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Guidelines for Technical MaterialUnified English BrailleGuidelines forTechnical Material2008 version updated August 2014i

Guidelines for Technical MaterialLast updated August 2014ii

Guidelines for Technical MaterialiiiAbout this DocumentThis document has been produced by the Maths Focus Group, a subgroup of theUEB Rules Committee within the International Council on English Braille (ICEB).At the ICEB General Assembly in April 2008 it was agreed that the documentshould be released for use internationally, and that feedback should be gatheredwith a view to a producing a new edition prior to the 2012 General Assembly.The purpose of this document is to give transcribers enough information andexamples to produce Maths, Science and Computer notation in Unified EnglishBraille.This document is available in the following file formats: pdf, doc, dxp or brf.These files can be sourced through the ICEB representatives on your localBraille Authorities.Please send feedback on this document to ICEB, again through the BrailleAuthority in your own country.Last updated August 2014

Guidelines for Technical MaterialivGuidelines for Technical Material1 General Principles. 11.1 Spacing . 11.2 Underlying rules for numbers and letters . 21.3 Print Symbols . 31.4 Format . 31.5 Typeforms . 41.6 Capitalisation . 41.7 Use of Grade 1 indicators . 52 Numbers and Abbreviations . 82.1 Whole numbers . 82.2 Decimals. 92.3 Dates . 92.4 Time . 102.5 Ordinal numbers . 102.6 Roman Numerals . 112.7 Emphasis of Digits . 112.8 Ancient Numeration systems . 112.9 Hexadecimal numbers . 122.10 Abbreviations. 123 Signs of Operation, Comparison and Omission . 153.1 Examples. 163.2 Algebraic Examples . 173.3 Use of the braille hyphen . 173.4 Positive and negative numbers . 183.5 Calculator keys . 183.6 Omission marks in mathematical expressions. 194 Spatial Layout and Diagrams . 204.1 Spatial calculations . 204.2 Tally marks . 254.3 Tables . 264.4 Diagrams . 275 Grouping Devices (Brackets) . 306 Fractions . 316.1 Simple numeric fractions . 316.2 Mixed numbers . 316.3 Fractions written in linear form in print . 326.4 General fraction indicators. 326.5 Extra Examples . 337 Superscripts and subscripts . 347.1 Definition of an item . 347.2 Superscripts and subscripts within literary text . 357.3 Algebraic expressions involving superscripts . 357.4 Multiple levels . 37Last updated August 2014

Guidelines for Technical Materialv7.5 Negative superscripts . 377.6 Examples from Chemistry . 387.7 Simultaneous superscripts and subscripts . 387.8 Left-displaced superscripts or subscripts . 387.9 Modifiers directly above or below . 398 Square Roots and other radicals. 408.1 Square roots . 408.2 Cube roots etc . 418.3 Square root sign on its own . 419 Functions . 429.1 Spelling and capitalisation . 429.2 Italics . 429.3 Spacing . 439.4 Trigonometric functions . 449.5 Logarithmic functions . 459.6 The Limit function . 469.7 Statistical functions . 469.8 Complex numbers . 4710 Set Theory, Group Theory and Logic . 4811 Miscellaneous Symbols . 5011.1 Spacing . 5111.2 Unusual Print symbols . 5111.3 Grade 1 indicators . 5111.4 Symbols which have more than one meaning in print . 5111.5 Examples. 5211.6 Embellished capital letters . 5511.7 Greek letters . 5612 Bars and dots etc. over and under . 5712.1 The definition of an item . 5712.2 Two indicators applied to the same item . 5913 Arrows . 6013.1 Simple arrows. 6013.2 Arrows with unusual shafts and a standard barbed tip . 6113.3 Arrows with unusual tips . 6214 Shape Symbols and Composite Symbols . 6514.1 Use of the shape termination indicator . 6614.2 Transcriber defined shapes . 6614.3 Combined shapes . 6715 Matrices and vectors . 6915.1 Enlarged grouping symbols . 6915.2 Matrices . 6915.3 Determinants . 7015.4 Omission dots. 7015.5 Dealing with wide matrices . 7115.6 Vectors . 7215.7 Grouping of equations . 73Last updated August 2014

Guidelines for Technical Materialvi16 Chemistry . 7416.1 Chemical names. 7516.2 Chemical formulae . 7516.3 Atomic mass numbers . 7616.4 Electronic configuration . 7616.5 Chemical Equations . 7716.6 Electrons . 7816.7 Structural Formulae . 7817 Computer Notation . 8317.1 Definition of computer notation . 8317.2 Line arrangement and spacing within computer notation . 8317.3 Grade of braille in computer notation . 86Last updated August 2014

1 General Principles1Guidelines for Technical Material1 General Principles1.1 Spacing1.1.1 The layout of the print should be preserved as nearly as possible. Howevercare should be taken in copying print spacing along a line as this is often simplya matter of printing style. Spacing should be used to reflect the structure of themathematics. Spacing in print throughout a work is often inconsistent and it is notdesirable in the braille transcription that this inconsistency should be preserved.1.1.2 For each work, a decision must be made on the spacing of operation signs(such as plus and minus) and comparison signs (such as equals and less than).When presenting braille mathematics to younger children, include spaces beforeand after operation signs and before and after comparison signs. For olderstudents who are tackling longer algebraic expressions there needs to be abalance between clarity and compactness. A good approach is to have theoperation signs unspaced on both sides but still include a space before and aftercomparison signs. This is the approach used in most of the examples in thisdocument.1.1.3 There are also situations where it is preferable to unspace a comparisonsign. One is when unspacing the sign would avoid dividing a complex expressionbetween lines in a complicated mathematical argument. Another is when thecomparison sign is not on the base line (for example sigma notation where iequals 1 is in a small font directly below).1.1.4 When isolated calculations appear in a literary text, the print spacing canbe followed.Last updated August 2014

1 General Principles21.2 Underlying rules for numbers and lettersListed below is a summary of the rules for Grade 1 mode and Numeric mode asthey apply to the brailling of numbers and letters in mathematics. Refer to thecomplete versions of these rules for more detail.1.2.1 Grade 1 modeA braille symbol may have both a grade 1 meaning and a contraction (i.e.grade 2) meaning. Some symbols may also have a numeric meaning. A grade 1indicator is used to set grade 1 mode when the grade 1 meaning of a symbolcould be misread as a contraction meaning or a numeric meaning.Note that if a single letter (excluding a, i and o) occurs in an algebraicexpression, it can be misread as a contraction if it is "standing alone" so mayneed a grade 1 indicator. The same is true of a sequence of letters in braille thatcould represent a shortform, such as ab or ac, if it is "standing alone".A letter, or unbroken sequence of letters is "standing alone" if the symbols beforeand after the letter or sequence are spaces, hyphens, dashes, or anycombination, or if on both sides the only intervening symbols between the letteror sequence and the space, hyphen or dash are common literary punctuation orindicator symbols. See the General Rule for a full definition of "standing alone".1.2.2 Numeric modeNumeric mode is initiated by the "number sign" (dots 3456) followed by one ofthe ten digits, the comma or the decimal point.The following symbols may occur in numeric mode: the ten digits; full stop;comma; the numeric space (dot 5 when immediately followed by a digit); simplenumeric fraction line; and the line continuation indicator. A space or any symbolnot listed here terminates numeric mode, for example the hyphen or the dash.A numeric mode indicator also sets grade 1 mode. Grade 1 mode, when initiatedby numeric mode, is terminated by a space, hyphen or dash. Therefore whilegrade 1 mode is in effect, a grade 1 indicator is not required except for any oneof the lowercase letters a-j immediately following a digit, a full stop or a comma.(Note that Grade 1 mode, when initiated by numeric mode, is not terminated bythe minus sign, "-.)Last updated August 2014

1 General Principles31.3 Print SymbolsOne of the underlying design features of UEB is that each print symbol shouldhave one and only one braille equivalent. For example the vertical bar is used inprint to represent absolute value, conditional probability and the words "suchthat", to give just three examples. The same braille symbol should be used in allthese cases, and any rules for the use of the symbol in braille are independent ofthe subject area. If a print symbol is not defined in UEB, it can be representedeither using one of the seven transcriber defined print symbols in Section 11, orby using the transcriber defined shape symbols in Section 14.1.4 Format. "continuation indicator1.4.1 In print, mathematical expressions are sometimes embedded in the textand sometimes set apart. When an expression is set apart, the braille formatshould indicate this by suitable indentation, for example cell 3 with overruns in 5or cell 5 with overruns in 7. An embedded expression which does not fit on thecurrent braille line should only be divided if there is an obvious dividing point.Often it is better to move the whole expression to the next braille line.1.4.2 When dividing a mathematical expression, choice of a runover site shouldfollow mathematical structure: before comparison signsbefore operation signs (unless they are within one of the mathematicalunits below)before a mathematical unit such aso fractions (and within the fraction consider the numerator anddenominator as units)o functionso radicalso items with modifiers such as superscripts or barso shapes or arrowso anything enclosed in print or braille grouping symbolso a number and its abbreviation or coordinatesUsually the best place to break is before a comparison sign or an operation sign.Breaking between braille pages should be avoided.Last updated August 2014

1 General Principles41.4.3 When an expression will not fit on one braille line and has to be divided,the use of indentation as suggested in 1.4.1 should make it clear that the overrunis part of the same expression. However in the unlikely case where the twoportions could be read as two separate expressions the continuation indicator(dot 5) should be placed immediately after the last cell of the initial line.(a b c d e)(f g h i j) (1 2 3 4 5)(6 7 8 9 10) 600" A"6b"6c"6d"6e" " f"6g"6h"6i"6j" "7 " #a"6#b"6#c"6#d"6#e" "" #f"6#g"6#h"6#i"6#aj" "7 #fjj1.5 TypeformsIn mathematics, algebraic letters are frequently italicised as a distinction fromordinary text. It is generally not necessary to indicate this in braille. However,when bold or other typeface is used to distinguish different types of mathematicalletters or signs from ordinary algebraic letters, e.g. for vectors or matrices, thisdistinction should be retained in braille by using the appropriate typeformindicator. See Section 2.7 for the emphasis of individual digits within numbers.1.6 CapitalisationIn mathematics and science, strings of capital letters often occur, for example ina geometrical name, in a physics formula or in genetics. Such strings shouldalways be uncontracted. Capital word indicators (double caps) are normallyused. See Section 16 for advice on capital letters in chemical formulae. It ispreferable to also use this approach in genetics or other topics where there arefrequent changes of case within a sequence of letters.rectangle ABCDV IRTriangle RSTAB2AB, BC and ACIIIrdHHHhrectangle ,,abcd;,v "7 ,,ir,triangle ,,rst not ,,r/,,ab;9#b;,,ab1 ,,bc & ;,,ac,,iii,'rd,h,h,hhLast updated August 2014

1 General Principles1.7 Use of Grade 1 indicators. ;. ;;. ;;;. ;'. "" ;;;. "" ;'grade 1 symbol indicatorgrade 1 word indicatorgrade 1 passage indicatorgrade 1 passage terminatorgrade 1 passage indicator on a line of its owngrade 1 passage terminator on a line of its own1.7.1 Grade 1 indicators will not be needed for simple arithmetic problemsinvolving numbers, operation signs, numerical fractions and mixed numbers.Evaluate the following:3 2½ ,evaluate ! foll[ 3#c "- #b#a/b "71.7.2 Simple algebraic equations which include letters but no fraction orsuperscript indicators may need grade 1 symbol indicators where letters standalone or follow numbers. (See Section 1.2 for the underlying rules and Section3.2 for more examples.)y x 4c;y "7 x"6#d;cLast updated August 20145

1 General Principles1.7.3 More complex algebraic equations are best enclosed in grade 1 passageindicators. This will ensure that isolated letters and indicators such assuperscript, subscript, fractions, radicals, arrows and shapes are well definedwithout the need for grade 1 symbol indicators.Consider the following equation:3x 4y y² x²,3sid] ! foll[ equa;n3;;;#cx"-#dy"6y9#b "7 x9#b;'Note that this particular equation could also be written#cx"-#dy"6y9#b "7 x;9#bbecause the left hand side of the equation is in grade 1 mode following thenumeric indicator (see Section 1.2).Similarlyx2 2 x 11 x2(fraction: x squared plus 2x all over 1 x squared close fraction)can be safely written as;;;(x9#b"6#bx./#a"6x9#b) "7 #a;'but could also be written;;(x9#b"6#bx./#a"6x9#b) "7 #aSee Section 11.5 for more examples of the use of grade 1 passage indicators.1.7.4 If a complex algebraic expression does not include a comparison sign(such as an equals sign) then it is unlikely to include interior spaces in braille(see Section 1.1.2). In this case a grade 1 word indicator will be enough toensure that superscript, subscript, fractions, radicals, arrows and shapeindicators are well defined without the need for grade 1 symbol indicators.Evaluate( y x2 ) .,evaluate ;;%" y"-x9#b" 4See Section 7.3 for more examples of the use of grade 1 word indicators.Last updated August 20146

1 General Principles71.7.5 When entire worked examples or sets of exercises are enclosed in grade 1passage indicators, the grade 1 indicators can be preceded by the "use indicator"and placed on a line of their own.Solve the following quadratic equations:1. x² x 2 02. x² 4x 3 03. 2x² x 1,solve ! foll[ quadratic equa;ns3"" ;;;#a4 x9#b"-x"-#b "7 #j#b4 x9#b"-#dx"-#c "7 #j#c4 #bx9#b"-x "7 #a"" ;'1.7.6 When only a few contracted words are involved, the grade 1 passageindicator can be used to enclose entire worked examples and sets of exercises.In this situation any words occurring in the exercises will be written inuncontracted braille and isolated letters will not need letter signs. Where there ismore text involved it is better to stay in grade 2 and use grade 1 passage, wordor symbol indicators only as required.1.7.7 In the examples in this document, grade 2 mode is assumed to be in effect,and grade 1 indicators have been included according to the guidelines in thissection. Minimising the number of indicators must be balanced against reducingclutter within the expression itself. A grade 1 symbol indicator which occurs halfway through an expression may be more disruptive to the reader than a word orpassage indicator, even if these take up more cells. It is also important to use aconsistent approach when transcribing a particular text. Overall the focus shouldbe on mathematical clarity for the reader.Further guidance will be given when more feedback has been received fromstudents.Last updated August 2014

2 Numbers and Abbreviations82 Numbers and AbbreviationsRefer to Section 1.2 for a summary of the rules for Grade 1 mode and Numericmode as they apply to the brailling of numbers and letters in mathematics.The braille representation of numbers such as dates and times should reflect thepunctuation used in print.2.1 Whole numbers456#def3,000#c1jjj5 000 000#e"jjj"jjjCalling seat numbers 30-59.,call s1t numb]s #cj-#ei4In the 60's,9 ! #fj'sIn the 60s,9 ! #fjsIn the '60s,9 ! '#fjsPhone 09-537 0891,ph"o #ji-#ecg"jhiaFor negative numbers see 4.2.Last updated August 2014

2 Numbers and Abbreviations2.2 Decimals8.93#h4ic0.7#j4g.7#4gIs the number in the range 2-5.5?,is ! numb] 9 ! range #b-#e4e8.8 is a decimal fraction.#4h is a decimal frac;n4For recurring decimals see Section 12 (bars, dots etc. over and under)2.3 28#bjja /#e /#bh2001.5.28#bjja4e4bh28/5-31/5#bh /#e-#ca /#eLast updated August 20149

2 Numbers and Abbreviations2.4 Time5:30 pm#e3#cj pm5.30#e4cj08.00#jh4jj1300#acjj6-7 a.m.#f-#g a4m46:15-7:45#f3#ae-#g3#de2.5 Ordinal numbers1st#ast2nd or 2d#bnd or #b;d3rd or 3d#crd or #c;d4th#dth1er#a;erLast updated August 201410

2 Numbers and Abbreviations112.6 Roman NumeralsRoman numerals should be brailled as if they were normal letters using the rulesfor grade 1 mode. Note that "v" and "x" will have grade 1 indicators but "i" will not.Read parts I, II and V.,r1d "ps ,i1 ,,ii & ;,v4Answer questions i, vi and x.,answ] "qs i1 vi & ;x4CD;,,cd2.7 Emphasis of DigitsIf a typeform indicator applies to a digit or digits within a number, the numericindicator needs repeating after any typeform indicator. If the first digit is affectedthen the typeform indicator should be placed before the numeric indicator.678456784567845#fg 2#hde#fg 1#hde#fg 1#hd '#e67845678451#fgh '#de2#fghdeFor recurring decimals see Section 12 (bars, dots etc. over and under)2.8 Ancient Numeration systemsBraille symbols to represent numerals from other number systems may bedevised for each situation using transcriber defined print symbols. These shouldbe defined either on the special symbols page or in a transcriber's note. (Seeexample in Section 11.5.7)Last updated August 2014

2 Numbers and Abbreviations122.9 Hexadecimal numbersHexadecimal numbers occur in a computer setting and are made up of the digits0 to 9 and the letters A to F. They should be treated the same as any other stringof letters and numbers.Fatal exception 0E has occurred at 0028:C00082CD,fatal excep;n #j,e has o3urr at#jjbh3,c#jjjhb,,cd2.10 AbbreviationsThe following signs are used for special print symbols:. @c. @e. @f. @l. @s. @y. @n cent euro franc pound (sterling) dollar yen (Japan) naira (Nigeria). .0. j. 7. 77. , a%percent degree′foot or minute (shown as a prime sign)′′inch or second (shown as a double prime sign)Åangstrom (A with small circle above)Note that the Rand (South Africa) is written in print as a normal capital R sowould be brailled as such.Note that the foot or minute may be shown in print by an apostrophe (') and theinch or second by a nondirectional double quote ("). This usage can be followedin braille.Last updated August 2014

2 Numbers and Abbreviations13Follow print for order, spacing, capitalisation and punctuation of abbreviations. (Ifit is unclear in print whether there is a space between a number and its unit, or ifprint spacing is inconsistent, then it is recommended that a space is inserted inthe braille.)Where should I write the dollar sign, US or US?,": %d ,i write ! doll sign1,,us@s or @s,,us830 cents can be written as 0.30, 30c or 30 .#cj c5ts c 2 writt5 z @s#j4cj1#cj;c or #cj@c4In South Africa, this would cost R13.51.,9 ,s\? ,africa1 ? wd co/ ,r#ac4ea4Before decimalisation, 1.75 was 1 15s so half of it was 17s 6d or 17/6.,2f decimalisa;n1 @l#a4ge 0@l#a #aes s half ( x 0 #ags #f;d or#ag /#f4Half a yard is 1 ft 6 in or 1′ 6′′ which is about 45 cm or 0.45 m.,half a y d is #a ft #f 9 or #a7#f77 : is ab #de cm or #j4de ;m41 L of water weighs 1000 g which is about 2 lbs 4 oz.#a ;,l ( wat] wei s #ajjj ;g :is ab #b lbs #d oz4Is the speed limit 30 mph or 50 km/h?,is ! spe limit #cj mph or #ejkm /h8Water freezes at 0 C or 32 F.,wat] freezes at #j j,c or #cb j,f4Last updated August 2014

2 Numbers and Abbreviations14To decrease by 15% multiply by 0.85.,to decr1se by #ae.0 multiply by#j4he4Add 1 can of beans, 1 c of flour, 2 T of oil and 1 tsp of baking powder.,add #a c ( b1ns1 #a ;c ( fl\r1 #b;,t ( oil & #a tsp ( bak p[d]4There are 360 in a revolution, 60′ in a degree and 60′′ in a minute.,"! e #cfj j 9 a revolu;n1 #fj7 9a degree & #fj77 9 a m9ute4One complete orbit lasts 2yr 5m 15d 7h 17min and 45s.,"o complete orbit la/s #byr #em#ae;d #g;h #agmin & #des4A 6 V battery will cause a current of 3 A to flow through a resistance of 2 Ω.,a #f ;,v batt]y w cause a curr5t (#c ,a to fl[ "? a resi/.e ( #b ,.w4The reading was 15 mHz.,! r1d 0 #ae m,hz4The pattern says k4 p1 sl1 k1 psso.,! Patt]n says k#d p#a sl#a k#apsso41Å 1µ10, 000#a , a "7 #a/aj1jjj .mLast updated August 2014

3 Signs of Operation, Comparison and Omission153 Signs of Operation, Comparison and OmissionOperation signs:. "6. ". "8. "/. 6. . "4 plus–minus (when distinguished from hyphen) times (multiplication cross) divided by (horizontal line between dots) plus or minus (plus over minus) minus or plus (minus over plus) multiplication dotComparison signs:. "7. @ . @ . @ . @ . "7@:. 9. 9 equals less than, or opening angle bracket greater than, or closing angle bracket less than or equal to greater than or equal to not equal to (line through an equals sign)approximately equal to (tilde over horizontal line) approximately equal to (tilde over tilde)Less common signs of comparison: is much less than. .@ . .@ . " 9. ."7. "7 is much greater than tilde over equals sign (approximately equal) . equals sign dotted above and below (approximately equal)equals sign with bump in top bar (difference between orapproximately equal)equivalent to (three horizontal lines). "7 is proportional to:ratio sign (represented by a colon as in print) Ratio. 3Last updated August 2014

3 Signs of Operation, Comparison and Omission(see also Section 11 for signs of operation and comparison used in set theory,group theory and logic)3.1 ExamplesIn most of the examples below, operation signs are unspaced from precedingand following terms but comparison signs are spaced. The first two examplesshow the use of extra space for the younger learner. Follow the guidelines inSpacing (Section 1.1.2).3 5 8#c "6 #e "7 #h8 5 3#h "- #e "7 #c3 5 5 3 15#c"8#e "7 #e"8#c "7 #ae2 cm 4 cm 6 cm#b cm"6#d cm "7 #f cm200g 5 1kg#

A braille symbol may have both a grade 1 meaning and a contraction (i.e. grade 2) meaning. Some symbols may also have a numeric meaning. A grade 1 indicator is used to set grade 1 mode when the grade 1 meaning of a symbol could

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