ABBREVIATED FUNCTION NAME	UNABBREVIATED FUNCTION NAME	BRAILLE
amp	amplitude
antilog	antilogarithm
arc(no abbreviation)	arc
arg	argument
colog	cologarithm
cos	cosine
cosh	hyperbolic cosine
cot	cotangent
coth	hyperbolic contangent
covers	coversine
csc	cosecant
csch	hyperbolic cosecant
ctn	contangent
ctnh	hyperbolic contangent
det	determinant
erf	error function
exp	exponential
exsec	exsecant
grad	gradient
hav	haversine
im	imaginary part
inf	infimum
lim	limit
	upper limit
	lower limit
ln	natural logarithm
log	logarithm
max	maxium
min	minimum
mod	modulo
re	real part
sec	secant
sech	hyperbolic secant
sin	sine
sinh	hyperbolic sine
sup	supremum
tan	tangent
tanh	hyperbolic tangent
vers	versine

Contractions With Function Names: Contractions must not be used in any abbreviated function name. Although contractions may be used with unabbreviated function names, an unabbreviated function name must not be contracted when used in conjunction with an abbreviated function name or with related mathematical symbols.

Spacing and Nonuse of the English Letter Indicator With Function Names and Their Abreviations:

a. A space must be left after a function name or its abbreviation. A space must be left even when the function name or its abbreviation is irectly followed by a sign of operation. The expression following the function name or its abbreviation must be spaced in accordance with its appropiate spacing rules.

No space should be left before a function name or its abbreviation unless it is preceded by a symbol requiring a space.

The English letter indicator must not be used with an English letter, a short-form combination, or a Roman numeral in regular type following a function or its abbreviation.

1. sin x
   

2. arc ab
   

3. Sine I
   

4. 
   

5. ctn -A = -ctn A
   

6. 1/cos -cos=tan * sin
   

7. 
   

8. 
   

9. y = 3tan 2x
   

10. 
    

11. y= 2sin x +3cos y
    

12. 2sin x +3cos y
    

13. sine x-sine y
    

14. 
    

15. 
    

16. 
    

17. 
    

18. 
    

19. 
    

20. 6sin 2A cos 4A
    

21. 
    

22. 
    

23. 
    

24. 
    

25. 
    

26. 
    

27. 
    

28. 
    

29. 
    

30. 
    

b. A space must be left between two or more consecutive abbreviated or unabbreviated function names unless they are clearly unspaced in the print text.

1. y = arc sin x
   

2. y = arcin x
   

3. What is the arc sine function?
   

4. Arc Sine x
   

5. 
   

6. 
   

7. 
   

8. cos arctan (-1)
   

9. cos[2 Arc csc (-7/5)]
   

10. The logarithm of sin 18* is written log sin 18*
    

Modifiers With Function Names: Modified function names must be transcribed according to the five-step rule for the transcription of modified expressions. When a function name or its abbreviation carries a modifier, The required space after the function name must follow the termination of the modifier.

However, the horizontal bar directly over or under the function name "lim" or "limit" must not be treated as a modifier. The symbols and must be used to denote "upper limit," and the symbols and must be used to denote "lower limit."

Superscripts and Subscripts With Function Names:

a. When a function name or its abbreviated carries a superscript or subscript, the required space after the function name must follow the superscript or subscript. A letter, numeral, or other mathematical expression following this space assumes the same level as the function name.

The superscript indicator must not be used when an abbreviated function name carries a mumeric superscript only on the first level below the baseline of writing. However, the subscript indicator must be used if an unabbreviated function name carries a numeric subscript.

1. sin²x
   

2. sine²x
   

3. sin²A+cos²(B+A)
   

4. 
   

5. 
   

6. (1-sin²x)² cos²x
   

7. 
   

8. 
   

9. log_n .125= -.6>
   

10. antilog_a x = N
    

11. 
    

12. log_2ex = -1.4
    

13. log₃ 81 = 4
    

14. log_.0543x = -.7
    

15. logarithm₃ 81 = 4
    

b. When a function name or its abbreviation occures in a superscript or subscript, the required space following it does not terminate the effect of the level indicator. In such cases, the space reinstates the level where the function name or its abbreviation appears. Thus, the letter, numeral, fraction, or other mathematical expression immediately following the space assumes the same levelas its related abbreviated or unabbreviated function name.

1. y = e^{sin x}
   

2. y = e^{sine x}
   

3. y = (sin x)^{tan x}
   

4. e^{ln x-2ln y}
   

5. e^{x+ln x}
   

6. y = e^cos2x
   

7. a^{g(x)log_af(x)}
   

8. 3^{log₃ 9}
   

9. e^{sin x}+e^{sin y}
   

10. 3^{log3 7}+2^{log₂ 5}
    

11. 2^{sec x} = y
    

12. a^{log_a x} = x
    

13. e^{sin x = a}>  y
    

Enclosed Lists With Abbreviated or Unabbreviated Function Names: Although the numeric indicator must be used at the beginning of an item in an enclosed list, it must be used before a numeral or decimal point and a numeral in regular type following a function name or its abbreviation.

1. (2sin 30', 3cos 60')
   

2. (cos .8000, 2cos .8000)
   

Punctuation With Function Names: An abbreviated function name is a mathematical expression and must be punctuated accordingly. An unabbreviated function name must be punctuated in the mathematical or literary mode according to its content.

1. Some trigonometric functions are sin, cos, and tan.
   
2. "Arcsin" is the "inverse sine."
   
3. What is the meaning of logsine?
   
4. Some trigonometric functions are sine, cosine and tangent.
   
5. Arc ACB is a major arc.
   

General Principles: Unless absolutely unavoidable, a mathematical expression must not be divided between braille lines or braille pages. Inorder to avoid a divsion, if there is insufficient space on the remainder of a line to accommodate the expression, the space must be left blank, and the entire expression must be brought down to the next line.

However when a mathematical expression is too long to be contained within the margins in effect, a division is unavoidable, and the expression may be divided between braille lines. Such expressions may begin in the remaining space on the line, provided the division in made in accordance with the principles defined below. The placement of all runovers must conform to the margin requirements for itemized, instructional, explanatory, labeled, subdivided, and displayed material.

Division of Long Numerals: A long numeral which cannot be contained on one braille line within the marginin effect must be divided aftera comma if a comma is present, and a hyphen must be inserted. If the numeral does not contained a comma, the hyphen may be inserted after any digit.

When a numeral is divided between braille lines, the numeric indicator must be used before the first digit of the numeral on the next line.

1. 100,000,000,000,000,000,000,000
   

2. 100000000000000000000000
   

3. A decillion is 1,000,000,000,000,000,000,000,000,000,000,000
   

Division of Long Mathematical Expressions:

a. Mathematical expressions, such as enclosed list, formulas, equations, etc., which cannot fit on one braille line within the boundaries of the margins used may be divided between lines in the following order of preference:

i. A division may be made after a comma between items in an enclosed list. When the items in an list must be divided between braille lines, neither the numeric indicator nor the English letter indicator must be used before a runover on the new line.

1. {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
   
2. 
   
3. Does {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .} represent a set of integers?
   
4. (. . . ,.10, .09, .08, .07, .06, .05, .04, . . .)
   

1. 1778 + 1294 + 865 + 905 + 1574 + 485 = 7901
   
2. 2 x 423 = (2x400)+(2x20)+(2x3)
   
3. 
   
4. 
   
5. 
   
6. 
   

1. 
   
2. 14x³-15y³+5x²-4y²+16x-10y+31
   
3. 
   
4. 
   
5. (3x10⁴)+(4x10³)+ (7x10²)+(5x10)+(4x1) = N
   
6. 
   
7. 
   
8. 
   
9. 
   

1. (49x⁷y⁶-63x⁶y⁴+56x ⁵y⁵+64x⁴y²)/7x²3y²2
   
2. (16x⁴ + 8x³y + 4x²y²+2xy³+ y⁴) (2x-y) = 32x⁵-y⁵
   
3. 144 ft²+122 ft²+
   145 ft²-131 ft²=N
   
4. 
   
5. 
   
6. 
   
7. ((((((Z₂)₃)₅)₇) ₁₁)₁₅)₁₇
   
8. a₁b₂c₃+ a₃b₁c₂+ a₂b₃c₁- a₃b₂c₁- a₁b₃c₂- a₂b₁c₃
   

When an expression is divided before a superscript or subscript change-of-level indicator, the indicator must be placed before the continuation of the expression on the new braille line.

If transition to a new braille line is made within a superscript or subscript, the level in effect is not changed when the division is made before a sign of operation or a fraction line. However, if the transition is made before a sign of comparison, the level in effect is terminated, and the level must be restated before the sign of comparison on the new braille line.

1. (3x²y)(3x²y)(3x²y)(3x²y)= 81x⁴y⁴
   

a. The componets of the following signs must not be divided between braille lines:
i.Plus or minus
ii.Minus or plus
iii.Plus followed by minus
iv.Minus followed by Plus
v.Signs of comparison compounded vertically or horizontally
vi.Superposed signs
vii.Tally marks belonging to the same group
viii.Sings of the shape with structural or interior modification

b. The following expressions must not be divided between braille lines:
i.A hyphenated expressions containing one or more mathematical componets
ii.An abbreviation and its related preceding or following numeral or letter
iii.A sign of shape and its identifying numeral, letter, or sequence of letters
iv.A function name or its abbreviation and the sign following it