RULE XIII--SUPERSCRIPTS AND SUBSCRIPTS

• Base Line
  "

  verbose

  Baseline

• Superscript
  ^

  verbose

  Superscript

• Superscript with Superscript
  ^^

  verbose

  SuperSuperscript

• Superscript with Subscript
  ^;

  verbose

  SuperSubscript

• Superscript with Superscript with Superscript
  ^^^

  verbose

  SuperSuperSuperscript

• Superscript with Superscript with Subscript
  ^^;

  verbose

  SuperSuperSubscript

• Superscript with Subscript with Superscript
  ^;^

  verbose

  SuperSubSuperscript

• Superscript with Subscript with Subscript
  ^;;

  verbose

  SuperSubSubscript

• Subscript
  ;

  verbose

  Subscript

• Subscript with Superscript
  ;^

  verbose

  SubSuperscript

• Subscript with Subscript
  ;;

  verbose

  SubSubscript

• Subscript with Superscript with Superscript
  ;^^

  verbose

  SubSuperSuperscript

• Subscript with Superscript with Subscript
  ;^;

  verbose

  SubSuperSubscript

• Subscript with Subscript with Superscript
  ;;^

  verbose

  SubSubSuperscript

• Subscript with Subscript with Subscript
  ;;;

  verbose

  SubSubSubscript

verbose

comma

§71. Nature of Superscripts and Subscripts: It is characteristic of mathematical expressions to employ signs, usually in smaller type, which are elevated or depressed relative to the base line. A sign which is elevated relative to the base line is called a superscript; one which is depressed relative to the base line is called a subscript. When an entire expression is at the superscript or subscript level, it should be written without an indicator in braille, but its position must be explained to the reader by a transcriber's note.

(1)

'9 x^2' #2 ',elevat$ 9 pr9t', is a sup]script4

verbose semantics

In x squared comma squared is a superscript period

(2)

'9 x;n' ',depress$ 9 pr9t',

is ! subscript4

verbose semantics

In x Subscript n Baseline comma Subscript n Baseline is the subscript period

(3)

'! sign = .degree is ,*_4

',elevat$ 9 pr9t',

verbose semantics

The sign for degree is degree period

 

§72. Hierarchy of Superscripts and Subscripts: Superscripts or subscripts may carry superscripts or subscripts of their own; the latter are then referred to as superscripts or subscripts of second order, and are thus distinguished from the former, which are called superscripts or subscripts of first order. Second order superscripts or subscripts may, in turn, carry superscripts or subscripts of their own, which are then called superscripts or subscripts of third order. While it is theoretically possible for a superscript or subscript to be of order higher than the third, this situation rarely arises in practice.

§73. Level Indicators: A level indicator other than the base-line indicator identifies the symbols which follow it as representing a superscript or subscript. The base-line indicator identifies the symbols which follow it as representing signs on the base line. The degree of elevation or depression specified by a level indicator is always with respect to the base line; the symbol which precedes the indicator, if it represents a sign at some other level, plays no role in this regard.

§74. Orientation by Level Indicator:

a. The effect of a level indicator with one component is to direct the reader's attention upward or downward from the base line according as that component is the superscript or the subscript indicator.

(1)

x^2

(x squared)

verbose semantics

x squared

(2)

y^3

(y cubed)

verbose semantics

y cubed

(3)

x^@#

(x carries an asterisk as a superscript)

verbose semantics

x Superscript asterisk

(4)

x^-2

(x carries minus 2 as a superscript)

verbose semantics

x Superscript negative 2

(5)

x;a

(x sub a)

verbose semantics

x Subscript a

(6)

x;-2

(x carries minus 2 as a subscript)

verbose semantics

x Subscript negative 2

b. The effect of a level indicator with two components may be analyzed as follows:

(1)

n^x^^y

(n carries a superscript x which carries a superscript y)

verbose semantics

n Superscript x Super Superscript y

(2)

x^n^;a

(x carries a superscript n which carries a subscript a)

verbose semantics

x Superscript n Super Subscript a

(3)

x;n;^a

(x carries a subscript n which carries a subscript a)

verbose semantics

x Subscript n Sub Superscript a

(4)

n;x;;y

(n carries a subscript x which carries a subscript y)

verbose semantics

n Subscript x Sub Subscript y

c. The effect of a level indicator with three components may be analyzed as follows:

(1)

n^x^^y^^^z

(n carries a superscript x which carries a superscript y which carries a superscript z)

verbose semantics

n Superscript x Super Superscript y Super Super Superscript z

(2)

n^x^^y^^^z^^^^,,,

(the ellipsis indicates the presence of superscripts of increasingly higher order; the dots are printed obliquely)

verbose semantics

n Superscript x Super Superscript y Super Super Superscript z Super Super Super Superscript diagonal ellipsis

(3)

x^y^^z^^;a

(x carries a superscript y which carries a superscript z which carries a subscript a)

verbose semantics

x Superscript y Super Superscript z Super Super Subscript a

(4)

x^y^;a^;^n

(x carries a superscript y which carries a subscript a which carries a superscript n)

verbose semantics

x Superscript y Super Subscript a Super Sub Superscript n

(5)

n^x^;a^;;j

(n carries a superscript x which carries a subscript a which carries a subscript j)

verbose semantics

n Superscript x Super Subscript a Super Sub Subscript j

(6)

x;a;^r;^^n

(x carries a subscript a which carries a superscript r which carries a superscript n)

verbose semantics

x Subscript a Sub Superscript r Sub Super Superscript n

(7)

x;a;^n;^;b

(x carries a subscript a which carries a superscript n which carries a subscript b)

verbose semantics

x Subscript a Sub Superscript n Sub Super Subscript b

(8)

x;p;;a;;^m

(x carries a subscript p which carries a subscript a which carries a superscript m)

verbose semantics

x Subscript p Sub Subscript a Sub Sub Superscript m

(9)

n;x;;y;;;z

(n carries a subscript x which carries a subscript y which carries a subscript z)

verbose semantics

n Subscript x Sub Subscript y Sub Sub Subscript z

(10)

n;x;;y;;;z;;;;,,,

(the ellipsis indicates the presence of subscripts of increasingly higher order; the dots are printed obliquely)

verbose semantics

n Subscript x Sub Subscript y Sub Sub Subscript z Sub Sub Sub Subscript diagonal ellipsis

d. The effect of a level indicator with more than three components may be analyzed in the same manner suggested for level indicators with two or three components.

§75. Left Superscripts and Subscripts: A superscript or subscript may occupy a position to the left, as well as to the right, of the sign to which it applies. The words left or right are then used with the words superscript or subscript to make the distinction in position.

            A right or left superscript or subscript is represented as such merely by preserving the relative horizontal positions of the superscript or subscript symbol to which it applies. Each must be preceded by its appropriate level indicator.

            Left superscripts or subscripts of the third or higher order, although rare, are treated in the manner suggested by the examples below.

(1)

^x"n

(x is a left superscript to n)

verbose semantics

Superscript x Baseline n

(2)

^-"x

(the minus sign is a left superscript to x)

verbose semantics

negative x

(3)

;x"n

(x is a left subscript to n)

verbose semantics

Subscript x Baseline n

(4)

;x"n;y

(x is a left subscript to n, y is a right subscript to n)

verbose semantics

Subscript x Baseline n Subscript y

(5)

#10^^-^4

(10 to the minus 4 power; the minus is superscript to the 4)

verbose semantics

10 Superscript negative 4

(6)

^n^;a"x

(n sub a is a left superscript to x)

verbose semantics

Superscript n Super Subscript a Baseline x

(7)

^;a^n"x

(a is a left subscript to n, the combination is a left superscript to x)

verbose semantics

Super Subscript a Superscript n Baseline x

(8)

;n;^a"x

(n to the a power is a left subscript to x)

verbose semantics

Subscript n Sub Superscript a Baseline x

(9)

;^a;n"x

(a is a left superscript to n, the combination is a left subscript to x)

verbose semantics

Sub Superscript a Subscript n Baseline x

(10)

;x;;y"n

(x sub y is a left subscript to n)

verbose semantics

Subscript x Sub Subscript y Baseline n

(11)

;;y;x"n

(y is a left subscript to x, the combination is a left subscript to n)

verbose semantics

Sub Subscript y Subscript x Baseline n

(12)

p^b'c"x

(p carries a right superscript b and c is a left superscript to x)

verbose semantics

p Superscript b Superscript c Baseline x

§76. Direct Superscripts and Subscripts: A superscript or subscript which occupies, respectively, a position directly over or under the sign to which it applies is called a modifier (see Rule XIV).

§77. Numeric Subscripts: The subscript indicator must not be used to indicate a numeric subscript provided that all of the following conditions hold:

(1)

x1

(x sub 1; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x 1

(2)

x11

(x sub 1 1; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x 11

(3)

_,a1

(German capitalized ah sub 1; subscript indicator not required because all conditions i-iv hold)

verbose semantics

German Upper A 1

(4)

x'1

(x prime sub 1; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x prime 1

(5)

x''2

(x double prime sub 2; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x double-prime 2

(6)

;3"x

(3 is a left subscript to x; subscript indicator is required because condition i does not hold)

verbose semantics

Subscript 3 Baseline x

(7)

x;i;;1

(x sub i sub 1; subscript indicator is required because condition ii does not hold)

verbose semantics

x Subscript i 1

(8)

log;2 x

(log to the base 2 of x; subscript indicator not required because all conditions i-iv hold)

verbose semantics

log Subscript 2 Baseline x

(9)

#12;7

(12 sub 7; subscript indicator is required because condition iii does not hold)

verbose semantics

12 Subscript 7

(10)

(,c,o3);2

(the carbonate radical taken twice; subscript indicator is required before the 2 because condition iii does not hold)

verbose semantics

left-parenthesis Upper C Upper O 3 right-parenthesis Subscript 2

(11)

,na2,c,o3

(sodium carbonate; subscript indicator not required because all conditions i-iv hold)

verbose semantics

Upper N a 2 Upper C Upper O 3

(12)

seven;3

(seven sub 3; subscript indicator is required because condition iii does not hold)

verbose semantics

seven Subscript 3

(13)

x;1;;j

(x sub 1 sub j; subscript indicator is required because condition iv does not hold)

verbose semantics

x Subscript 1 Sub Subscript j

(14)

x;2;^n

(x carries a subscript 2 which carries a superscript n; subscript indicator is required because condition iv does not hold)

verbose semantics

x Subscript 2 Sub Superscript n

(15)

x;2'

(x sub 2 prime; subscript indicator is required because condition iv does not hold)

verbose semantics

x Subscript 2 prime

(16)

x;2+k

(x carries a subscript of 2 plus k; subscript indicator is required because condition iv does not hold)

verbose semantics

x Subscript 2 plus k

(17)

x;1_/2

(x sub one-half; subscript indicator is required because condition iv does not hold)

verbose semantics

x Subscript one-half

(18)

;3"x1

(3 is a left subscript to x, 1 is a right subscript to x; subscript indicator is required before 3 because condition i does not hold)

verbose semantics

Subscript 3 Baseline x 1

(19)

,a;x1

(A sub x 1; subscript indicator is required because condition iv does not hold)

verbose semantics

Upper A Subscript x Subscript 1

(20)

x10,000

(x sub 10,000; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x 10,000

(21)

x1.2

(x sub 1.2; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x 1.2

(22)

x.6

(x sub .6; subscript indicator not required because all conditions i-iv hold)

verbose semantics

x .6

(23)

.,s0^n"a;k

(the summation from zero to n of a sub k; subscript indicator is not required because all conditions i-iv hold)

verbose semantics

sigma-summation Subscript 0 Superscript n Baseline a Subscript k

(24)

.,p0^n"a;k

(the product from zero from zero to n of a sub k; subscript indicator not required because all conditions i-iv hold)

verbose semantics

pi-product Subscript 0 Superscript n Baseline a Subscript k

(25)

#3af;16

(A and F represent a numeral in base 16; subscript indicator is required because condition iii does not hold)

verbose semantics

3 Upper A Upper F 16

(26)

!;0^>1-x^^2^]"f(x)dx

(the integral from 0 to the square root of 1-x2 of f of x dx; subscript indicator is required because condition iii does not hold)

verbose semantics

Integral Subscript 0 Superscript StartRoot 1 minus x squared EndRoot Baseline f left-parenthesis x right-parenthesis d x

§78. Comma at Superscript or Subscript Level: A commonly occurring superscript or subscript notation is the one in which two consecutive items are separated by a comma with an optional space following the comma. In this configuration, the symbol ?? (dots 2-4-6) must be used to replace the comma and the optional space used in this way. This contracted form must not be used to replace a comma and the optional space which follows it in a configuration which is on the base line.

(1)

x;i[j[k

(each comma is followed by a space in ink print)

verbose semantics

x Subscript i comma j comma k

(2)

x;(a[b)

(the comma is not followed by a space in ink print)

verbose semantics

x Subscript left-parenthesis a comma b right-parenthesis

(3)

x;1[2

(the comma is followed by a space in ink print)

verbose semantics

x Subscript 1 comma 2

(4)

,p;n;;x[y

(the comma is followed by a space in ink print)

verbose semantics

Upper P Subscript n Sub Subscript x comma y

(5)

x;n-1[n-1, x;n-1[n, x;n[n-1

(the comma and space between the items on the base line cannot be contracted)

verbose semantics

x Subscript n minus 1 comma n minus 1 Baseline comma x Subscript n minus 1 comma n Baseline comma x Subscript n comma n minus 1

(6)

(x, y)

(the comma and space between the items on the base line cannot be contracted)

verbose semantics

left-parenthesis x comma y right-parenthesis

§79. Circumstances Determining Changes of Level: The symbols and situations listed below have the following effect in determining changes of level.

a. A level indicator terminates the effect of a previous level indicator and initiates the level implied by the new indicator. In the case of the base-line level, the previous base-line indicator may only be implied.

(1)

x^2"+1

(the superscript indicator terminates the previous implied base-line level and initiates the superscript level, the base-line indicator terminates the previous superscript level and initiates the base-line level)

verbose semantics

x squared plus 1

(2)

x;a"+y^2

(the subscript indicator terminates the previous implied base-line level and initiates the subscript level, the base-line indicator terminates the previous subscript level and initiates the base-line level, the superscript indicator terminates the previous base-line level and initiates the superscript level)

verbose semantics

x Subscript a Baseline plus y squared

(3)

?e^x^^2"/2#

(the superscript indicator terminates the previous implied base-line level and initiates the first-order superscript level, the second order superscript indicator terminates the previous first-order superscript level and initiates the second-order superscript level, the base-line indicator terminates the previous second-order superscript level and initiates the base-line level)

verbose semantics

StartFraction e Superscript x squared Baseline Over 2 EndFraction

(4)

,a^n+n+ ''' ^to ^m ^n^_'s

.k a^mn

(the superscript indicator which follows the n preserves the effect of the preceding superscript indicator; otherwise, the punctuation indicator would terminate the effect of the previous level indicator and initiate the base-line level)

verbose semantics

Upper A Superscript n plus n plus n ellipsis to m n apostrophe s Baseline equals a Superscript m n

b. The punctuation indicator terminates the effect of any previous level indicator and initiates the base-line level. In addition, the comma, provided it is not a numeric symbol, terminates the effect of any previous indicator and initiates the base-line level. However, the comma, when it is a numeric symbol and the contracted form ?? (dots 2-4-6), preserves the level that is already in effect.

(1)

x^2_4

(the period is at the base-line level)

verbose semantics

x squared period

(2)

x^2, x^3

(the comma is at the base-line level)

verbose semantics

x squared comma x cubed

(3)

x^10,000

(the comma is a numeric symbol and preserves the superscript level)

verbose semantics

x Superscript 10,000

(4)

x;i[j

(the contracted form for a comma and optional space preserves the subscript level that is already in effect)

verbose semantics

x Subscript i comma j

(5)

,p;n;;1;[n;;2;[ '''

(the contracted form for a comma and optional space preserves the indicated subscript level)

verbose semantics

Upper P Subscript n 1 comma n 2 comma ellipsis

c. A space or the transition to a new braille line which is followed by literary text or unrelated mathematical text terminates the effect of any previous level indicator and initiates the base-line level. However, if a space occurs between the parts of an abbreviation or phrase, the appropriate level indicator must be restated before each part.

(1)

#2p^2 is alw ev54

(the space before the literary text terminates the previous superscript level and initiates the base-line level)

verbose semantics

2 p squared is always even period

(2)

#3@*10^4 ]gs

(the space before the literary text terminates the previous superscript level and initiates the base-line level)

verbose semantics

3 times 10 Superscript 4 Baseline ergs

(3)

#6,696@*10^8 mph

(the space before the literary text terminates the previous superscript level and initiates the base-line level)

verbose semantics

6.696 times 10 Superscript 8 Baseline miles per hour

(4)

(x^2 y^2")

(these items are entries in a matrix and hence unrelated; the space terminates the previous superscript level and initiates the base-line level)

verbose semantics

left-parenthesis x squared y squared right-parenthesis

(5)

$t;reg4 ;polygon_4

(level indicators are required between each part of this abbreviation to show that they are both at the same subscript level)

verbose semantics

triangle Subscript reg period polygon period

(6)

$t;regular ;polygon

(level indicators are required between each part of this phrase to show that they are both at the subscript level)

verbose semantics

triangle Subscript regular polygon

d. The space which immediately follows a symbol of shape, an abbreviated function name, or an unabbreviated function name, provided the latter is in a mathematical context, preserves the level that is already in effect. If these items carry a superscript or subscript, the space which follows such a superscript or subscript reinstates the level that was in effect before.

(1)

b;$t ,a,b,c

(the space preserves the subscript level at which the triangle appears)

verbose semantics

b Subscript triangle Upper A Upper B Upper C

(2)

sin x

(the space preserves the base-line level of sin)

verbose semantics

sine x

(3)

cos^2 x

(the space reinstates the base-line level of cos)

verbose semantics

cosine squared x

(4)

e^sin x

(the space preserves the superscript level at which sin appears)

verbose semantics

e Superscript sine x

(5)

e^sin x+icos x

(each space preserves the superscript level at which the abbreviated function names appear)

verbose semantics

e Superscript sine x plus i cosine x

(6)

e^x+ln x

(the space preserves the superscript level at which ln appears)

verbose semantics

e Superscript x plus l n x

(7)

e^cos^^2 x

(the space reinstates the superscript level at which cos appears)

verbose semantics

e Superscript cosine squared x

(8)

e^sin^^2 x+sin^^2 y

(each space reinstates the superscript level at which the abbreviated function names appear)

verbose semantics

e Superscript sine squared x plus sine squared y

(9)

q^log^;q a

(the space reinstates the superscript level at which log appears)

verbose semantics

q Superscript log Super Subscript q Superscript a

(10)

,v;max (m[n)

(the space preserves the subscript level at which max appears)

verbose semantics

Upper V Subscript max left-parenthesis m comma n right-parenthesis

e. The space which occurs in a numeral for the purpose of dividing it into short regular segments preserves the level already in effect.

(1)

e^3.14159 ^26535

verbose semantics

e Superscript 3.1415926535

 

f. The space which precedes an ellipsis or long dash preserves the effect of any previous level indicator. The space which follows the ellipsis or long dash preserves the level that is already in effect. However, if such a space is followed by literary text, unrelated mathematical text, or a sign of comparison, this space initiates the base-line level.

(1)

x^1+1_/2+1_/3+ ''' +1_/n

(both spaces preserve the superscript level)

verbose semantics

x Superscript 1 plus one-half plus one-third plus ellipsis plus 1 slash n

(2)

s1 ''' s;n

(both spaces preserve the implied base-line level)

verbose semantics

s 1 ellipsis s Subscript n

(3)

#10^3+ ---- is equal to #10^5_4

(the space before the dash preserves the previous superscript level, and the space following the dash terminates the effect of the previous superscript level and initiates the base-line level)

verbose semantics

10 Superscript 3 plus large-dash Baseline is equal to 10 Superscript 5 Baseline period

(4)

#10^3+ ---- .k #10^5

(the space before the dash preserves the previous superscript level, and the space following the dash terminates the effect of the previous superscript level and initiates the base-line level)

verbose semantics

10 Superscript 3 plus large-dash Baseline equals 10 Superscript 5

g. The space or transition to a new braille line which is followed by a comparison symbol terminates the effect of a level indicator already in effect and initiates the base-line level. The space after a comparison symbol preserves the level that is already in effect.

(1)

x^2"+y^2"+z^2 .k r^2

(the space which is followed by the equals symbol terminates the effect of the preceding superscript level and initiates the base-line level, the space after the equals symbol preserves the base-line level)

verbose semantics

x squared plus y squared plus z squared equals r squared

(2)

#2^x "k #3^x

(the space which is followed by the less than symbol terminates the effect of the preceding superscript level and initiates the base-line level, the space after the less than symbol preserves the base-line level)

verbose semantics

2 Superscript x Baseline less-than 3 Superscript x

(3)

q^log^;q a .k a

(the space which is followed by the equals symbol terminates the effect of the preceding superscript level and initiates the base-line level, the space after the equals symbol preserves the base-line level)

verbose semantics

q Superscript log Super Subscript q Superscript a Baseline equals a

(4)

!;u ;.k a

(the subscript indicator before the equals symbol keeps this symbol at the subscript level; the space after the equals symbol preserves the level that is already in effect)

verbose semantics

Integral Subscript u equals a

(5)

(1-sin^2 x)^2

.k cos^4 x

(the transition to a new braille line before the equals symbol terminates the previous superscript level and initiates the base-line level)

verbose semantics

left-parenthesis 1 minus sine squared x right-parenthesis squared equals cosine Superscript 4 Baseline x

h. Any symbol or situation other than those in a to g preserves the level that is already in effect.

§80. Use of Level Indicators:

a. A level indicator must be used before any braille indicator or grouping symbol whenever this braille indicator or grouping symbol applies to a level other than the one currently in effect.

(1)

>x^2"+y^2"]

(the termination indicator applies to the base line, therefore the base-line indicator is required)

verbose semantics

StartRoot x squared plus y squared EndRoot

(2)

e^>x^^2^+y^^2^]

(the termination indicator applies to the first-order superscript level, therefore the superscript indicator is required)

verbose semantics

e Superscript StartRoot x squared plus y squared EndRoot

(3)

?1/x^2"#

(the closing simple-fraction indicator applies to the base line, therefore the base-line indicator is required)

verbose semantics

StartFraction 1 Over x squared EndFraction

(4)

,?d(?x/y#),/1+(?x/y#)^2"'#

(the closing complex-fraction indicator applies to the base line, therefore the base-line indicator is required)

verbose semantics

StartStartFraction d left-parenthesis StartFraction x Over y EndFraction right-parenthesis OverOver 1 plus left-parenthesis StartFraction x Over y EndFraction right-parenthesis squared EndEndFraction

(5)

"x^2"<:]

(the directly-over indicator applies to the base line, therefore the base-line indicator is required)

verbose semantics

ModifyingAbove x squared with bar

(6)

x^2"[y^2"]

(the opening and closing cancellation indicators apply to the base line, therefore the base-line indicator is required)

verbose semantics

x squared CrossOut y squared EndCrossOut

(7)

(x^2"+y^2")

(the closing parenthesis applies to the base line, therefore the base-line indicator is required)

verbose semantics

left-parenthesis x squared plus y squared right-parenthesis

(8)

x^(m^^n^)

(the closing parenthesis applies to the first-order superscript level, therefore the superscript indicator is required)

verbose semantics

x Superscript left-parenthesis m Super Superscript n Superscript right-parenthesis

b. The superscript indicator must be used to restate the superscript level when two superscripts are consecutive but one applies to the expression which precedes it and the other applies to the expression which follows it. Similarly, the subscript indicator must be restated when two subscripts are consecutive and one applies to the expression preceding it and the other applies to the expression following it. A superscript or subscript indicator must be restated before a modified expression which is interior to the superscript or subscript expression, provided that the multipurpose indicator is also used.

(1)

p^b^c"q

verbose semantics

p Superscript b Superscript c Baseline q

(2)

,p;b;b",q

verbose semantics

Upper P Subscript b Subscript c Baseline Upper Q

(3)

,p1;2",q

verbose semantics

Upper P 1 Subscript 2 Baseline Upper Q

(4)

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

(the subscript indicator after the plus sign must be restated before the multipurpose indicator)

verbose semantics

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

c. The appropriate level indicator must be used before each part of an abbreviation or phrase which is at a level other than the base line.

(1)

$t;regular ;polygon

verbose semantics

triangle Subscript regular polygon

(2)

a^n+n+n ''' ^to ^m ^n^_'s

verbose semantics

a Superscript n plus n plus n ellipsis to m n apostrophe s

 

d. Whenever spaces are left for the purpose of achieving alignment, level indicators must be used as though such spaces were not present.

(1)

2x^3"- x^2"+ x+ 1

3x^3"+4x^2"-10x+ 7

5x^2" +12

-2x^3" - 6x

333333333333333333333

3x^3"+8x^2"-15x+20

verbose semantics

StartLayout 1st Row 1st Column 2 x cubed 2nd Column minus x squared 3rd Column plus x 4th Column plus 1 2nd Row 1st Column 3 x cubed 2nd Column plus 4 x squared 3rd Column minus 10 x 4th Column plus 7 3rd Row 1st Column Blank 2nd Column 5 x squared 3rd Column Blank 4th Column plus 12 4th Row 1st Column minus 2 x cubed 2nd Column Blank 3rd Column minus 6 x 4th Column Blank 5th Row 1st Column 3 x cubed 2nd Column plus 8 x squared 3rd Column minus 15 x 4th Column plus 20 EndLayout

??!??

e. The appropriate level indicator must be used before any symbol or situation in which a change of level is required but the change is not effected by any of the conditions of §79.

(1)

#360^.*"-9t]val

(the base-line indicator places the hyphen at the base-line level)

verbose semantics

360 degree hyphen interval

(2)

!;u ;.k a

(the subscript indicator before the equals symbol keeps this symbol at the subscript level; otherwise it would be base-line level)

verbose semantics

Integral Subscript u equals a

(3)

t@);t ;.k a^t ^.k b .k b-a

(the subscript and superscript indicators before the first two equals symbols keep these at the subscript and superscript levels respectively, while the space before the last equals symbol places it at the base-line level)

verbose semantics

t right-bracket Subscript t equals a Superscript t equals b Baseline equals b minus a

(4)

e^sin x ^.k a .1 y

(the superscript indicator before the equals symbol keeps this symbol at the superscript level)

verbose semantics

e Superscript sine x equals a Baseline greater-than y

(5)

.p;s;;1;''' s;;n

(the subscript indicator before the ellipsis places the ellipsis at the first-order subscript level)

verbose semantics

Upper P Subscript s 1 ellipsis s Sub Subscript n

(6)

.p1^.a^;1"''' .p;r^.a^;r

(the base-line indicator places the ellipsis at the base-line level)

verbose semantics

Upper P 1 Superscript alpha 1 Baseline ellipsis Upper P Subscript r Superscript alpha Super Subscript r

(7)

(1-sin^2 x)^2

"cos^2 x

(it is assumed that this expression has had to be run over to another braille line at the place indicated; the base-line indicator at the beginning of the braille line places cos at the base-line level; otherwise it would have remained at the superscript level initiated on the previous braille line)

verbose semantics

left-parenthesis 1 minus sine squared x right-parenthesis squared cosine squared x

§81. Non-Use of Level Indicators:

a. The base-line indicator must not be used to return to the base line from a numeric subscript if the subscript indicator has not been used before the numeric subscript.

(1)

(x1+1)

(base-line indicator not required before the plus symbol)

verbose semantics

left-parenthesis x 1 plus 1 right-parenthesis

(2)

(x1y1+x2y2)

(base-line indicator not required after any of the numeric subscripts)

verbose semantics

left-parenthesis x 1 y 1 plus x 2 y 2 right-parenthesis

b. The base-line indicator must not be used before a right enlarged grouping symbol if this symbol either is separated from its preceding material by one or more spaces, or if the material which precedes the right grouping symbol is not the end of an expression.

(1)

.,(u .k x^2 .,)

.,(v .k x^2"+y^2".,)

verbose semantics

StartLayout Enlarged left-brace 1st Row u equals x squared 2nd Row v equals x squared plus y squared Enlarged right-brace EndLayout

(2)

,\f(x, y) x^2jk,\

,\ "+y,\

,\g(x, y) x^2jk,\

,\ "-y,\

verbose semantics

Start 2 By 2 Determinant 1st Row 1st Column f left-parenthesis x comma y right-parenthesis 2nd Column x Superscript 2 j k Baseline plus y 2nd Row 1st Column g left-parenthesis x comma y right-parenthesis 2nd Column x Superscript 2 j k Baseline minus y EndDeterminant

 

c. A level indicator must not be used before any closing grouping symbol which is drawn in.

(1)

#1 #1 #1

x y z

x^2 y^2 z^2

verbose semantics

Start 3 By 3 Determinant 1st Row 1st Column 1 2nd Column 1 3rd Column 1 2nd Row 1st Column x 2nd Column y 3rd Column z 3rd Row 1st Column x squared 2nd Column y squared 3rd Column z squared EndDeterminant

 

d. A level indicator must not be used to change the level if any symbol or situation specified in §79 has already effected the change to the desired level.

§82. Simultaneous and Non-Simultaneous Superscripts and Subscripts:

a. When an expression simultaneously carries a superscript and subscript, the subscript must be indicated first, even if the subscript is numeric and does not require the subscript indicator. However, if this sign carries one or more primes in addition, see §83.

(1)

x;a^n

(x carries simultaneously a subscript of a and a superscript of n)

verbose semantics

x Subscript a Superscript n

(2)

;a^n"x

(x carries simultaneously a left subscript of a and a left superscript of n)

verbose semantics

Subscript a Superscript n Baseline x

(3)

x1^2

(x carries simultaneously a subscript of 1 and a superscript of 2)

verbose semantics

x 1 Superscript 2

b. When the same expression carries a superscript and a subscript which are not simultaneous, the relative horizontal position of the signs must be retained in the transcription, but the base-line indicator must be inserted before making the transition to the other level.

(1)

a^n";m

(the superscript is closer to the a than the subscript)

verbose semantics

a Superscript n Baseline Subscript m

(2)

a;m"^n

(the subscript is closer to the a than the superscript)

verbose semantics

a Subscript m Baseline Superscript n

(3)

^a";b"x

(the left subscript is closer to the x than the left superscript)

verbose semantics

Superscript a Baseline Subscript b Baseline x

(4)

;b"^a"x

(the left superscript is closer to the x than the left subscript)

verbose semantics

Subscript b Baseline Superscript a Baseline x

(5)

x1"^2

(the subscript is closer to the x than the superscript)

verbose semantics

x 1 Baseline Superscript 2

(6)

x';a"^b

(the subscript is closer to x prime than the superscript)

verbose semantics

x prime Subscript a Baseline Superscript b

§83. Primes in Addition to Superscripts or Subscripts:

a. The prime symbol must never be preceded by the superscript indicator.

(1)

x'

verbose semantics

x prime

 

b. When an expression carries one or more primes in addition to superscripts or subscripts, the prime symbol or symbols must be indicated first unless such symbols do not occur at the beginning of the superscript or subscript, in which case they must retain the same position as in ink print.

(1)

x';a

verbose semantics

x prime Subscript a

(2)

x'^2

verbose semantics

x prime squared

(3)

x';a^b

verbose semantics

x prime Subscript a Superscript b

(4)

x''1^3

verbose semantics

x double-prime 1 Superscript 3

(5)

x'^@#

verbose semantics

x prime Superscript asterisk

(6)

x^@#'

verbose semantics

x Superscript asterisk prime

(7)

,a;ue^@#'

verbose semantics

Upper A Subscript u e Superscript asterisk prime

(8)

,a';ue^@#

verbose semantics

Upper A prime Subscript u e Superscript asterisk

 

c. For primes in other roles see §172.

§84. Plurals and Possessives: For plurals or possessives of mathematical expressions which end with a superscript or subscript see §39.