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Nemeth Braille Code for Mathematics and Science
1972 Revision
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RULE XIV--MODIFIERS

Modification Indicators
  • Directly Over
    • First order
      <
      verbose
      OverScript
      Listen to MathSpeak
    • Second order
      <<
      verbose
      OverOverScript
      Listen to MathSpeak
  • Directly Under
    • First order
      %
      verbose
      UnderScript
      Listen to MathSpeak
    • Second order
      %%
      verbose
      UnderUnderScript
      Listen to MathSpeak
  • Multipurpose
    "
  • Superposition
    @
    verbose
    Overstrike
    Listen to MathSpeak
  • Termination
    ]
    verbose
    EndScripts
    Listen to MathSpeak
Modifiers
  • Arc
    • Concave upward
      $a
      Example as described in the content
      verbose
      concave-arc
      Listen to MathSpeak
    • Concave downward
      $'
      Example as described in the content
      verbose
      convex-arc
      Listen to MathSpeak
  • Arrow
    • Barbed at both ends
      $[33o
      Example as described in the content
      verbose
      two-way-arrow
      Listen to MathSpeak
    • Barbed at left
      $[33
      Example as described in the content
      verbose
      left-arrow
      Listen to MathSpeak
    • Barbed at left and dotted at right
      $[33*
      Example as described in the content
      verbose
      left-arrow-dotted-at-right
      Listen to MathSpeak
    • Barbed at right
      • Contracted form
        $o
        Example as described in the content
        verbose
        right-arrow
        Listen to MathSpeak
      • Uncontracted form
        $33o
        Example as described in the content
        verbose
        right-arrow
        Listen to MathSpeak
    • Dotted at both ends
      $*33*
      Example as described in the content
      verbose
      arrow-dotted-at-both-ends
      Listen to MathSpeak
    • Dotted at left (no barb)
      $*33
      Example as described in the content
      verbose
      arrow-dotted-at-left
      Listen to MathSpeak
    • Dotted at left and barbed at right
      $*33o
      Example as described in the content
      verbose
      right-arrow-dotted-at-left
      Listen to MathSpeak
    • Dotted at right (no barb)
      $33*
      Example as described in the content
      verbose
      arrow-dotted-at-right
      Listen to MathSpeak
    • Hollow dot at both ends
      $.*33.*
      Example as described in the content
      verbose
      arrow-with-hollow-dots-at-both-ends
      Listen to MathSpeak
    • Hollow dot at left (no barb)
      $.*33
      Example as described in the content
      verbose
      arrow-with-hollow-dot-at-left
      Listen to MathSpeak
    • Hollow dot at left and barbed at right
      $.*33o
      Example as described in the content
      verbose
      right-arrow-with-hollow-dot-at-left
      Listen to MathSpeak
    • Hollow dot at right and barbed at left
      $[33.*
      Example as described in the content
      verbose
      left-arrow-with-hollow-dot-at-right
      Listen to MathSpeak
    • Hollow dot at right (no barb)
      $33.*
      Example as described in the content
      verbose
      arrow-with-hollow-dot-at-right
      Listen to MathSpeak
  • Bar
    • Horizontal (macron)
      :
      Example as described in the content
      verbose
      bar
      Listen to MathSpeak
    • Vertical
      \
      Example as described in the content
      verbose
      vertical-line
      Listen to MathSpeak
  • Caret (circumflex)
    _<
    Example as described in the content
    verbose
    caret
    Listen to MathSpeak
    • Inverted
      _%
      Example as described in the content
      verbose
      inverted-caret
      Listen to MathSpeak
    • Left-pointing
      ;<
      Example as described in the content
      verbose
      less-than
      Listen to MathSpeak
    • Right-pointing
      ;%
      Example as described in the content
      verbose
      greater-than
      Listen to MathSpeak
  • Dot
    *
    Example as described in the content
    verbose
    dot
    Listen to MathSpeak
  • Hollow dot
    .*
    Example as described in the content
    verbose
    hollow-dot
    Listen to MathSpeak
  • Question Mark
    _8
    Example as described in the content
    verbose
    question-mark
    Listen to MathSpeak
  • Tilde
    • Extended
      @,:
      Example as described in the content
      verbose
      extended-tilde
      Listen to MathSpeak
    • Simple
      @:
      Example as described in the content
      verbose
      tilde
      Listen to MathSpeak
  • Triangle (equilateral)
    $t
    Example as described in the content
    verbose
    triangle
    Listen to MathSpeak

§85. Modifiers: A modifier is a superscript or subscript which occupies, respectively, a position directly over or directly under the sign to which it applies. The modifiers in the list at the beginning of this rule are those most commonly used, but other modifiers must be treated in the same manner.

§86. Modified Expressions:

a. The Five-Step Rule for Transcribing Modified Expressions: The components of a modified expression must appear in the following order:

i. Multipurpose indicator "
ii. Expression being modified.
iii. Directly-over indicator < or directly-under indicator %
iv. Modifier.
v.Termination indicator ]
These five components may never be separated from each other by transition to another braille line. The termination indicator terminates only the modified expression; it does not affect the level at which the modified expression occurs.

b. When the expression being modified is a single digit or a letter, lower-case or capitalized, from any alphabet, and in any type form, and when the modifier is the horizontal bar directly above such a single digit or letter, the digit or letter, followed by the bar, serves to express the modification. This construction should be regarded as a contracted form of expression and must be used whenever applicable. If the modification includes a superscript, subscript, or prime, the five-step rule of a above must be followed. The five-step rule may be used in conjunction with the contracted form without fear of confusion.

§87. Modifiers of Higher Order:

a. A modifier of the second order must be preceded by the second-order directly-over or directly-under indicator, and a modifier of the third order must be preceded by the third-order directly-over or directly-under indicator. The termination indicator, however, must be used only once, after the last modifier symbol.

b. A modifier of order higher than the third must be treated in the manner suggested in a above.

c. A modifier, to be of order higher than the first, must be associated with the same expression as a modifier of lower order. In §86b(11) above, the long bar is not a modifier of the second order because no modifier of the first order is associated with teh same expression as the long bar.

§88. Simultaneous Modifiers: When a mathematical expression is simultaneously modified above and below, the modifier below must be indicated first. The termination indicator, however, must be used only once, after the last modifier symbol. If the modifiers involved are of order higher than the first, they are treated as described in §87.

§89. Parallel Horizontal Bars: Parallel horizontal bars must not be regarded as the equals sign or the identity sign when they occur above or below a mathematical expression other than a comparison sign. Furthermore, the bar which is more remote from the mathematical expression being modified must not be regarded as a modifier of second or third order; the double or triple bar must be regarded as a single modifier.

§90. Binomial Coefficient: The two expressions which constitute a binomial coefficient must be separated by the directly-under indicator. The expression which follows the opening parenthesis and precedes the directly-under indicator corresponds to the upper sign in the binomial coefficient; the expression which follows the directly-under indicator and precedes the closing parenthesis corresponds to the lower sign of the binomial coefficient.

§91. Modified Expressions in Superscripts and Subscripts: If a modified expression is part or all of a right superscript or subscript, the multipurpose indicator must be preceded by the appropriate level indicator. This will automatically be the case if the modified expression occurs at the beginning of the superscript or subscript; but the appropriate level indicator must be restated if the modified expression occurs at an interior position of the superscript or subscript. If the contracted form for a modified expression is used so that the multipurpose indicator does not appear, the appropriate level indicator must not be restated.

§92. Plural Modified Expressions: (See §39).

§93. Modification by Superposition: When one sign modifies another by superposition, in deciding which is the basic sign and which is superposed sign, the following hierarchy, in descending order, should be used as a guide:


A sign belonging to a category lower on the list must be regarded as superposed on a sign higher on the list, and the superposition transcribed accordingly. If two signs belong to the same category, it is permissible to represent the superposition in either order, provided that the same order is used consistently throughout the entire transcription. The components of a sign compounded by superposition must be joined by the superposition indicator and transcribed unspaced, and without transition to another braille line. The termination indicator must follow the second component. (For other examples, see "Comparison Signs Compounded by Superposition" pages 140-141, and 143.)

§94. Interior Modifiers with Signs of Shape: See §111.

§95. Arc:

§96. Arrows: Arrows must not be regarded as modifiers when they occur directly over or directly under a comparison sign. In that event, they become a component of a sign of comparison compounded vertically.

§97. Horizontal Bar:

a. The horizontal bar must not be regarded as a modifier when it occurs directly over or directly under a comparison sign. In that event, it becomes a component of a sign of comparison compounded vertically (see §147). When the horizontal bar is itself modified by a dot under it or a caret directly over or under it, the combination is a modified sign of comparison (see §146). When the horizontal bar is itself modified by a dot over it, the combination is a sign of operation.

b. The horizontal bar is often used to indicate the recurrence of one or more digits in a decimal numeral by placing it over the digits which recur.

c. When the horizontal bar occurs over or under the integral sign, or over or under the abbreviated or unabbreviated function name for limit, the bar must not be treated as a modifier (see §171 and §118, respectively).

§98. Caret:

§99. Dot:

a. The dot is frequently used to indicate the recurrence of one or more digits in a decimal numeral. When used for this purpose, a dot is usually placed in print over each digit of the recurring sequence. In braille, however, only a single dot must be used as a modifier.

b. Although there is theoretically no limit to the number of dots which may be placed over or under a single mathematical expression, in practice the number rarely exceeds three dots. However, as many dots must be used in the transcription as are present in the printed text, except in the case of recurring decimals as in a.

§100. Hollow Dot:

§101. Question Mark:

§102. Tilde: The tilde, simple or extended, must not be regarded as a modifier when it occurs directly over or under a comparison sign. In that event, it becomes a component of a sign of comparison compounded vertically (see sect;147). When the tilde, simple or extended, is itself modified by a dot or a caret directly over or under it, the combination is a modified sign of comparison (see §146).