

        braille mathematics notation

              issued by
       the mathematics committee of the
     braille authority of the united kingdom

            ------------
            in 1 volume
            ------------













    the royal national institute for the blind
       224 great portland street
        london w1n 6aa




  note: the braille authority of the united
kingdom was formerly the british national uniform
type committee: the previous 1980 edition
of the code was published under the former name.
            ------------























               contents
                                 page
    members of the mathematics committee
    introduction ------------------ iii
table of braille mathematical signs ----- 1
  modification of signs ------------ 10
braille mathematics notation ------------ 12
  1 general remarks -------------- 12
  2 numeral and letter signs --------- 14
  3 the spacing of braille signs ------- 21
  4 the oblique stroke and fraction
      line ---------------------- 25
  5 brackets ------------------ 26
  6 indices -------------------- 27
  7 special functions and other words
      and abbreviations -------------- 37
  8 delimiting the argument of a function
  9 coordinates and sets --------- 43
  10 matrices, determinants and
      other arrays ---------------- 46
  11 worked calculations; logical
      figures ------------------ 51
  12 miscellaneous notation ------- 54
  13 layout of mathematical text -- 57
  14 units -------------------- 60
  15 alphabets used in mathematics
    index ----------------------- 73
            ------------



    members of the mathematics committee
a. will. chatters (chairman), bristol
  university
just. a. allnut, central electricity
  research laboratory
that. knowledge. devonald, royal national college
  for the blind
just. fullwood, marconi systems
that. do. maley, braille house, rnib
miss do. more. nicholl, braille house, rnib
so. just. phippen, braille house, rnib
will. but. like. poole, chairman braille authority
  of the united kingdom
miss just. short, chorleywood college
do. spybey, worcester college
people. just. talbot, principal, queen alexandra
  college
            ------------











             introduction
  this edition of the braille mathematics notation has
been prepared by the mathematics committee of the
braille authority of the united kingdom. it arose from
the need to extend and amend the 1980 edition, and
this also seemed to be a good occasion to rewrite the
code book in a more convenient form. it should be
emphasized that no changes have been made in
such basic notations as those for addition,
subtraction, multiplication, division,
brackets, equality, unions,
intersections, inequalities and so on.
  two of the most important changes made in
this edition are concerned with unit abbreviations and with
letter fount conventions. the abbreviations for such
units as "metre", "second" and "gram" will
now follow the number of units rather than
precede it, so that braille will follow the standard
print convention. the rules about fount indicators
have been changed so that letters will now be assumed
to be small latin unless shown otherwise.
another important change is the extension of the
use of the ed sign to code any sort of
special function, e.g. lim, exp, det,
etc., as well as the trigonometric and
hyperbolic functions. at the same time some

             introduction
of the old word abbreviations such as edr for
"rectangle", have been deleted.
  a few alterations have been made in order
to simplify the rules and increase conceptual
clarity. for example, the numerator of a
fraction will always be shown even when it is 1,
so that one-half will now be written 1; rather than
bb. similarly the exponent 2 will always be
shown, so that x-squared will be written xing; rather
than xing. it will no longer be necessary to decide
whether an upper symbol in print should be
treated as a superscript or as an
exponent; the same superscript sign +
will now serve in both cases. the special
method for dealing with limits of integrals has
been deleted. some new symbols have been added
to the code to deal with certain print symbols which
were not catered for before.
  there was a strong feeling in the committee that the
time had come to rewrite and restructure the
code book. the two part structure of the old
code has been abandoned, and it is hoped that the
way the new edition is organized will make it
much easier to use.
  i would like to thank all the members of the
committee for being so generous with their time and
expertise and for making our discussions lively but

             introduction
good humoured. i am sure that all the other
members will join me in expressing our
appreciation of the work done by the braille house staff
in producing all the documents so efficiently
in both braille and print.
                        a. will. chatters
                             bristol,
                      november 1986.
            ------------





     table of braille mathematical signs
  the braille signs in the following table should
generally be used to represent the corresponding
print signs whatever their meaning. in a few
cases, however, more than one braille sign has the
same print equivalent (e.g. those
representing the print dot): for these the
correct braille sign should be used according to the meaning
or use stated. the braille signs are arranged in
"alphabetic" order.
  [in the table, signs are preceded by the
full cell for to make the position of the dots
clear: it should not be read as being part of the
sign.]

fori: left half arrow
forl pound sterling
foro greater than
foro right brace bracket
foroo much greater than
foroow greater than or less than
forocco right arrow with heads at front and rear
foro) greater than or equal to
foro) greater than sign with swung dash beneath
    (greater than or approximately equal
    to)

     table of braille mathematical signs
for and ampersand (and)
forfor matrix bracket
forfor infinity
forforfor end of proof
forof left square bracket
for the integral
forwith right square bracket
forch subscript sign
forgh left round bracket
forsh root
forwh superscript bar
fored "word" indicator
foredc cos
foredd triangle symbol
foredg grad (gradient)
foredhc cosh
foredhs sinh
foredht tanh
foredhgh cosech
foredhou coth
foredh- sech
foredl log
foredq square symbol, vector or wave
    operator
foreds sin
foredt tan
foredgh cosec

     table of braille mathematical signs
foredsh curl or rot
foredth div (divergence)
foredou cot
fored: circle symbol
fored8c edhcing-,, arccosh
fored8s edhsing-,, arcsinh
fored8t edhting-,, arctanh
fored8gh edhghing-,, arccosech
fored8ou edhouing-,, arccoth
fored?- edh-ing-,, arcsech
fored- sec
foredcents edcing-,, arccos
fored@l edling-,, antilog
fored@s edsing-,, arcsin
fored@t edting-,, arctan
fored@gh edghing-,, arccosec
fored@ou edouing-,, arccot
fored@- ed-ing-,, arcsec
fored^l colog
forer index termination sign
forou matrix line indicator
forouo contains
forouo) contains or equal to
forouu reverse ouing
forouu) reverse ouing)
foroufor star, small circle, etc. (operation
    sign)

     table of braille mathematical signs
forouwith reverse ouin
forouch backward oblique stroke (set
    difference)
forouow contained in
forouow: double shaft left arrow
forouowcco double shaft leftstright arrow
forouow) contained in or equal to
forou, inverted A: for all
foroucco double shaft right arrow
forouen reverse E: there exists
forou! union
forou) star, small circle, etc. (second
    operation sign)
forou? intersection
forouin is an element of
forou" crossed zero: empty set
forouing is normal subgroup of
forouing) is normal subgroup of or equal
    to
forou- not
forow less than
forow left brace bracket
forowow much less than
forowow: double headed left arrow
forow: left arrow
forowcco leftstright arrow
forowccow left arrow with heads at front and rear

     table of braille mathematical signs
forow:: long shaft left arrow
forow) less than or equal to
forow) less than sign with swung dash beneath
    (less than or approximately equal
    to)
for, decimal point
forcce right half arrow
forcci upward arrow
forccl reverse :
forcco right arrow
forccoo double headed right arrow
forccp percent sign
forcccco long shaft right arrow
forccen downward arrow
for:) varies as, proportional to
forccin right half arrow
for$ dollar sign
forencce right half or full arrow with left
    half or full arrow beneath
forenccee long right half or full arrow with
    short left half or full arrow beneath
forenencce short right half or full arrow with
    long left half or full arrow beneath
for-- equivalent to
for" degree sign: small superscript
    circle
forst oblique stroke, fraction line sign

     table of braille mathematical signs
forsten continued fraction sign
foring superscript sign
forble numeral sign
forble' perpendicular symbol
forar right round bracket
for' comma or space marking thousands, etc.
for' unit separation sign
for' differentiation dot
for- double differentiation dots
for- radian sign: superscript c
for-' triple differentiation dots
for@ small bold latin letter sign
for@ index separation sign
for@d curly d, partial derivative d
for@the closed line integral
for@ch bottom index sign
for@th actuarial symbol (years): top and
    right-hand sides of square enclosing
    preceding term
for@wh superscript hat
for@. superscript dagger
for`en superscript star
forinch superscript dash (prime)
for@st since
for@ing top index sign
for@ar inverted superscript hat
for@. small letter sign

     table of braille mathematical signs
for@ small letter sign
for^ capital bold latin letter sign
for^ bar over (following) negative number
for^a @angstr@om symbol (capital a with
    small circle above)
for^wh tilde (superscript swung dash)
for^cco right arrow with vertical bar on shaft
for^! v shaped symbol (or, join)
for^) inverted v shaped symbol with
    horizontal bar above (is projective
    with)
for^? inverted v shaped symbol (and, meet,
    vector product)
for^. capital letter sign
for^ capital letter sign
for vertical bar (modulus, determinant,
    such that, divides, restricted to,
    evaluated at, etc.)
for seconds (angle) or inches unit
    symbol (superscript double dash)
for capital greek letter sign
forv inverted capital people (coproduct)
for{ angle symbol
for, is a theorem: t shaped symbol on
    its side
for: is valid: t shaped symbol on its
    side with two horizontal lines

     table of braille mathematical signs
for) approximately equal to: double
    swung dash, horizontal line with
    "humped" line above, equals sign with
    dots above and below
for? hash
for" nabla, del: inverted d
for_ double vertical bars (e.g. norm,
    matrix brackets, parallel symbol)
for. recurring decimal sign
for. mathematical hyphen
for. differential operator separation sign
for. crossed symbol sign
forlord reverse ,
for.- swung dash (dash with alternate
    curves)
for.-were equals symbol with swung dash above
for. minutes (angle) or feet unit
    symbol (superscript dash)
for. small greek letter sign
for.of left angle bracket
for.with right angle bracket
for.ow: superscript or subscript left
    arrow indicating vector (e.g.
    AB.owcc[
for.cco superscript or subscript right arrow
    indicating vector (e.g. AB.cco[
for.) horizontal line with swung dash above

     table of braille mathematical signs
for.' point notation of partial correlation or
    regression, in statistics
for small latin letter sign
for. divided by
for! plus
for!- plus or minus
for) equals
for? multiplied by, cross symbol
for' dot (multiplication dot, and (in
    logic--
for- minus
for-to minus or plus
for@ small letter sign
for. small letter sign
for) is defined as: equals sign with
    triangle above or subscript "def",
    etc.
for capital latin letter sign
for separation sign
forChild therefore
forTo exclamation mark: factorial
for^ capital letter sign
for double capital greek letter sign
for. capital letter sign
            ............

         modification of signs
         modification of signs
  (a) when it is required to show that a non-
alphabetic symbol is printed in bold or
large type as distinct from the usual print
sign, a dot 4 is inserted after the initial
cell of compound signs which begin with the out sign,
dots 5-6 or dots 4-5.
    ex1. ou@! ou@? @? ^@?
  this indication is not made when it is just a
style of printing and of no mathematical
significance. it is not usually shown for bold
" or edq signs in braille.
  (b) a symbol consisting of a struck out letter
or sign is coded by inserting a dot 5: this
follows the out sign for signs prefixed
by ou, replaces the dots 5-6 for signs
prefixed by dots 5-6, and precedes all
other signs (i.e. precedes a letter fount
sign if present). if a dot 5 is already
present at this point, it must be retained.
    ex2. .Rather .ow) .cco
    @ .(0
    A.xBut
    .a --ach.mer.ging.m (feynmann
      "slash")
  analogously, "greater than but not equal
to" is coded as o.) and "contains but not

         modification of signs
equal to" as ouo.gg.
  (c) symbols printed in a circle or a
square may be coded by placing the sign in round
or square brackets respectively, with the
compound sign obeying the usual rules for the
enclosed sign.
    ex3. it gh8ary
    a ghffarb
            ------------



        braille mathematics notation
          1. general remarks
  11. set out mathematical expressions
are generally brailled beginning in cell 5 with
runovers in cell 7, whatever the setting in
print. (see s'13 for further details
concerning layout of mathematical text.)
  12. dot 6 precedes any punctuation
sign except a hyphen or dash, used within
or immediately following a mathematical
expression. (see for example s'34,
62 and 911, but note the examples in
s'7, and s'1410 for unit abbreviations.)
  13. dot 5 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
point in the print. the following rules for
dividing mathematical expressions should be
observed:
  (a) do not separate letters from their letter signs
or numbers 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

14 general remarks
long).
  (c) do not divide immediately after a sign which
is normally spaced before but not after (e.g. ff,
gg, --, owgg, ouow, .-, cono, etc.
(see s'32--; or after the signs '
(multiplication dot), still (oblique
stroke), or an opening bracket.
  (d) do not divide immediately before a er
sign, or before a closing bracket.
  (e) do not divide an expression at a
point where a space is specially omitted
to unite it (e.g. in indices (s'65) or
coordinates (s'912--.
  it is generally not good practice to divide a
short expression (e.g. x (1[ which could be
conveniently brailled complete on a new line.
  14. if the print text contains symbols
which are not listed or otherwise covered in this
code, special braille signs may be devised
to represent them, or else the meaning of existing
signs may be changed for the purpose. any such
non-standard notation should be stated beforehand (e.g.
at the beginning of the work in which it appears, or at
the point of use). (see in particular
s'223.)
            ------------

215 numeral and letter signs
       2. numeral and letter signs
  21. the numeral sign # indicates that
immediately following letters a-just represent
numerals one-nine and zero until the sequence
is ended by a space, a letter k-z, or any
other sign except those given in s'211-
216 below.
  211. the decimal point is coded as
dot 2.
    ex1. 1,2 12,34 ,56
      3,14gh159ar
  212. the print comma or space
separating groups of digits such as thousands or
thousandths is coded as dot 3.
    ex2. 1'jjj'jjj 1,000'jej
  213. recurring decimals are coded
by inserting a dot 5 before the recurring sequence,
that sequence being indicated in print by dots
placed above its first and last digits (or its
single digit).
    ex3. 3,.adbheg 2,009father
  214. a bar indicating a negative
integer part of a number (a notation used for
logarithms) is coded by inserting the dots 4-5
sign between the numeral sign and the number.
    exbled. ^beaaceg
  215. a number should only be divided

218 numeral and letter signs
(using a mathematical hyphen, dot 5) if
it is too long for a whole line. in such cases
the force of the numeral sign carries over the
dot 5 and is not reasserted at the beginning of the
new line.
    ex5. 10'jjj'jjj'jjj'jjj'jjj'.
      jjj
  216. the force of the numeral sign
carries over the literary hyphen - if the
two numbers are on the same braille line. if
this is not the case the numeral sign must be
repeated.
    ex6. may 1-7 1,2-2,3

  217. numeral signs may be omitted
altogether from tables or worked calculations to save
space or to allow an uncluttered
presentation: in such cases advance notice should
be given.
  218. simple numerical fractions are
coded by stating the denominator as a lower number
following the numerator in the upper position.
    ex7. 1; 3.
  if either the numerator or denominator has
a comma or space in the print indicating
thousands, etc., the comma may be omitted in the
braille. alternatively, if the comma is to be

22 numeral and letter signs
retained, an oblique stroke still and a
second numeral sign should be inserted, with the
denominator written in the upper position. (this
latter method retaining the oblique stroke
may also be used for fractions without separating
dot 3's when it is considered desirable
to emphasize the fraction line (e.g. in
calculations).)
    ex8. 23/10'jjj
  219. in mixed numbers the numeral
sign is repeated for the fractional part.
    ex9. 23.
  22. a letter fount sign gives the fount of the
immediately following letter only. [letter fount signs
are preceded by the full cell for in the
following table:]
                       small capital
latin ..................' for for
greek .................. for. for
bold (or underlined) latin
  (see s'224) ......' for@ for^
bold greek ............' for@. for^
bold numeral (two kinds rather
  than small and capital) for@ble for^ble
other founts (see s'223) for. for.
                         for@. for^.
                         for@ for^
                         for@ for^

221 numeral and letter signs
  221. the small latin letter sign dots
5-6 is generally omitted, all letters being
assumed to be small latin unless shown
otherwise. it must, however, be stated when the letter
    (a) stands alone or begins a
mathematical expression in ordinary text;
    (b) is an a-just immediately following a
number (but not a lower number, e.g. the
denominator of a simple fraction), even at
the beginning of a line;
    (c) immediately follows a sequence of letters
brailled with a double letter fount sign (s'222);
    (d) immediately follows a vertical bar
(dots 4-5-6);
    (e) is an o within brace brackets;
    (f) is an a immediately after an opening
square bracket in ordinary text;
    (g) is the first letter of a small roman
numeral, or a capital roman numeral
coded as such (s'23);
    (h) immediately follows a ed prefixed
function or expression (see s'71).
  (see s'143 for unit abbreviations.)
    ex10. it Fghxar )y 5c
      3.c people 2.pr ^a^b
    owxx o no
    owa every i o uo

222 numeral and letter signs
    ofpHr.with
    the interval ofa bwith

    ex11. a ladder ABut of length 2l,
      width w and mass 3f makes an
      angle this with a wall BCan and an
      angle 2.th with the ground at A.
  222. if three or more letters of the same
fount other than small latin (or roman
numbers treated as such (s'23-- occur 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. to avoid misreading (as bar and
ampersand respectively), the greek letters which
(eta) and and (chi) (small or capital)
should always be preceded by their single letter fount
signs, and therefore terminate a double greek letter
fount sign sequence.
    ex12. ABC abgde ABCD.
      EFGHIJ BEABBBBCC`ENB.
      ABCde abgst..prs
      abg.wh abgwh (abg bar)
      abg.wh.oundsr
  exceptionally (dots 64-5-6)
is used as the double capital greek letter

225 numeral and letter signs
fount sign to distinguish it from double bars.
    ex13. gdps _s (mod spirit[
      _spiritfghxar_ (norm spiritfghxar[
  223. undesignated letter fount signs
are used for other founts according to the particular
requirements of the text. when they are used their
meaning should be stated. if the fount is
particularly predominant and frequent, it
may in some cases be advantageous to adopt
a single cell letter fount sign for the purpose,
to avoid over-use of a two cell fount sign.
such non-standard use should be explained.
  224. letters printed or written with an
underline or under-tilde used as a fount sign
may generally be coded with the dot 4 or dots
4-5 letter fount signs without further comment,
unless bold letters also occur in the text, or
if the underline or tilde needs special or
individual indication.
  dot 4 is also used to show the bold fount of
other signs (see paragraph (a) p10),
and is the standard fount sign used to indicate
curly d in partial derivatives, etc.
without further comment.
    ex14. @dfst@dx @d@ast@dx
  225. a special script l is
sometimes used in print to distinguish the letter from

23 numeral and letter signs
number 1. this is unnecessary in braille: code the
letter as an ordinary letter l.
  226. capital greek sigma and pi
indicating summation and product are conventionally
normally coded as spirit and p respectively,
whether or not written as bold letters in print.
  227. aleph and the symbol denoting the
weierstrassian elliptic function (an
unusual letter p[ are coded as letters a and
p respectively with suitable letter fount
signs. this notation should be explained when first
used.
  228. the dirac (or planck) constant
symbol, a crossed h, used in physics
should be coded as hwh unless this notation would be
ambiguous in the particular context. otherwise
it may be sufficient to code it simply as h
if that letter is not otherwise used, or else a
particular letter fount sign should be adopted for the
purpose: in both these latter cases the
notation should be explained when first used.
  23. roman numerals, whether small or
capital, are normally written preceded by the
dots 5-6 letter sign as in literary braille.
if, however, they are indistinguishable in the print
from ordinary algebraic letters, they may be
coded as such according to the rules in the rest of this

31 the spacing of braille signs
section.
  when it is necessary to show that roman numerals are
capital they should be coded as ordinary
capital algebraic letters.
    ex15. ii iii aiii
      (small latin a, roman iii[
      Xingiii fingiverghxar
  24. special letter fount rules are
adopted for chemical formulae: the force of the
dot 6 capital sign carries over all
letters in an unspaced chemical formula until
another letter fount sign or numeral sign
intervenes; and dot 5 is used to indicate a
two letter element symbol, letters following that
symbol being assumed to be capital unless
shown otherwise.
    ex16. nameaoh H.cl Hbbo
  for further details see braille chemistry
notation (rnib 1985).
            ------------
           3. the spacing of
             braille signs
  31. 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
coordinates, etc: these cases are dealt with

321 the spacing of braille signs
in the particular sections (see for example
s'6, 9, 10). an initial space is
also omitted when the sign immediately follows an
opening bracket, and similarly a following
space is omitted when the sign immediately
precedes a closing bracket.
  32. operation and relation signs, etc.
prefixed by dots 5-6, dots 4-5, or
the out sign (except ' and ou"[; all
arrows; the signs o) ow) -- ) were
.-were ow) o) :) .- _ (parallel
to) 'Be and the negation of these signs, are
spaced before but not after.
    ex1. it ffy )z
    not .(0
    A .-But
    it ouinIt
    A ouowBut
    ouenx oueau out-ghu ouinxar
    ghoueaxarghouenyar.fghx yar
    It ouwitha ^ccob ouinally
  321. the sign out- for not may be
immediately preceded by a sign which would normally be
unspaced after (e.g. cono ' ^! and,
etc.). in such cases out- should be brailled
unspaced. the same also applies to the signs
.- ! - etc.; however, when two signs

34 the spacing of braille signs
beginning with dots 5-6 occur in succession, the
dots 5-6 is omitted before the second
sign. (exceptionally, brackets should be used
for the juxtaposition of plus and minus signs,
as !- means "plus or minus", etc.
see s'53.)
    ex2. people ^8ou-quite conoou-rather
    people ^ff.-people
    1 ffgh-2ar
    a ggff1
    it cono-2
    eds-it
  322. arrows between ordinary words (e.g.
oucco "implies") should remain spaced.
  33. the signs o oo ow owow oow
Child @st , : lord conl forfor, and the
negation of these signs are spaced on both
sides.
    ex3. more o not
    people conoq Child out-q conoou-people
  34. punctuation is generally spaced according
to print, though it should not usually be left spaced
both before and after in braille.
    exbled. fConIt conoally (the colon may be
      unspaced or spaced on both sides
      in print)
    6Con3 (2

35 the spacing of braille signs
    it con)x ffh
    ofGConHwith
    con.fghfar.fghgar:
    H": more o more"
  35. other signs are freely brailled
unspaced, subject to satisfying the constraints
on adjacent signs given by the above rules
(s'32-34). see, however, s'72 for the
treatment of adjacent word segments representing
special functions, etc.
    ex5. a'b'c
    x/y
    shx
    2shx
    pandq ^ffr
    3To n
    PghEFar
    owxx o 5o
    f"ingbb.y
    _x_
    2teds.th
    edledsx
  (refer also to the various examples throughout this
code.)
            ------------



41 oblique stroke and fraction line
        4. the oblique stroke
            and fraction line
  41. the sign still (dots 3-4) is
used in braille to represent both the oblique
stroke and the horizontal fraction line in
print (except where it is omitted for simple
numerical fractions (s'218,
219--. when it represents the
horizontal fraction line, brackets must be
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 still sign.
    ex1. gha ffbarstc
    astghb -car
    gha ffbarstgha -car
    astb ffcstd ggghad ffbcarstbd
    ghx 82y 83.zarst5
    ghx/yarstgha ffbar
  (if the numerator or denominator consists
only of terms multiplied together and not
separated by spaces, extra brackets are not
required.)
    ex2. gha ffbarghb ffcarstd
    xyzstghx ffyarghy ffzarghz ffxar
    nmst1'2'3

54 brackets
    (b) to separate a fraction brailled with an
oblique stroke from a following unspaced
multiplying term, unless the fraction is an
index and thus terminated by a er sign (see
s'61).
    ex3. ghastbarc
    x.pstbled
    3.x
    ghghx ffallyar/xzarx
    xingastbery
            ------------
            5. brackets
  rules concerning brackets occur in various
sections of the code (e.g. s'bled the oblique
stroke and fraction line; s'6 indices;
s'9 coordinates and sets; s'10
matrices). the following should also be observed.
  51. all brackets in the print should be
represented by brackets of the same type in
braille.
  52. modulus bars serve as brackets.
    ex1. a ffbstghc ffdar
  53. brackets should be inserted to enclose
a signed number or term when preceded by a
plus or minus sign.
    ex2. 1 ffgh-2ar
  54. insertion of brackets may be

61 indices
desirable in the braille to add clarity, even when not
specifically required by the rules.
            ------------
             6. indices
  61. the superscript sign +
indicates that the expression which follows is
printed in the superscript position. unsigned
or negative simple whole number
superscripts are written in the lower part of the
cell without a numeral sign; except for this
case, superscript expressions must be
terminated by a er sign if not by a space
or a closing bracket enclosing the whole
term. arrows such as cono coni or conen
immediately following the superscript sign should be
preceded by the dot 4 separator to avoid
misreading of the dots 2-5 cell.
    ex1. 2ing; 2ing-con 2ingff3
      3ing1,5 xingbby
      ghx ffyaring; ggxing; ffying; ff2xy
      xingnery xingnmery ^aingThat
      Aingx/yerBut ghaingbarc
      aingghbcarerd (the brackets are present
      in print) xingbbstying:
      xingperstyingq ffaingperstbingq
      dingbbx/dying; dingnerystdtingn
      @dingbbfst@dx@dy aing@cce Eingble'

62 indices
      ^aing_ It conoingferally
  62. the subscript sign child (dots
1-6) indicates that the expression which follows
is printed in the subscript position.
unsigned or negative simple whole
number 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 cases
subscript expressions must be terminated
by a er sign if not by a space or a
closing bracket enclosing the whole term. as in
print, x,, may represent x-oneOne or
x-eleven, etc. arrows such as cono coni or
conen immediately following the subscript sign should
be preceded by the dot 4 separator to avoid
misreading of the dots 2-5 cell.
    ex2. it; it;: xbby: y-,
      xchff1 zch0,5 245ch?
      xchnery Xchn1 xch2n
      ghxchiarb gh^a^bar,, xchnering;
      xchneringm
      Tch.rlessering.msionerSchsioneringless
      Tch.rBelessering.m Schsion,less
      achn@th
      it ffchmery

64 indices
      spiriteaingationerning;
      the"ingeafghxardx
      theeaingforerfghxardx
      thechEerfd.m
      thechaeringberthechceringdergghx yardxdy
      spiritchierxchi
      ofxingbbwitheaingn
  621. root exponents are coded as
lower (right-hand) indices.
    ex3. shx shccx shchnerx
  63. lower numbers are used with the sign
.' (dots 4-F3) representing the point in
partial regression coefficients and partial
correlations etc. in statistics, but are not
used within indices in conjunction with any other
sign except the minus sign as in s'61 and
s'62 above.
    exbled. iteabb.':
  64. expressions with several levels of
indexation are coded by asserting the superscript
or subscript signs as required,
remembering that a superscript or subscript
sign does not cancel the force of a preceding such
sign, but indicates a deeper level.
  the er index terminator only cancels 1
level of indexation, i.e. the last uncancelled
superscript or subscript sign (lower

65 indices
numbers act so as to cancel immediately preceding
superscript or subscript signs without
a er sign). if more than one level needs
to be cancelled at a point in an expression,
the appropriate number of er signs should be
used.
    ex5. xcha, xchabbery
      xchaing; (i.e. the whole term aing; is
      subscript to x[ xchaingb
      xingachb (achb is superscript to
      x[ xchachberery xingachnerbingnerery
      (achnerbingn is superscript to x[
      xchnering;
  65. indices consisting of two or more terms
etc. which would normally be spaced in braille (see
s'3), should be united by omitting the spaces
between those terms (if not already united by being
enclosed in brackets). in such cases, a
dot 4 separator should be inserted before signs
such as cono coni conen ouo ouo) for which
ambiguities could otherwise arise. as
exceptions, the sign o) and the signs which are
normally spaced (e.g. ow o[ 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 s'72), so

652 indices
brackets are again required. (subscript
coordinates or sets enclosed in brackets
are coded as in s'91.)
    ex6. xing1ff2
    xingaffb
    xinggha ffbar
    xchaffb
    xingastbffc (i.e. astb ffc is the
      index)
    xingastghb ffcar (astghb ffcar is the
      index)
    achn-1erxingnff1
    xcha,b
    xchm,nff1
    spiritchnggaeringberxchildren
    spiritchiouinZerxingi
    pchghi o 0arerxchi
    spiritchgh0 ow i owggationarerAchi
    spiritchi,jerxchij
    dfstdxchx)a
  651. when omitting spaces it will be necessary
in some cases to insert the er sign to terminate
an index within the superscript or
subscript.
    ex7. xingachnerff1
    spiritchEchierouowggEerfghEchiar
  652. brackets should be used to enclose

661 indices
an index consisting of two or more separate
expressions.
    ex8. spiritchghi o(0 just o(1arerxchij
    edlimchghx@ccofor y@ccoforarerfghx yar
    edlimchghz@cco1 zouinIarerfghzar
  (as shown in the last example, when
brackets are used to enclose the index and the
individual expressions are brailled unspaced,
a separating comma may be omitted as is done
with coordinates, etc. (s'9).)
  653. brackets (rather than the omission of
spaces) should be used to unite more complex
indices and those in which spacing needs to be
retained, as well as to unite indices which are
to be divided at the end of a braille line.
    ex9. achghnff1 mff1ar xchghn ffm.
      ff1ar
    spiritchghi, j (1areringnerxchij
    edmaxchghi (1, 2, ..., narerxchri
  66. indices on the left are preceded
by the appropriate + or child signs and
followed, if necessary, by er signs.
    ex10. ingnerA +;cc8chinbbUs
      ingnerchmeration childrenerCchr chccP;
      chterVchn@th
  661. additional brackets may be
required in braille to make clear the proper

68 indices
attachment of such indices in an unspaced
sequence.
    ex11. 5ghchnerCchrar aghchccPbbar
      6s6pghingccPeaar
  67) the signs @ing (dots 43-4-6)
and @ch (dots 41-6) are used to indicate
indices directly above or below a term
respectively, when the distinction between these and
standard indices is necessary. the same rules are
employed as for the + and child index signs;
if a combination of such signs occurs in an
expression, each must be cancelled separately.
    ex12. E@chierchr f@ing"chiering;
  68. when a superscript or subscript
arrow extends over several signs, that extent
may generally be indicated in braille by using
brackets. (lower numbers are not regarded as
full signs here, but as being part of the sign to which
they are an index, so that cases such as
accing@cco and aingbbing@cco may be coded without
brackets. special functions (see s'7)
are also regarded as single entities in this
connection.)
    ex13. ghabcdaring@cco abcding@cco
      accing@cco achnering@cco (brackets
      unnecessary)
  in the particular case of vectors or

69 indices
directed line segments (determined by two
points), both ghABaring@cco and ghABarch`cco
may be compactly coded as AB.cco and both
ghABaringow: and ghABarchow: as AB.ow: etc.
    ex14. ABbb.cco ffBbbC.cco.
      ggAC.cco
    AXea.cco'Aallyea.cco (0
  69. if a dash, star, hat, tilde,
dagger, etc. is in its usual superscript
position in print, it is brailled without an index
sign.
    ex15. xinch xeainch mingninch
      xingnerinch ychner@ining; z`en `enRather
      zwh zchnerwh zingbbwh zwhing;
  the er index terminator is not required for
such signs.
    ex16. thewhfdx x@iny f`enghxar
  if such a sign is printed in the
subscript position, the subscript sign
should be used unless this degree of explicitness
is not required (in which case it may be coded
without as above). when the subscript sign is
used, the er index terminator should be inserted
before following unspaced expressions not in the
same subscript position, as usual. (see
also the note in s'224 concerning subscript
bars and tildes used as fount signs.)

693 indices
    ex17. fch`energhxar xchwh xchwhery
      xchwherchildren thechwherfdx
  691. in geometry, the print notation
ABC with a large hat sign above, or
AB@whCan (the angle ABC[ is
exceptionally more conveniently coded as
{ABC, rather than using the hat sign.
  692. bars, hats, etc. in braille are
to be understood as referring only to the immediately
preceding symbol or bracketed group;
brackets should thus be inserted in braille to show that
such a bar, hat, etc. extends over several
signs in print. (as in s'68, lower
numbers are not regarded as full signs here but
as being part of the sign to which they are an index, so
that a: bar is coded accwh and aing; bar as
aingbbwh, etc. special functions (see
s'7) are also regarded as single entities in
this connection.)
    ex18. xywh ghABarwh ghx ffyarwh
      xchnerwh ach3wh xchnwh 3.wh
  693. an uninterrupted string of dashes
or stars is coded by giving only the initial
dot 4.
    ex19. x@* f@*inghxar y`enen
      x@wh@*
  for other such combinations each dash, star, etc.

611 indices
should have its own dot 4 or dots 4-5.
    ex20. x@in`en x@wh@wh x@whinch
  610. the signs ' (dot 3), -
(dots 3-6), -' (dots 3-F3),
etc. are used to indicate respectively
single, double, triple dots, etc. above a
letter or symbol. a er terminator is not
required after such signs.
    ex21. it' this-@r aing@cce'
  611. a plus or minus sign in an
index position may be coded without the
superscript or subscript sign by placing
the plus or minus close up to the term.
the er sign should be used to separate such a
term from an immediately following expression.
    ex22. e- p! pffer.p-
      ffferghxchrar lfferghkar xea!
      Mffering; Rffer`en
  the superscript or subscript sign should,
however, be used for such expressions if
ambiguities arising from non-index plus or
minus signs could occur (e.g. to distinguish from the
expression x cono0! where the plus is not an
index), or if greater explicitness is
required.
    ex23. Lchffering@cci fchff,perghxar
      Rchffinch Rchffchm

71 special functions
  612. when several indices are attached
to a term they should be brailled in the most natural
order according to their meaning, i.e. as they would
normally be read. in particular, for the the
integral sign, spirit summation sign, p
product sign, ou@! union sign, ou@?
intersection sign, etc., subscripts should be
brailled before superscripts. (see the various
examples in s'62.) when such considerations do
not clearly apply, indices should then simply
be stated in the most convenient order for coding. in
particular, it is generally neatest to braille
subscript numbers attached to letters before
dashes, bars, etc., as the subscript sign
can thereby be dispensed with, e.g. braille x-one-dash
as xeainch rather than x@inchea. the order of
indices should not otherwise be regarded as
critical, though once a conventional order for a
type of expression has been established in a
particular piece of work, it should be adhered to
throughout for consistency.
            ------------
        7. special functions
       and other words and abbreviations
  71. these are words, parts of words, or
sets of initials used to denote functions,
etc., and are generally printed so as to be

71 special functions
distinguishable from algebraic letters (i.e. usually
printed in ordinary type rather than italic
type for algebraic letters). such functions,
etc. are indicated in braille by placing
the ed sign before the letter or letters which denote it.
some of the more common functions (e.g. sin, cos,
log) have special braille abbreviations as given
in the table of signs; in other cases the word
segment is brailled in full without literary
contractions. a ed prefixed function or
expression may be immediately followed by any
mathematical sign; an immediately following
small latin letter thus requiring its dots
5-6 letter sign. (this latter rule applying
also to the signs edd for the triangle symbol
and edq for the square symbol.)
    ex1. eds2 edsx ffedtA edhcx
      edh-x edsgha ff.thar ggedsaedc.th.
      ffedcaeds.th 2edlx ggedlxing;
      edlghedsxar edg.f gg".f
      edlimxchildren eds-limAchi edlnx
      edexpx theedexp-xingbbdx
      edprghEar as ggedrez ffiedimz
      neddimEchi ggeddimEvery
      not ggwedordness edspecA
      edcardghnar edtrghedexpBar
      tedtrA edgcdgh6 9ar (3

72 special functions
      edg.c.d.gha bar )d
  711. a ed function with a special
braille abbreviation may be used to represent a
variety of print forms with the given meaning. thus
    ed@s represents edsing-,, arcsin,
                    arc sin;
    ed8s ., edhsing-,, arcsinh,
                    arsinh;
    ed@l ., edling-,, antilog,
      etc.
  72. a sequence of two or more spaced word
segments (which may denote separate functions,
or together a single function), not interrupted
by any other mathematical sign, is generally
coded with a single initial ed sign andwiththe
spaces between word segments reproduced. a
function within the sequence which has a special
braille abbreviation should, however, be prefixed by its
own ed sign, with the space separating it from the
preceding word segment omitted.
    ex2. edleds.th
    2edexpeds.th
    2teds ln.thchi
    rather ffedt arg.z, ffedt arg.z;
    edtr expA
    edledc rezing;
    edimf ggedker cokerf

74 special functions
    from ggedess. supghfchiar
    Have ggedst. graph limHchildren
    eddetMore ggeingghedtr lnMar (brackets
      are required according to s'65)
    ghedlim supxchnarstghedlim infitychnar
      (brackets are required according to
      s'bled1 (a--
    ghedre traing.msion ffrarsteddetaing.msion
  73. indices are generally coded in the
usual way with the superscript or subscript
signs (the subscript sign should not be
omitted before lower numbers here, to avoid them
being misread as punctuation marks). the inverse
logarithmic, trigonometric and hyperbolic
functions, however, have special designated
braille signs (e.g. ed@s represents
edsing-,, etc.).
    ex3. edsingbb.th edsingner.th edlchea"x
      edlcheerx edlimchn@ccoforerxchildren
      edlimchwherchn@ccoforerxchildren
      edlddi.m.chn@ccoforerfchildren 7f
      ed@s1; ed8cx
    edcodimghally Xar ggedinfchZouowggallyer.
      edcodimghAs Xar
  74. functions beginning with an initial
capital in print may be brailled by inserting a
dot 6 before the initial letter after the ed sign,

81 delimiting the argument
when this use of capitals in print is
mathematically significant.
    exbled. edLz ed@Sx edArgz
  75. the ed sign may be used generally
to indicate and distinguish ordinary words or
abbreviations within mathematical expressions,
though a certain degree of discretion should be
exercised here, since in many cases
(particularly with indices) the expression is more
simply coded and read without the ed sign by just
leaving the expression uncontracted.
    ex5. Achedtr Acht (t is an
      algebraic letter as distinct from the
      previous example) muchmercury
      Vchin Vchout
    fghxar ggconstant
    ofnot ghA or Carwith and ofBut and
      ghnot Aarwith
            ------------
        8. delimiting the argument
             of a function
  81. if the argument of a function,
integral sign, summation sign, etc. is a
fraction brailled with a still sign (dots
3-4), the argument must be enclosed in
brackets in braille whether or not this is done so
in the print. brackets should not be inserted where the

812 delimiting the argument
print is ambiguous in such cases.
    ex1. edsghastbar edsastb
      edsghgha ffbarstcar edgalghE/far
      theghdx/xar
    spiritchghi o(1arergh1/2ingiar
    spiritghxchieringbbstnar
    spiritxchieringbbstn
    spiritchierxchierychierstn
    gh.farcht(0 ggspiritCchmneredsghm.px/aar.
      edsghn.pystaar
  811. 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 be shown in braille by enclosing the whole
argument in brackets. brackets should not be
inserted where the print is ambiguous in such
cases.
    ex2. edsgh.a ff.tharedcgh.b ff.thar
    thef ffg
    ou@ffchiouinationerEchi ouchA
  (cases such as edledc rezing; are brailled
without inserting brackets in accordance with
s'72.)
  812. it is not necessary to insert additional
brackets for arguments consisting of unspaced
terms in braille. the conventional print use or
non-use of brackets should be sufficient in

91 coordinates and sets
such cases.
    ex3. eds.wt edledc.th
      edcgh2.p -wart edsaedcb
      edsghaedcbar edlx'edlity
  82. the root sign shall is generally to be
understood as applying only to the immediately following
number, letter, function or bracketed group
(i.e. that following the root exponent if
present). thus if a large root sign is
used in print to show that it applies to a fraction
which would be brailled with the still sign, or if a
horizontal bar is used in print to show that the
argument extends beyond the first letter or number,
brackets should be placed around the whole argument
in braille if they are not already present in the
print. indices may, however, be 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).
    exbled. shxy shghxyar shchnerxy
      sh3. sh3/4 shastb shghastbar
      shedsx shspiritxchiering; shx; shachildren
      shghxingbbar
            ------------
       9. coordinates and sets
  91. commas shown in print to separate
elements (or arguments) of sets,

912 coordinates and sets
coordinates, functions, etc. displayed in
brackets, are generally omitted in braille, the
spacing between those elements being sufficient. a
comma before an element preceded by an operation
sign (usually plus or minus) should, however,
be retained to avoid ambiguity.
    ex1. gh1 2ar
    gh-1 2ar
    gh2, -3ar
    gh1, -2 4ar
    fghx you zar
    ow1 3 5 7o
    ofa bwith
    witha bof
    xchgha bar
    ofRather ! '.with
    ofp tHpinch t@in.with
  911. when the arguments of a function
generally printed with commas are separated
into groups by a different punctuation mark (e.g.
a semicolon), this latter punctuation mark
should be retained in the braille.
    ex2. fghx you zBe so tar
  912. elements consisting of two or more
terms which would normally be spaced in braille, may
be united in such expressions by omitting the
spaces between terms (if not already united by being

92 coordinates and sets
enclosed in brackets). in omitting spaces
it will be necessary in some cases to insert
a er sign to terminate an index.
    ex3. ghxff1 yff1ar
    ghxfft, -you zar
    ghastbffx cstdffyar
    fghxeaff.a xbbff.a ... xchnerff.aar
    ghxingbbffying; zingbbfftingbbar
    ghghx ff.aar ghx -aarar
  in particularly complex cases it may be
preferable to preserve the spaces between terms
by retaining the commas between elements. this method
may also be necessary when an element contains spaced
word segments (s'72), since these spaces
may not be omitted.
    exbled. ghghA ffBareds.th ffedc.f.
      -xingsfft, ghA ffBareds.f.
      ffedc.th ffxings-tar
  913. an element with spaces omitted
to unite it according to s'912 should not be split
between terms at the end of a braille line: the element
should rather either be brailled with spaces by enclosing it in
brackets (so that it can be split between terms),
or else it should be taken down complete onto
the next line.
  92. when sets or coordinates are
divided between elements at the end of a braille line,

101 matrices and other arrays
a dot 5 mathematical hyphen should not be used
(as it would imply that the element continued), nor
should a comma be used if commas are generally
omitted in the braille.
    ex5. consider the set Every ggowa but can do
      every from go
            ------------
           10. matrices,
       determinants and other arrays
  101. round or square matrix
brackets are brailled as columns
of for signs of the appropriate length. long
vertical bars (e.g. for determinants) or
double vertical bars are brailled as single or
double columns of signs (dots
4-5-6) respectively. columns of
elements are spaced from each other by 1 clear
cell running down between the columns. elements
within a column should be brailled with their left-hand
cells aligned unless prefixed by a plus
or minus sign which is normally brailled to stand out.
elements beginning with a small latin letter do not
need to be prefixed by a letter sign unless they
immediately follow a vertical bar sign.
    ex1. for1 0for for a -bfor forxfor
          for0 1for for-can dfor foryfor
                                forzfor

103 matrices and other arrays
    for a but cfor _x,, iteabb_
              _x;, itbbbb_
    for1 2 3forfor1 2for
    forbled 5 6forfor3 4for
              for5 6for
  102. elements consisting of two or more
terms which would normally be spaced in braille are
united by omitting the spaces as is done with
coordinates or sets (s'912).
    ex2. for aeaeaffb,, a,; for
          for a;, abbbbffbbbbbfor
  103. a site in a matrix marked with a
dot in print should be marked with a dot 3 in
braille. ellipses are brailled as ordinary
literary ellipses which should be aligned in an
appropriate matrix column, and a line of
dots is brailled as a line of dot 3's.
    ex3. forn, ' ' for
          for' not; ' for
          for' ' nccfor
    for0 a, ...... 0 for
    for a, 0 ...... 0 for
    for...............'for
    for......'' 0 achnfor
    for......'' achildren 0 for



105 matrices and other arrays
    forb,, but,; ... bch1sfor
    forb;, but;; ... bch2sfor
    for.....................for
    forbchs1 bchs2 ... bchss for
  104. partition lines in matrices may be
shown by a sequence of : signs (dots
2-5) for horizontal lines, and
by signs (dots 4-5-6) aligned
vertically for vertical lines.
    exbled. forx,, it,; 1 0for
          forx;, it;; 0 1for
          for:::::::::::::ccccfor
          for1 0 1 0for
          for0 1 0 1for
  105. matrices or determinants set
out on a separate line in print are normally
indented to begin in cell 5 (or cell 7 for
runovers to an equation) as usual for
mathematics. a cell 1 start may be used for
wide matrices if it allows the array to be
brailled without runovers when this would not be possible
with the normal indentation.
  in a matrix equation or for matrices within
ordinary text, all the matrices which fit
across the page should be brailled with their top rows and
any other intervening single line expressions or
text on the same braille line. single line

106 matrices and other arrays
expressions or text following a matrix should
be brailled on the bottom line of the matrix
unless another matrix appears on that same
line in which case the expression or text is
placed on the top line between the matrices.
    ex5. ABut ggfor 1 0 1forfor1 0for
                 for-2 1 3forfor0 2for
                             for5 1for.
      ggfor6 1for
        for13 5for ggCan
    ex6. matrices for1 0for, for1 0for
                   for0 1for for1 1for
      and for1 0for
        for0 0for are linearly independent.
  matrices should not be brailled directly beneath
one another with one or both of the long brackets
aligned. to avoid this one of the matrices can be
moved on 1 or 2 cells, or else a
blank line left between the matrices.
  106. a matrix which is too wide for the
braille page may be split between columns with the
remaining part of the matrix placed beneath and indented
2 cells from the start of the matrix. facing
pages may also be used for wide matrices
if convenient.

107 matrices and other arrays
    ex7.
for-eds.feds.yffedc.thedc.fedc.y
for-eds.fedc.y-edc.thedc.feds.y
for eds.thedc.f
  edc.feds.yffedc.theds.fedc.y -eds.thedc.yfor
  edc.fedc.y-edc.theds.feds.y eds.theds.yfor
  eds.theds.f edc.th for

  107. a linear method may
alternatively be used to represent
matrices and other such arrays. this method is
favoured for binomial coefficients, and for
christoffel symbols whereby the particular type
of bracket used in print can be represented in
braille, but it is not the primary method used for
representing matrices in general
transcription work. in this method the rows are
brailled in order, from the topmost downwards, with
an unspaced out sign used to indicate the
start of a new row. the whole sequence is
enclosed in the appropriate standard
mathematical brackets, and may be divided
at any convenient point at the end of a braille
line, a dot 5 mathematical hyphen not being
used if the division is between spaced elements.
    ex8. ghnourar represents the binomial


111 worked calculations
      coefficient fornfor
                forrfor
    ex9. owioujko (a christoffel symbol)
    ex10. ghaeaeaffc a,; a,: aea.oua;,
      abbbbffc a;: abb.oua:, a:; accccffc
      acc.oua., a.; a.: a..ffcar
            ------------
        11. worked calculations;
          logical figures
  111. horizontal lines used in worked
calculations are brailled as a line
of : signs (dots 2-5) of the
appropriate length. digits, tens and
hundreds, etc. should generally be vertically
aligned, and operation signs placed as in
print. it may be convenient to omit numeral
signs so as to leave the braille less cluttered,
but this convention should be explained beforehand in general
transcription work.
    ex1. 1532
           619 !
          :::::

          :::::




112 worked calculations
    ex2. [numeral signs are omitted]
        cieafh
       ::::::
    after fce
       dh
       ::
       aee
       add
       :::
        aaj
         if
        :::
         ad
    etc.
  112. worked calculations with algebraic
expressions should generally be arranged so that the
plus and minus signs and terms of the same
degree are vertically aligned. this may be
achieved compactly by omitting spaces (as
well as possibly leaving or inserting spaces
where necessary), or less compactly by just inserting the
necessary spaces.






113 worked calculations
    ex3.
                 xingbb- xffbled
          :::::::::::::::::::
    x-2 xingcc-3xingbbff6x-8
          xingcc-2xing;
          ::::::::::
             - xingbbff6x
             - xingbbff2x
             ::::::::::::
                      4x-8
                      4x-8
                      :::::::
                      ......'
  113. horizontal lines in logical
figures may also be represented by a sequence
of : signs.
    exbled. D.f.w D.y.w
          :::::::::::::
          :::::::::::::
            DK..fyw








122 miscellaneous notation
    ex5.
     D.f.w D.y.w
    ::::::::: iic :::::::::
    DNN.F.W DNN.Y.W
    ::::::::::::::::::::::::: iib
         DNDN.FATION.Y.W
         :::::::::::::: by definition of Knowledge
            DK..fyw

            ------------
         12. miscellaneous
               notation
  121. the signs oufor and out) are used
to represent a variety of operator symbols
as required.
    ex1. it oufory (x star y[
    from ou)g (f circle g: function
      composition)
  122. continued fractions are coded using the
sign sten (dots 3-D2-6) as a
modified fraction line sign indicating that
all the terms following it are part of the
denominator. (this notation should not be used in other
contexts.)
    ex2. a ffbstenc ffdstene !...
    ex3. 1/en4 ff1/en1 ff1/en1.
      ff1/en2 (5;:

125 miscellaneous notation
  123. the differential operators
dstdx, @dst@dx, @dingbbst@dx@dy, etc.
should be separated from their following operands by a
dot 5 if they (the operators) are not
enclosed in brackets.
    exbled. dstdxfatherghxar
    @dst@dt.gh.f ff.yar
    gh@i@dst@dx ff@j@dst@dyar.fghx yar
  (dfstdx, @dfst@dx, etc. are coded as
ordinary fractions.)
  124. a dot or blank in print
indicating an unspecified argument, etc.
may be shown in braille as a single dot 3.
    ex5. fgh'ar fghx 'ar
  1241. dashes are also freely used for
similar purposes.
    ex6. -- ?--: ghA Bar conoA 8But
    --st5 (2,"
  125. ellipses representing omitted
or unspecified terms in an expression, are
spaced in the same way as the terms that they
represent would be. (an ellipsis is coded
as an unspaced sequence of three dot
3's.)
    ex7. edsx )x -xingccst3To .
      ffxingenst5To -...


129 miscellaneous notation
    fghzar ggghz -aeaarghz -abbar....
      ghz -achnar
    fghx, it; ... xchnar
  126. the (landau) notation Oghyar and
oghyar used in analysis, etc. should be coded
with letter o's (i.e. not zeros).
    ex8. edsx )x -xingccst3To .
      ffOghxingenar
    edsx )x -xingccst3To ffoghxingccar
      (for small x[
  127. in expressions such as
    ex9. 5 (2 (mod 3[,
the bracketed remark should be spaced from the
preceding term, and literary brackets used.
  128. in the notation (below) for cyclic
permutations, the numbers or other symbols
used may be brailled spaced or unspaced according
to print. each number should have its own numeral
sign, whether spaced or unspaced.
    ex10. gh134ar
    ghaeaount abbount ... achmerountar
  the notation for a general permutation may be
brailled as a matrix.
    ex11. for1 2 3 4for
           for3 2 4 1for
  129. the notation for "edcx or edsxWas,
etc. appearing as one function written above

132 layout of text
the other, may be coded on one line by using
brackets.
    ex12. It ggghedc edsarpx
  1210. the terminal letters in expressions
such as 1:rd and 3enth's should generally be
omitted in braille after fractions (but not after whole
numbers).
    ex13. 1: 2nd
            ------------
            13. layout of
           mathematical text
  the following remarks are additional to those in
s'1.
  131. centred headings should not begin before
cell 9 to avoid confusion with set out
mathematics.
  132. equation numbers should be placed in
literary brackets (whether or not brackets
are used in the print), starting in cell 5 of the
first line of the equation or equations to which they
refer. if any further line of an equation
has a reference number, it should be placed (in
brackets) at the beginning of the line to which it
refers, starting in cell 5, and followed by two
blank spaces before the equation continues.
  if the reference symbol is not a simple
number but wholly or partly some other sign, it

1334 layout of text
is treated in the same way, with a dot 6
separation sign inserted before the closing bracket
if necessary.
    ex1. (1@in[ (*) (11)
  133. the following are cases where it may
be necessary or advantageous to differ from the usual
cell 5 with runovers in cell 7
mathematics layout:
  1331. wide arrays (s'10).
  1332. worked calculations (s'11).
  1333. to enable simultaneous equations
to all start in the same cell if the first equation
is preceded by an equation number or otherwise
does not begin in cell 5.
    ex2. it ffy (3
          2x ff3y (8
  1334. to allow cell 5 or cell 7
indented text indicators (e.g. question numbers
or letters) to stand out in parts of text consisting
principally of such indicators and set out
equations:
    (a) the layout used in s'1333 above
may be used where applicable; or
    (b) the equations may each be started in
cell 1; or
    (c) blank lines may be inserted
to separate the numbered or lettered sections of

134 layout of text
text, retaining the usual cell 5 and 7
layout of equations and those numbers or letters.
  note. set out equations not beginning in
cell 5 by virtue of s'1333 or
s'1334 above should not be allowed runovers.
if equations are thereby too long (i.e. any
equation of a set of simultaneous equations),
a standard cell 5 and 7 format should be adhered
to instead. (equations simply preceded by cell
5 equation numbers are normally allowed cell
7 runovers.)
  134. if the right-hand side of an equation
has several options subject to different
conditions (often grouped with a large bracket in
print), the large bracket is omitted in
braille, and one of the following alternative
layouts may be used:
    (a) the left-hand side and first option
may be brailled as a complete equation, and the
second and further options placed beneath with their
equals signs in the same cell as that of the first
option, if the options are all short enough to fit
without runovers; or
    (b) the left-hand side of the equation (if
it is reasonably short) may be repeated for
each option, so that the options are each brailled as
complete equations; or

141 units
    (c) the left-hand side may be brailled
once followed by the first option, with the subsequent
options each begun on a separate line with their
equals signs in cell 5, and with all
runovers in cell 7.
    ex3.
    x ggx, x o(0
         )-x, x ow 0
  135. if conditions or words of comment are
associated with and interrupt a mathematical
expression, they may be inserted as in print at
the appropriate place, but should not be preceded
or followed by a dot 5 hyphen. the
mathematical expression should continue in cell
7 of a new line.
            ------------
             14. units
  (tables of standard unit abbreviations are
given at the end of this section for reference
[p68-72].
  the following guidelines apply
to scientific and mathematical text.
  (note. the examples in this section have been
chosen to illustrate a variety of print forms
as found: this choice is not intended to indicate
recommended practice.)
  141. units are placed before or after the

143 units
number to which they refer, according to print. units should
be spaced in braille, apart from the signs in
s'146 denoting units of angle and length in
feet and inches, and monetary unit abbreviations
preceding the number or carrying an initial letter
sign.
  142. unit abbreviations are coded according
to the print by placing a dot 6 before each
capital letter, with greek "mu" coded as more,
the @angstr@om symbol (in print, a
capital a with a small circle above) coded
as ^a, greek "omega" (ohm) coded as world,
the percent symbol coded as conp, the pounds
sterling symbol coded as letter l, and the dollar
symbol as lower d. (see however s'1421,
s'144.)
  1421. mmhereg should be coded with a dot 5
before the abbreviation hereg for mercury, rather than
using a dot 6.
  143. the dots 5-6 letter sign is only
required:
    (a) before a lower case single letter unit
symbol standing alone or with an index or
punctuation, or immediately following and unspaced
from the number (e.g. before p for pence).
    (b) before the abbreviation s for second in
combined units, where necessary to avoid ambiguity

144 units
(see s'148).
    (c) before a unit abbreviation consisting of a
sequence of initials without full stops (e.g.
mph), if the dot 6 letter sign is not used
(see s'144). this does not refer
to abbreviations such as cm, kg, etc., in which the
first letter stands for a prefix, not a word; or
to combined units brailled with dot 3 separators
according to s'148.
  144. the dot 6 letter sign is not used
to show capitals in conventional informal
abbreviations such as mph, m.p.h., mpg,
etc. these are coded in accordance with s'143
(c).
    ex1.
    3 metres
    6 m (6 metres)
    2, 3 m (2, 3 metres)
    2 810ing; m (2 810ing;
      metres)
    l6 (6 pounds sterling)
    lx (x pounds sterling)
    l5,30p (5 pounds 30 pence)
    l60m (60 million pounds)
    25 conp (25 percent)
    3 s (3 seconds)
    3 sec (3 seconds)

146 units
    1 mol (1 mole)
    20 km (20 kilometres)
    5 ft 10 ins (5 feet 10 inches)
    4 That (4 teslas)
    5 mA (5 milliamperes)
    10 Hz (10 hertz. in print, h
      is a capital letter, z is a lower
      case letter.)
    3,3 Pa (3,3 pascals)
    9 GeVery (9 gigaelectronvolts)
    6 MWill (6 megawatts)
    8 ^a (8 @angstr@oms)
    3 ms (3 microseconds)
    20 ml (20 millilitres)
    10 cc (10 cubic centimetres)
    12 c.c. (12 cubic centimetres)
    40 m.p.h. (40 miles per hour)
    60 mph (60 miles per hour)
    6 your. (6 years)
  145. unit abbreviations should generally be
coded in the same way, whether or not
accompanying a number.
  the pounds sterling symbol (spaced) should,
however, be coded as lBe and the dollar symbol
(spaced) coded as $. see also
s'1462.
  146. in simple expressions of angle,

1461 units
or length in feet and inches,
    superscript 0 (degrees) is coded
      as was (lower j)
    single dash (minutes, or feet)
      as . (dots 4-6)
    double dash (seconds, or inches)
      as (dots 4-5-6)
    superscript c (radians, when used
      instead of rad) as - (dots 3-6)
and follow the number to which they apply, with the whole
group unspaced. when the degree sign
follows a lower number it should be preceded by the
superscript sign + to avoid ambiguity.
    ex2.
    6" (6 degrees)
    30. (30 minutes, or 30 feet)
    10 (10 seconds, or 10 inches)
    2.p- (2.p radians)
    6"30. (6 degrees 30 minutes)
    5.10 (5 minutes 10 seconds,
      or 5 feet 10 inches)
    6"30.10 (6 degrees 30
      minutes 10 seconds)
    1bbing" (1; degree)
  1461. in expressions of temperature,
and in bearings, the letters Can, F; Not, So, Every,
Will, are brailled unspaced from the number to which they

148 units
apply.
    ex3.
    30"From (30 degrees fahrenheit)
    10"Can (10 degrees celsius)
    50"So (50 degrees south)
    N30"Will (north 30 degrees west)
  1462. in combined units (see
s'148), the symbols for degrees,
degrees From and degrees Can are coded as
    dg dgFrom dgCan
respectively.
  these abbreviations are also used in braille when the
unit symbol is not attached to a number.
  147. indices attached to unit
abbreviations or words, should be shown as lower
numbers immediately following the + superscript
sign.
    exbled.
    5 kming; (5 kilometreingbb[
    6 sing-, (6 seconding-,[
  148. in combined units, a dot 3 should be
inserted between the individual units unless an
index or stroke is present at that point. a
group consisting of a multiplying prefix
attached to a basic unit symbol (e.g.
kg, cm, etc.) is to be regarded as a single
unit, and so a dot 3 separator should not

1481 units
intervene.
  the stroke should be brailled as still (dots
3-4).
  1481. in print, the individual units
in combined units are usually separated
by spaces or half-spaces if not by a stroke
(dots are also occasionally used). the dot 3
separator should, however, always be used between
individual units in braille according to s'148,
even when no separation is shown between them in
print.
    ex5.
    3 N'm (3 newton metres)
    6 Nstming; (6 newtons per
      metreingbb[
    4 more's (4 metre seconds)
    10 ms (10 milliseconds)
    metre'ness (metre seconds. this
      example shows the letter sign used before
      s for second when the dot 3 might
      otherwise be read as an apostrophe.)
    5 g'ming-, (5 gram metreing-,[
    30 msts (30 metres per second)
    mstsing; (metres per secondingbb. the
      fact that no number is present does
      not affect the coding: according to s'143,
      no letter sign is required, even when

1410 units
      the unit occurs within ordinary text.)
    4 rad'sing-, (4 radians
      seconding-,[
    10ing-con Not's'ming-be (10ing-con
      newton second metreing-be[
    5 mingbbsing-, (5 metreing;
      seconding-,[
    x coulombsts (x coulombs per
      second)
  149. long combined units may be split
at the end of a braille line using the dot 5
mathematical hyphen. a dot 3 if present
at that point, remains before the dot 5 hyphen.
short unit expressions should not be divided.
  it is preferable for spaced units not to be
separated from their preceding number at the end of a
braille line.
  1410. the dot 6 mathematical separation
sign is not required after a unit abbreviation
before following punctuation unless the abbreviation
ends with an index or one of the angle symbols
given in s'146.
    ex6.
    2 m. (2 metres.)
    7 mingbb. (7 metreingbb.)
    5". (5 degrees.)
            ............

14t units
                tables
  (si units and prefixes are taken from the
association for science education booklet:
si units, signs, symbols and
abbreviations (1981). other non-si
units are taken from nuffield advanced
science book of data (1982).)
             si units
name symbol name symbol
metre m coulomb Can
kilogram kg volt Very
second s ohm world
ampere A siemens So
kelvin Knowledge farad From
candela could weber Wb
mole mol tesla That
radian rad henry Have
steradian sr lumen lm
hertz Hz lux lx
newton Not becquerel Bq
pascal Pa gray Gy
joule Just sievert Sv
watt Will
            ............

14t units
         multiplying prefixes
  these may be attached to any of the above units.
(exceptionally kg already has a prefix
attached, but other multiples e.g. mg, g,
are formed in the obvious way.)
sub-multiple prefix symbol
10ing-, ...'' deci '' do
10ing-be ...'' centi '' can
10ing-con ...'' milli more
10ing-to ...'' micro more
10ing-in ...'' nano '' not
10ing-,; ...' pico '' people
10ing-,en ...' femto from
10ing-,? ...' atto '' a

multiple prefix symbol
10ing, ......' deca '' da
10ing; ......' hecto have
10ing: ......' kilo '' knowledge
10ing! ......' mega '' More
10ingin ......' giga '' Go
10ing,; ...... tera ... That
10ingeaen ...... peta '' People
10ing,? ...... exa ... Every
            ............
  (these prefixes are also sometimes used before
units in the following table.)

14t units
      other non-si unit symbols
name symbol
@angstr@om ......'' ^a
astronomical unit au (also AUs)
atmosphere ......'' atm
atomic mass unit u
biot ............' Bi
british thermal unit Btu
calorie .........' cal
curie ............ Ci
day ............... d
debye ............ Do
decibel .........' dBut
degree (angular) '' was (superscript
                     0[
degree celsius ... by Can (or dgCan)
degree fahrenheit by From (or dgFrom)
dyne ............' dyn
electronvolt ...'' eVery
foot ............' ft (also a dash)
franklin .........' Friend
gallon .........'' gal
gauss ............ Go
hectare (100 ares) ha
horsepower ......... hp
hour ............'' h
hundredweight ...... cwt

14t units
name symbol
inch ............' in (also a double dash)
kilogram-force ... kgf
  (kilopond) ... kp
knot .........'' kn
litre .........' l (also Like)
micron ......... mm (also more)
mile (nautical) not. mile
minute (angle) '' . (a single dash)
minute (time) ...'' min (m is also used in
                   time of day, e.g.
                   18 h 23 m)
oersted .........' Oe
ounce .........'' oz
ounce (fluid) ... fl oz
pint ............ pt
poise .........' People
pound (weight) ...' lb
pound-force ......'' lbf
rad or r@ontgen Rather
rem ............ Rem
second (angle) (a double dash)
stokes .........' St
ton-force ......'' tonf
tonne .........' t
torr .........'' Torr
It unit ......'' Xu

15 alphabets used in mathematics
name symbol
yard ............ yd
year ............ a
            ------------
        15. alphabets used in
             mathematics
         the greek alphabet
  [small and capital letters are given
respectively.]
alpha '' a a nu ...'' not n
beta ... but b xi ...'' it x
gamma '' go g omicron o o
delta '' do d pi ...'' people p
epsilon every e rho ...' rather r
zeta ... as z sigma '' so spirit
eta ...' which tau ...' that t
theta ...' this th upsilon us u
iota ... i i phi ...' from f
kappa '' = k chi ...'' and 
lambda like l psi ...' you y
mu ...'' more many omega '' will world
            ............
  german (gothic) letters are coded using the latin
letter equivalents given in the print version of this
code, prefixed by a (small or capital)
letter fount sign from those given in s'22.
            ------------



                index
           (including index of
         mathematical signs)
  [in the index, mathematical signs are
preceded by the full cell for to make the
position of the dots clear: it should not be read as
being part of the sign. references refer to sections
and examples except where stated as page
references. no references are given on spacing
for individual mathematical signs: for this
refer to the general entry on spacing of
signs.]
                 a
actuarial symbol, (years) for@th 62
          ex2, 66 ex10
  superscript circle, 67 ex12
aleph, 227
all, for forou,
alphabets, 15
ampersand for and
and (ampersand) forandBe (inverted v symbol)
          for^?
angle, symbol for{ 691
  units of, 146
@angstr@om unit for^a 142
antilog fored@l 71, 711

a index
approximately equal to for)
arccos foredcents 71, 711
arccosec fored@gh 71, 711
arccosech fored8gh 71, 711
arccosh fored8c 71, 711
arccot fored@ou 71, 711
arccoth fored8ou 71, 711
arcsec fored@- 71, 711
arcsech fored?- 71, 711
arcsin fored@s 71, 711
arcsinh fored8s 71, 711
arctan fored@t 71, 711
arctanh fored8t 71, 711
argument of a function, 8
  omitted, 124, 1241
arrow
  down forccen
  left forow:
    double headed forowow:
      head at each end of shaft forowccow
    double shafted forouow:
    half fori:
    indicating vector for.ow: 68
    long forow::
  leftstright forowcco
    double shafted forouowcco

a-but index
  right forcco
    double headed forccoo
      head at each end of shaft forocco
    double shafted foroucco
    half, lower forccin
      upper forcce
        long with short left half arrow beneath
          forenccee
        short with long left half arrow beneath
          forenencce
        with left half arrow beneath forencce
    indicating vector for.cco 68
    long forcccco
      with short left arrow beneath forenccee
    short with long left arrow beneath forenencce
    with left arrow beneath forencce
    with vertical bar on shaft for^cco
  up forcci
arrows, after subscript sign, 62
  after superscript sign, 61
asterisk: see star
                 b
bar, horizontal
    indicating negative integer part of
          number for^ 214
    over term forwh 69, 692, 612
    subscript, 224, 69

b index
  vertical
    double for_ 221 (d), 222
          ex13
    single for 221 (d), 222
          ex13
because for@st
binomial coefficients, 107
bold or large symbols (indicated by
          for@[: modification of signs
          (a)
brackets, 5
  angle [left and right] for.of for.with
  brace (curly) forow foro 221 (e),
          107 ex9
  determinant for for 221 (d), 101
  insertion of, 53, 54, 81, 811,
          812, 82, 913
    for fractions, 41
    for indices, 65, 652, 653,
          661, 692
  matrix forfor forfor, for_ for_ 221
          (d), 101
  modulus for for 221 (d), 222
          ex13, 52
  norm for_ for_ 221 (d), 222
          ex13
  round forgh forar

b-can index
  square forof forwith 221 (f)
                 c
centred headings, 131
chemical elements and formulae, 24
christoffel symbols, 107
circle symbol fored:
  as an operator foroufor, forou) 121
  enclosing a symbol: modification of signs
          (c)
colog fored^l 71
colon within mathematical expression, 34
          exbled
comma, in numbers indicating thousands, etc.
          (indicated by for'[ 212,
          218
  omission of, 218, 652, 91
  retention of, 218, 91, 912
  within mathematical expression, 12, 62
          ex2, 65 ex6
comment, words of, 135
contained in forouow
  or equal to forouow)
contains forouo
  or equal to forouo)
coordinates and sets, 9
coproduct forv
cosec (cosecant) foredgh 71

c-do index
cosech foredhgh 71
cos (cosine) foredc 71
cosh foredhc 71
cot (cotangent) foredou 71
coth foredhou 71
cross (e.g. multiplication sign) for?
crossed-out symbols (indicated by for.[:
          modification of signs (b)
  zero (empty set) forou"
curl or rot foredsh
                 d
@d (curly d), 224
dagger for@. 69
dash, horizontal bar representing a blank
          for-- 1241
  superscript (prime) forinch 222,
          69, 693, 612
  swung for.-
    superscript, subscript (tilde)
          for^wh 224, 69
decimal point for, 211
decimal, recurring (indicated by for.[,
          213
defined as for)
degree for" 146, 1461, 1462
del (nabla) for" 71 ex1
derivative, 61 ex1, 123

d-every index
  partial, 224 ex14, 61 ex1,
          123
determinant: see brackets; matrices
difference, set forouch
differential operators, 123
dirac (planck) constant, 228
div (divergence) foredth
divided by for., forst (oblique stroke)
divides for
division of mathematical expressions, 13,
          215, 653, 913,
          92, 106, 149
dollar for$ 142, 145
dot, differentiation for' 610
    double for- 610
    triple for-' 610
  indicating position in matrix for' 103
  indicating unspecified argument for' 124
  multiplication for'
  partial correlation or regression notation
          for.' 63
                 e
element of, is forouin
ellipsis for... 103, 125
empty set forou"
"end of proof" indicator forforfor
equal to for)

e-have index
  related signs (see table) for.-were for.)
equation numbers, 132, 1334 note
equations, simultaneous, 1333
  with options, 134
equivalent to for--
evaluated at for
exists, there forouen
                 f
factorial forTo 35 ex5
foot (single dash unit symbol) for. 146
fraction line forst 4
  within an index, 61 ex1
  within the argument of a function, 81, 82
fractions, continued, 122
  simple numerical, 218, 219
                 g
german (gothic) alphabet, 15
grad (gradient) foredg
greater than foro
  much greater than foroo
  or approximately equal to foro)
  or equal to foro)
  or less than foroow
greek alphabet, 15
  double capital sign for 222
                 h
hash for?

h-i index
hat for@wh 69, 691, 692, 693
          ex19 and 20, 612
hyperbolic functions, 71
  inverse 71, 711, 73
  sequences of, 72
hyphen, literary, 12, 216
  mathematical for. 13, 215, 92,
          107, 135, 149

inch (double dash unit symbol) for 146
indices, 6, 711, 73
  arrows, 61, 62, 68
  dashes, stars, bars, etc., 69,
          691, 692, 693,
          146
  directly above (indicated by for@ing[,
          67
  directly below (idicated by for@ch[,
          67
  left-hand, 66, 661
  multiple levels, 64
  multiple term, omission of spaces in,
          65, 651, 652,
          653
  plus and minus signs, 611
  subscript (indicated by forch[ 62,
          63

i-like index
  superscript (indicated by foring[, 61
  termination of (indicated by forer[, 61,
          62, 64, 651, 66,
          67, 69, 610, 611,
          912
infinity forfor
integral for the 62 ex2, 69 ex16 and
          17, 612
  closed line for@the
intersection forou?
inverted
  A forou, (for all)
  hat for@ar
  ford for" (del, nabla)
  forp forv (coproduct)
  v symbol for^? (and, meet, vector
          product, etc.)
                 j
join for^!
                 l
landau O, o notation, 126
layout of mathematical text, 1, 13
less than forow
  much less than forowow
  or approximately equal to forow)
  or equal to forow)
letter fount signs, 22, 221, 223-

l-more index
          228, 23, 24, 71,
          142, 1421, 143,
          144
  double, 222
limits, 652 ex8, 71 ex1, 72
          ex2, 73 ex3
logical figures, 113
log (logarithm) foredl 71
                 m
matrices, 10
  blank entries in, 103
  division of, 106
  layout of text, 105
  linear representation of (using forou sign),
          107
  partitions in, 104
meet for^?
minus for- 53, 611
minus or plus for-to 
minute (single dash unit symbol) for.
          146
mod, 127
modification of signs, p10
modulus bars for for 221 (d),
          222 ex13, 52
multiplication, cross for?
  dot or scalar for'

m-people index
  vector (inverted v symbol) for^?
                 n
nabla for" 71 ex1
norm bars for_ for_ 221 (d), 222
          ex13
normal subgroup of forouing
  or equal to forouing)
not forou-
null set forou"
numbers, 21
  division of, 215
  lower, 218, 61, 62, 63
  mixed, 219
  roman, 23
  signed, 53
numeral sign forble 21
  omission of, 217, 111
                 o
O, o: see landau notation
oblique stroke forst: see fraction line
operation symbols foroufor, forou) 121
or (v symbol) for^!
                 p
parallel to for_
percent symbol forccp 142, 144 ex1
permutation, cyclic, 128 ex10
  general, 128 ex11

p-so index
perpendicular to forble'
plus for! 53, 611
plus or minus for!-
pound sterling forl 142, 145
prime: see dash
projective with for^)
proportional to for:)
punctuation, separation of, 12
  spacing of, 34
  within mathematical expressions: see
          colon, comma
                 r
radian for- 146
restricted to for
reversed
  Every forouen (there exists)
  forouing forouu
  forouing) forouu)
  forouin forouwith
  forst forouch
  for: forccl
  for, forlord
root forsh 82
  exponents of, 621
                 s
sec (secant) fored- 71
sech foredh- 71

s index
second (double dash unit symbol) for
          146
separator, differential operator for. 123
  index for@ 61, 62, 65
  mathematical for 12, 1410
  unit for' 148, 1481
sets: see coordinates and sets
sin (sine) foreds 71
since for@st
sinh foredhs 71
square, symbol foredq 71
  enclosing a symbol: modification of signs
          (c)
spacing of signs, 3
  omission of, 13 (e), 31, 321
    in coordinates and sets, 912,
          913
    in indices, 65, 651, 653
    in matrices, 102
    in worked calculations, 112
spacing in print of argument of functions,
          811
special functions and other word segments, 7
  capital initial, 74
  sequences of, 72
star (operator) foroufor, forou) 121
  superscript, subscript for`en 222,

s-very index
          69, 693, 612
                 t
tan (tangent) foredt 71
tanh foredht 71
temperature, 1461, 1462
therefore forChild
theorem, is a for,
tilde for^wh 224, 69, 692,
          612
triangle symbol foredd 71
trigonometric functions, 71
  inverse, 71, 711, 73
  sequences of, 72
                 u
union forou!
units, 14
  combined, 148, 1481
  of angle, 146, 1461
  position of, 141
  spacing of, 141
  tables
    multiplying prefixes, p69
    other non-si unit symbols, p70
    si units, p68
unlisted signs, 14, 223

valid, is for:

v-will index
varies as for:)
vector, directed line segment, 68
vector or wave operator (represented by a
          square symbol) foredq
                 w
weierstrass function, 227
word segment indicator fored 7
worked calculations, 111, 112
            ------------
                the end



 
