RULE XII--FRACTIONS

• Simple
• Opening
  ?

  verbose

  StartFraction

• Closing
  #

  verbose

  EndFraction

• Complex
• Opening
  ,?

  verbose

  StartStartFraction

• Closing
  ,#

  verbose

  EndEndFraction

• Hypercomplex
• Opening
  ,,?

  verbose

  StartStartStartFraction

• Closing
  ,,#

  verbose

  EndEndEndFraction

• Fractional Part of a Mixed Number
• Opening
  _?

  verbose

  StartFraction

• Closing
  _#

  verbose

  EndFraction

• Used with Simple-Fraction Indicators
• Diagonal line or slash
  _/

  	verbose	Over

• Horizontal
  /

  	verbose	Over

• Used with the Fractional Part of a Mixed Number
• Diagonal line or slash
  _/

  	verbose	Over

• Horizontal
  /

  	verbose	Over

• Used with Complex-Fraction Indicators
• Diagonal line or slash
  ,_/

  	verbose	OverOver

• Horizontal
  ,/

  	verbose	OverOver

• Used with Hypercomplex-Fraction Indicators
• Horizontal
  ,,/

  	verbose	OverOverOver

• Used with a Spatial Arrangement
• Horizontal (varying in length)
  33333

  	verbose	OverOverOver

§61. Simple Fractions: For the purposes of this Code, a simple Fraction is one whose numerator and denominator contain no Fractions except possibly at the superscript or subscript level.

§62. Use of Simple-Fraction Indicators:

a. Simple-Fraction indicators must be used, except in the case of mixed numbers, to enclose a simple Fraction whose numerator and denominator are separated by a horizontal Fraction line in ink print.

(1)

?1/3#

verbose semantics

one-third

(2)

x^?1/3#

verbose semantics

x Superscript one-half

(3)

?a+b/c#

verbose semantics

StartFraction a plus b Over c EndFraction

(4)

?x^?1/2#"/2#

verbose semantics

StartFraction x Superscript one-half Baseline Over 2 EndFraction

(5)

rate .k ?distance/time#

verbose semantics

rate equals StartFraction distance Over time EndFraction

b. Simple-Fraction indicators must be used to enclose a simple Fraction whose numerator and denominator are separated by a diagonal line in ink print, when the expressions on either side of the diagonal line appear at different levels relative to it, or in different type size than is normal for the purpose for which these expressions are used.

(1)

?a+b_/c+d#

(in ink print, the numerator is written near the top of the diagonal line and the denominator is written near the bottom)

verbose semantics

StartFraction a plus b Over c plus d EndFraction

(2)

#3?x_/y#

(in ink print the 3, x, and y are at the same level, but the x and y are in smaller type than the 3)

verbose semantics

3 StartFraction x Over y EndFraction

§63. Non-Use of Simple-Fraction Indicators:

a. Simple-Fraction indicators must not be used to enclose the Fractional part of a mixed number.

(1)

#4_?3/8_#

verbose semantics

4 and three-eighths

(2)

#2_?3_/4_#x

verbose semantics

2 and three-fourths x

b. Simple-Fraction indicators must not be used to enclose a simple Fraction whose numerator and denominator are separated by a diagonal line in ink print when the expressions on either side of the diagonal line appear at the same level relative to it, or are of the

same type size as the surrounding Mathematical text. Sometimes the expressions on either side of the diagonal line are not the terms of a Fraction at all. Even when they are, the transcriber cannot always be certain of where the Fraction begins or ends. Accordingly, it is better to avoid the use of indicators altogether in these cases and permit the braille reader to make a judgment based on the same information that is available to the sighted reader.

(1)

#1_/3

verbose semantics

one-third

(2)

x^1_/2

(in ink print, 1 and 2 are at the same level; although the 1 and 2 are in smaller type, they are of normal size for printing superscripts)

verbose semantics

x Superscript one-half

(3)

x^?1/2#"_/2

(in ink print, the x and 2 are at the same level and are of normal size for printing base-line signs)

verbose semantics

x Superscript one-half Baseline slash 2

(4)

x^1_/2"_/7

(in ink print, 1 and 2 are at the same level and x and 7 are at the same level; each pair of signs is of normal size for printing at its respective level)

verbose semantics

x Superscript one-half Baseline slash 7

(5)

a+b_/c+d

(in ink print, all letters are of normal size and at the same level on either side of a diagonal line)

verbose semantics

a plus b slash c plus d

(6)

(a+b)_/(c+d)

(in ink print, all letters are of normal size and at the same level on either side of a diagonal line)

verbose semantics

left-parenthesis a plus b right-parenthesis slash left-parenthesis c plus d right-parenthesis

(7)

,i_/(n)

(the expressions on either side of the diagonal line are not the terms of a Fraction)

verbose semantics

Upper I slash left-parenthesis n right-parenthesis

(8)

#1_/31_/70

(the expression represents a date)

verbose semantics

1 slash 31 slash 70

§64. Mixed Numbers: For the purposes of this Code, a mixed number is an expression which begins with a numeral and is followed, usually in smaller type, by a simple Fraction whose numerator and denominator are both numerals. The Fraction line of this simple Fraction may be either horizontal or diagonal in ink print. The mixed-number indicators must be used to enclose the Fractional part of a mixed number. An expression is not a mixed number if it contains any letter, even though such an expression is of the same form as a mixed number in every other respect.

(1)

#4_?3/8_#

verbose semantics

4 and three-eighths

(2)

#4_?3_/3_#

verbose semantics

4 and three-eighths

(3)

x?3/8#

(this is not a mixed number; Fraction in smaller type than the x)

verbose semantics

x three-eighths

(4)

x?3_/8#

(this is not a mixed number; Fraction in smaller type than the x)

verbose semantics

x three-eighths

(5)

#3?1_/y#

(this is not a mixed number; Fraction in smaller type than the 3)

verbose semantics

3 StartFraction 1 Over y EndFraction

§65. Complex Fractions: For the purposes of this Code, a complex Fraction is one whose numerator, denominator, or both, contains at least one simple Fraction. A Fraction is not a complex Fraction if the only simple Fractions it contains are at the superscript or subscript level.

§66. Use of Complex-Fraction Indicators: Complex-Fraction indicators must be used to enclose a complex Fraction.

(1)

,??3/8#,/5,#

verbose semantics

StartFraction three-eighths Over 5 EndFraction

(2)

,?1_/2,/2_?2/3_#,#

verbose semantics

StartFraction one-half Over 2 and two-thirds EndFraction

(3)

,?2_/3,/3_/2,#

verbose semantics

StartFraction two-thirds Over three-halves EndFraction

(4)

,?5,/4_?3/8_#,#

verbose semantics

StartFraction 5 Over 4 and three-eighths EndFraction

(5)

,?3_/4,/5,#

verbose semantics

StartFraction three-fourths Over 5 EndFraction

(6)

,??1/2#,_/?3/4#,#

verbose semantics

StartFraction one-half Over three-fourths EndFraction

§67. Hypercomplex Fractions: For the purposes of this Code, a hypercomplex Fraction is one whose numerator, denominator, or both, contains at least one complex Fraction. A Fraction is not a hypercomplex Fraction if the only complex Fractions it contains are at the superscript or subscript level.

(1)

?a/b^,??3/4#,/?5/6#,#"#

(this is not a hypercomplex Fraction)

verbose semantics

StartFraction a Over b Superscript StartFraction three-fourths Over five-sixths EndFraction Baseline EndFraction

§68. Use of Hypercomplex-Fraction Indicators:

a. Hypercomplex-Fraction indicators must be used to enclose a hypercomplex Fraction. The use of a linear arrangement within a spatial arrangement is preferable to an arrangement which is entirely linear or entirely spatial.

(1)

   ,?1_?1/4_#,/1_?3/5_#,#   
,,?3333333333333333333333,,#
             #5             

(preferred method of transcribing a hypercomplex Fraction)

verbose semantics

StartStartFraction StartFraction 1 and one-fourth Over 1 and three-fifths EndFraction OverOver 5 EndEndFraction

(2)

   ,?(1-x)?d/dx#(2x)
    -#2x?d/dx#(1-x)
     ,/(1-x)^2",#
,,?333333333333333333,,#
    #1+(?2x/1-x#)^2

(preferred method of transcribing a hypercomplex Fraction)

verbose semantics

StartStartStartFraction StartStartFraction left-parenthesis 1 minus x right-parenthesis StartFraction d Over d x EndFraction left-parenthesis 2 x right-parenthesis minus 2 x StartFraction d Over d x EndFraction left-parenthesis 1 minus x right-parenthesis OverOver left-parenthesis 1 minus x right-parenthesis squared EndEndFraction OverOverOver 1 plus left-parenthesis StartFraction 2 x Over 1 minus x EndFraction right-parenthesis squared EndEndEndFraction

(3)

           ;d         ;d
     (1-x)?33#(2x)-2x?33#(1-x)
           ;x         ;x
   ,?3333333333333333333333333,#
              (1-x)^2
,,?33333333333333333333333333333,,#
                #2x
           #1+(?3333#)^2
                #1-x

(complete spatial arrangement)

verbose semantics

StartStartStartFraction StartStartFraction left-parenthesis 1 minus x right-parenthesis StartFraction d Over d x EndFraction left-parenthesis 2 x right-parenthesis minus 2 x StartFraction d Over d x EndFraction left-parenthesis 1 minus x right-parenthesis OverOver left-parenthesis 1 minus x right-parenthesis squared EndEndFraction OverOverOver 1 plus left-parenthesis StartFraction 2 x Over 1 minus x EndFraction right-parenthesis squared EndEndEndFraction

(4)

,,?,?(1-x)?d/dx#(2x)-2x?d/dx#(1-x) 
  ,/(1-x)^2",#,,/1+(?2x/1-x#)^2",,#

(complete linear arrangement)

verbose semantics

StartStartStartFraction StartStartFraction left-parenthesis 1 minus x right-parenthesis StartFraction d Over d x EndFraction left-parenthesis 2 x right-parenthesis minus 2 x StartFraction d Over d x EndFraction left-parenthesis 1 minus x right-parenthesis OverOver left-parenthesis 1 minus x right-parenthesis squared EndEndFraction OverOverOver 1 plus left-parenthesis StartFraction 2 x Over 1 minus x EndFraction right-parenthesis squared EndEndEndFraction

b. Hypercomplex Fractions of higher order may be transcribed in the manner suggested by a above. It is only necessary to use dot 6 the proper number of times before the Fraction indicators and their matching Fraction line.

§69. Continued Fractions: A continued Fraction is one in which each denominator, except possibly the last one, is the sum of a whole number and a Fraction. A spatial arrangement must be used for each Fraction. In this case, each Fraction line must have proportionately the length shown in print, and Fraction indicators of any kind must not be used with a continued Fraction.

(1)

               #1
>2] .k #1+333333333333
                #1
          #2+333333333
                  #1
             #2+333333
                   #1
                #2+333
                   #x+ '''_4

verbose semantics

StartRoot 2 EndRoot equals 1 plus ContinuedFraction 1 Over 2 plus StartFraction 1 Over 2 plus StartFraction 1 Over 2 plus StartFraction 1 Over 2 plus ellipsis

§70. Spatial Arrangement for Fractions:

a. Whenever a Fraction is transcribed spatially, all Fraction indicators, except for continued Fractions, must be shown, and each Fraction line must have precisely the length necessary to cOver the longest expression to which it applies, and the terms of the Fraction

must be centered on their Fraction lines. The runOver of an expression which is too long to be centered on the Fraction line which applies to it may be effected at suitable places in accordance with the rules for runOvers, but each portion of the divided expression must be centered on the Fraction line to which the expression, as a whole, applies.

b. In general, the linear arrangement for Fractions must be used when no expressly forbidden in the case of continued Fractions. However, when Fractional notation is first presented to the reader, as in the lower grades, or when there is any other special need, any Fraction may be represented spatially.

(1)

 #1+2
?3333#
 #2+4

verbose semantics

StartFraction 1 plus 2 Over 2 plus 4 EndFraction

(2)

 ;x
?33#
 ;y

verbose semantics

StartFraction x Over y EndFraction

(3)

         4t.e 
rate .k ?3333#
          "t

verbose semantics

rate equals StartFraction distance Over time EndFraction

(4)

 #5280 ft  #60 mi  #1 hr    #1 m9
?33333333#?333333#?333333#?3333333#
  #1 mi    #1 hr   #60 m9  #60 sec

      #88 ft
  .k ?333333# .k #88 ft_/sec4
      #1 sec

verbose semantics

StartFraction 5280 feet Over 1 miles EndFraction StartFraction 60 miles Over 1 hours EndFraction StartFraction 1 hours Over 60 minutes EndFraction StartFraction 1 minutes Over 60 seconds EndFraction equals StartFraction 88 feet Over 1 seconds EndFraction equals 88 feet slash seconds period

c. For spatial arrangement of Fractions in connection with cancellation see §60.

d. For spatial arrangement of hypercomplex Fractions see §68.

e. For spatial arrangements of continued Fractions see §69.