RULE XIV--MODIFIERS

• Directly Over
• First order
  <

  verbose

  OverScript

• Second order
  <<

  verbose

  OverOverScript

• Directly Under
• First order
  %

  verbose

  UnderScript

• Second order
  %%

  verbose

  UnderUnderScript

• Multipurpose
  "
• Superposition
  @

  verbose

  Overstrike

• Termination
  ]

  verbose

  EndScripts

• Arc
• Concave upward
  $a

  	verbose	concave-arc

• Concave downward
  $'

  	verbose	convex-arc

• Arrow
• Barbed at both ends
  $[33o

  	verbose	two-way-arrow

• Barbed at left
  $[33

  	verbose	left-arrow

• Barbed at left and dotted at right
  $[33*

  	verbose	left-arrow-dotted-at-right

• Barbed at right
• Contracted form
  $o

  	verbose	right-arrow

• Uncontracted form
  $33o

  	verbose	right-arrow

• Dotted at both ends
  $*33*

  	verbose	arrow-dotted-at-both-ends

• Dotted at left (no barb)
  $*33

  	verbose	arrow-dotted-at-left

• Dotted at left and barbed at right
  $*33o

  	verbose	right-arrow-dotted-at-left

• Dotted at right (no barb)
  $33*

  	verbose	arrow-dotted-at-right

• Hollow dot at both ends
  $.*33.*

  	verbose	arrow-with-hollow-dots-at-both-ends

• Hollow dot at left (no barb)
  $.*33

  	verbose	arrow-with-hollow-dot-at-left

• Hollow dot at left and barbed at right
  $.*33o

  	verbose	right-arrow-with-hollow-dot-at-left

• Hollow dot at right and barbed at left
  $[33.*

  	verbose	left-arrow-with-hollow-dot-at-right

• Hollow dot at right (no barb)
  $33.*

  	verbose	arrow-with-hollow-dot-at-right

• Bar
• Horizontal (macron)
  :

  	verbose	bar

• Vertical
  \

  	verbose	vertical-line

• Caret (circumflex)
  _<

  	verbose	caret

• Inverted
  _%

  	verbose	inverted-caret

• Left-pointing
  ;<

  	verbose	less-than

• Right-pointing
  ;%

  	verbose	greater-than

• Dot
  *

  	verbose	dot

• Hollow dot
  .*

  	verbose	hollow-dot

• Question Mark
  _8

  	verbose	question-mark

• Tilde
• Extended
  @,:

  	verbose	extended-tilde

• Simple
  @:

  	verbose	tilde

• Triangle (equilateral)
  $t

  	verbose	triangle

§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:

(1)

"x%:]

(x with subscribed bar)

verbose semantics

x under-bar

(2)

"x+y<:]

(x plus y, with superscribed bar)

verbose semantics

ModifyingAbove x plus y with bar

(3)

"lim%x $o #0] f(x)

(the limit, as x approaches 0, of f of x)

verbose semantics

limit Underscript x right-arrow 0 EndScripts f left-parenthesis x right-parenthesis

(4)

"x^2"<:]

(x squared, with a superscribed bar)

verbose semantics

ModifyingAbove x squared with bar

(5)

"x'<:]

(x primed, with superscribed bar)

verbose semantics

ModifyingAbove x prime with bar

(6)

"x1<:]

(x sub 1, with superscribed bar)

verbose semantics

ModifyingAbove x 1 with bar

(7)

"x;n"<:]

(x sub n, with superscribed bar)

verbose semantics

ModifyingAbove x Subscript n Baseline with bar

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.

(1)

x:

(x with superscribed bar)

verbose semantics

x over-bar

(2)

x:+y:

(x with superscribed bar plus y with superscribed bar)

verbose semantics

x over-bar plus y over-bar

(3)

x:y

(x with superscribed bar, times y)

verbose semantics

x over-bar y

(4)

xy:z

(x, times y with superscribed bar, times z)

verbose semantics

x y over-bar z

(5)

x:^2

(x with superscribed bar squared)

verbose semantics

x over-bar squared

(6)

x:'

(x with superscribed bar primed)

verbose semantics

x over-bar prime

(7)

x:;1

(x with superscribed bar, sub 1)

verbose semantics

x over-bar Subscript 1

(8)

x:;n

(x with superscribed bar, sub n)

verbose semantics

x over-bar Subscript n

(9)

_;,z:

(boldface capital Z with superscribed bar)

verbose semantics

Bold Upper Z over-bar

(10)

#3.54:

(3.54, bar superscribed to the 4)

verbose semantics

3.5 ModifyingAbove 4 with bar

(11)

"(a:_;,a+b:_;,b)<:]

(a with superscribed bar times boldface A plus b with superscribed bar times boldface capitalized B, the whole expression with superscribed bar)

verbose semantics

ModifyingAbove Above left-parenthesis a over-bar Bold Upper A plus b over-bar Bold Upper B right-parenthesis with bar

(12)

,a;x:

(A with a right subscript of x with superscribed bar)

verbose semantics

Upper A Subscript x over-bar

(13)

,a;x:+y:

(A with a right subscript of x with superscribed bar plus y with superscribed bar)

verbose semantics

Upper A Subscript x over-bar plus y over-bar

(14)

e^ax:

(e with a right superscript of a times x with a superscribed bar)

verbose semantics

e Superscript a x over-bar

(15)

x:_'s

(the plural of x with superscribed bar)

verbose semantics

x over-bar apostrophe s

§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.

(1)

"x+y<:<<a .k #3]

(x plus y superscribed by a bar, which in turn is superscribed by a equals 3)

verbose semantics

ModifyingAbove x plus y with bar Overscript a equals 3 EndScripts

(2)

"x+y%:%%a .k #3%%%b .k #2]

(x plus y subscribed by a bar, which in turn is subscribed by a=3, which is in turn subscribed by b equals 2)

verbose semantics

ModifyingBelow x plus y With bar Underscript a equals 3 UnderUnderscript b equals 2 EndScripts

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.

(1)

"x+y%:<:]

(x plus y, with subscribed and superscribed bars)

verbose semantics

ModifyingAbove ModifyingBelow x plus y With bar with bar

(2)

".,s%n .k #1<,=]?1/2^n"# .k #1

(the Greek capitalized sigma with subscribed n equals 1 and superscribed infinity sign)

verbose semantics

sigma-summation Underscript n equals 1 Overscript infinity EndScripts StartFraction 1 Over 2 Superscript n Baseline EndFraction equals 1

(3)

"x+y%:%%a .k #3<:<<b .k #2]

(x plus y subscribed by a bar which is in turn subscribed by a equals 3; superscribed by a bar which is in turn superscribed by b equals 2)

verbose semantics

ModifyingAbove ModifyingBelow x plus y With bar with bar Underscript a equals 3 Overscript b equals 2 EndScripts

§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.

(1)

"x<::]

(x superscribed by double horizontal bars)

verbose semantics

x over-bar over-bar

(2)

"x%::<::]

(x subscribed and superscribed by double horizontal bars)

verbose semantics

x under-bar under-bar over-bar over-bar

(3)

"x%::<:]

(x subscribed by double horizontal bars and superscribed by single horizontal bar)

verbose semantics

x under-bar under-bar over-bar

(4)

"x%:::]

(x subscribed by triple horizontal bars)

verbose semantics

x under-bar under-bar under-bar

§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.

(1)

(n%k)

(the binomial coefficient with n as the upper sign and k as the lower sign)

verbose semantics

StartBinomial n Choose k EndBinomial

(2)

(g;j"%a;j")

(the binomial coefficient with g sub j as the upper sign and a sub j as the lower sign)

verbose semantics

StartBinomial g Subscript j Choose a Subscript j EndBinomial

§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.

(1)

,a;"x<@:]

(A carries a subscript of x with superscribed tilde)

verbose semantics

Upper A Subscript x over-tilde

(2)

,a;"x<@:]+;"y<@:]

(A carries a subscript of x with superscribed tilde plus y with superscribed tilde; the subscript level after the plus sign must be restated before the multipurpose indicator)

verbose semantics

Upper A Subscript x over-tilde plus y over-tilde

(3)

,a;x:+y:

(A carries a subscript of x with superscribed bar plus y with superscribed bar)

verbose semantics

Upper A Subscript x over-bar plus y over-bar

§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:

(1)

!@$r]

(integral sign with superposed rectangle)

verbose semantics

integral superposed rectangle

(2)

:@$4]

(horizontal bar with superposed square)

verbose semantics

CrossOut square EndCrossOut

(3)

*@.K]

(dot between bars of equals sign)

verbose semantics

dot superposed equals

(4)

.k@_"k]

(equals sign with superposed inclusion sign)

verbose semantics

equals superposed subset

(5)

_"k@.k]

(inclusion sign with superposed equals sign)

verbose semantics

subset superposed equals

(6)

$[@$a]

(angle with superposed arc)

verbose semantics

angle superposed arc

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

§95. Arc:

(1)

",a%$a]

(A with subscribed arc concave upward)

verbose semantics

ModifyingBelow Upper A With convex-arc

(2)

",a,b<$a]

(AB with superscribed arc concave upward)

verbose semantics

ModifyingAbove Upper A Upper B with convex-arc

(3)

",a%$']

(A with subscribed arc concave downward)

verbose semantics

ModifyingBelow Upper A With concave-arc

(4)

",a,b%$']

(AB with subscribed arc concave downward)

verbose semantics

ModifyingBelow Upper A Upper B With concave-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.

(1)

",a,b<$o]

(AB with superscribed arrow barbed at right)

verbose semantics

ModifyingAbove Upper A Upper B with right-arrow

(2)

",a,b<$[33]

(AB with superscribed arrow barbed at left)

verbose semantics

ModifyingAbove Upper A Upper B with left-arrow

(3)

",a,b<$[33o]

(AB with superscribed arrow barbed at both ends)

verbose semantics

ModifyingAbove Upper A Upper B with two-way-arrow

(4)

",a,b<$[33*]

(AB with superscribed arrow barbed at the left and dotted at the right)

verbose semantics

ModifyingAbove Upper A Upper B with left arrow dotted at right

(5)

",a,b<$*33*]

(AB with superscribed arrow dotted at both ends)

verbose semantics

ModifyingAbove Upper A Upper B with arrow dotted at both ends

(6)

",a,b<$.*33.*]

(AB with superscribed arrow with hollow dots at both ends)

verbose semantics

ModifyingAbove Upper A Upper B with arrow with hollow dots at both ends

(7)

",a,b<$*33]

(AB with superscribed arrow dotted at left)

verbose semantics

ModifyingAbove Upper A Upper B with arrow dotted at left

(8)

",a,b<$*33o]

(AB with superscribed arrow dotted at left and barbed at right)

verbose semantics

ModifyingAbove Upper A Upper B with right arrow dotted at left

(9)

",a,b<$33*]

(AB with superscribed arrow dotted at right)

verbose semantics

ModifyingAbove Upper A Upper B with arrow dotted at right

(10)

,x "$33o<f.*g] ,y

(arrow with superscribed f hollow dot g between X and Y)

verbose semantics

Upper X right-arrow Overscript f ring g EndScripts Upper Y

§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.

(1)

#.3:

(decimal point 3, with a bar over the 3)

verbose semantics

point ModifyingAbove 3 with bar

(2)

#."7128<:]

(decimal point 7128, with a bar over the four digits)

verbose semantics

point ModifyingAbove 7 1 2 8 with bar

(3)

#3.57"29<:]

(3.5729, with a bar over the 29)

verbose semantics

3.57 ModifyingAbove 2 9 with bar

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:

(1)

"x<_<]

(x with superscribed caret)

verbose semantics

ModifyingAbove x with caret

(2)

".k<_<]

(equals sign with superscribed caret)

verbose semantics

ModifyingAbove equals with caret

(3)

"x%_%]

(x with subscribed inverted caret)

verbose semantics

x Underscript inverted caret EndScripts

(4)

".k<;<]

(equals sign with superscribed left-pointing caret)

verbose semantics

less-than-or-equal-to

(5)

".k<;%]

(equals sign with superscribed right-pointing caret)

verbose semantics

greater-than-or-equal-to

§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.

(1)

#."3<*]

(decimal point 3, with a dot over the 3)

verbose semantics

point ModifyingAbove 3 with dot

(2)

#."135<*]

(decimal point 135, with a dot over each of the three digits)

verbose semantics

point ModifyingEachAbove 1 3 5 with dot

(3)

#.13"5<*]

(decimal point 135, with a dot over the 5)

verbose semantics

.13 ModifyingAbove 5 with dot

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.

(1)

"x<**]

(x with two dots over it)

verbose semantics

ModifyingAbove x with two-dots

(2)

"x<***]

(x with three dots over it)

verbose semantics

x Overscript three dots EndScripts

(3)

"x%**]

(x with two dots under it)

verbose semantics

ModifyingBelow x With two-dots

§100. Hollow Dot:

(1)

".k<.*]

(equals sign with superscribed hollow dot)

verbose semantics

equals Overscript ring EndScripts

§101. Question Mark:

(1)

".k<_8]

(equals sign with superscribed question mark)

verbose semantics

equals Overscript question-mark EndScripts

(2)

".k%_8]

(equals sign with subscribed question mark)

verbose semantics

equals Underscript question-mark EndScripts

§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).