conjunction with letters in a

monetary unit See also x

Ex A

Australian dollars

In simple expressions of angle or length in feet and inches

degrees is co ded as

minutes or feet as

seconds or inches as

radians when used instead

of rad as

and follow the numb er to which they apply with the whole group unspaced When the

degree sign follows a lower numb er it should b e preceded by the sup erscript sign to

avoid ambiguity

Units

degrees

minutes or feet

seconds or inches

radians

degrees minutes

minutes seconds or feet inches

degrees minutes seconds

degree

In expressions of temp erature and in b earings the letters C F N S E W are

brailled unspaced from the numb er to which they apply

Ex F

degrees Fahrenheit

degrees Celsius

degrees south

N W

North west

In combined units see x Fand C are co ded as

resp ectively

Units

These abbreviations are also used in braille when the unit symb ol is not attached to

anumb er

Indices attached to unit abbreviations or words should b e shown as lower numbers

sup erscript sign immediately following the

Ex km

kilometre

second

In combined units a dot should b e inserted b etween the individual units unless an

index or stroke is present at that p oint A group consisting of a multiplying prex attached

to a basic unit symb ol eg kg cm etc is to b e regarded as a single unit and so a dot

separator should not intervene

The stroke should b e brailled as

In print the individual units in combined units are usually separated by spaces or

halfspaces if not by a stroke dots are also o ccasionally used The dot separator should

however always b e used b etween individual units in braille according to xeven when

no separation is shown b etween them in print

Ex Nm

newton metres

newtons p er metre

metre seconds

milliseconds

metre s

metre seconds This example shows the letter sign used

b efore s for second when the dot might otherwise b e read

as an ap ostrophe

gram metre

Units

metres p er second

rad s

radians second

Nsm

newton second metre

m s

metre second

x coulombs

x coulombs p er second

Long combined units may b e split at the end of a braille line using the dot math

ematical hyphen A dot if present at that p oint remains b efore the dot hyphen A

stroke at that p oint should b e taken onto the new line Short unit expressions should not

b e divided

It is preferable for spaced units not to b e separated from their preceding number at

the end of a braille line

The dot mathematical separation sign is not required after a unit abbreviation

b efore following punctuation unless the abbreviation ends with an index or one of the

angle symb ols given in x

Ex m

metres

metre

degrees

Units

TABLES

SI units and prexes are taken from The Asso ciation for Science Education b o oklet

SI Units Signs Symbols and Abbreviations Other nonSI units are taken from

Nueld Advanced ScienceBook of Data

SI UNITS

Name Symbol Name Symbol

metre m coulomb C

kilogram kg volt V

second s ohm

amp ere A siemens S

kelvin K farad F

candela cd web er Wb

mole mol tesla T

radian rad henry H

steradian sr lumen lm

hertz Hz lux lx

newton N b ecquerel Bq

pascal Pa gray Gy

joule J sievert Sv

watt W

MULTIPLYING PREFIXES

These may b e attached to any of the ab ove units Exceptionally kg already has a

prex attached but other multiples eg mg g are formed in the obvious way

Submultiple Prex Symbol Multiple Prex Symbol

deci d deca da

centi c hecto h

milli m kilo k

micro mega M

nano n giga G

pico p tera T

femto f p eta P

atto a exa E

These prexes are also sometimes used b efore units in the following table

Units

OTHER NONSI UNIT SYMBOLS

Name Symbol

angstrom A

astronomical unit au also AU

atmosphere atm

atomic mass unit u

biot Bi

British thermal unit Btu

calorie cal

curie Ci

day d

debye D

decib el dB

degree angular

degree Celsius C

degree Fahrenheit F

dyne dyn

electronvolt eV

fo ot ft also

franklin Fr

gallon gal

gauss G

hectare ares ha

horsep ower hp

hour h

hundredweight cwt

inch in also

kilogramforce kgf

kilop ond kp

knot kn

litre l also L

micron m also

mile nautical n mile braille as n mile

minute angle

minute time min m is also used in time of day eg h m

o ersted Oe

ounce oz

ounce uid oz A space is left b etween and oz in braille

pint pt

p oise P

p ound weight lb

p oundforce lbf

rad or rontgen R

rem Rem

Alphabets Used in Mathematics

second angle

stokes St

tonforce ton f braille as tonf

tonne t

torr Torr

X unit Xu

yard yd

year a

ALPHABETS USED IN MATHEMATICS

The following lists give the Greek and German Gothic alphab ets These are co ded

as the letters or signs indicated preceded by the small or capital Greek letter fount sign

or a small or capital unassigned fount sign chosen from those listed in x for German

letters

GREEK

Small Capital Small Capital

A a alpha N n nu

B b b eta x xi

g gamma o O o omicron

d delta p pi

E e epsilon P r rho

Z z zeta s sigma

H eta T t tau

theta u upsilon

I i iota f phi

K k kappa X chi

l lambda y psi

M m mu w omega

Alphabets Used in Mathematics

GERMAN

Small Capital Small Capital

a A a n N n

b B b o O o

c C c p P p

d D d q Q q

e E e r R r

f F f s S s

g G g t T t

h H h u U u

i I i v V v

j J j w W w

k K k x X x

l L l y Y y

m M m z Z z

INDEX

References refer to sections and examples except where stated as page references

Ex Ex curly d actuarial symb ol

Ex dashes representing a blank see

aleph also indices

alphab ets decimals

angle symb ol recurring

units of derivative Ex

argument of a function partial Ex Ex

omitted determinantssee matrices

arrows after subscript sign dierential op erators

after sup erscript sign Dirac Planck constanth

indicating vector division of mathematical expressions

bar indicating negativeinteger part of

numb er

dot dierentiation

subscript

indicating p osition in matrix

vertical d Ex

indicating unsp ecied argument