Our Nemeth symbol library allows individuals to look up Nemeth symbols and math-related terms using common words.

Symbols and terms are listed after this introduction. Once you have found the symbol or term in the list (please review Instructions and Tutorials to help you navigate the library), select that particular link, which will take you to a description of how the symbol, expression, or equation is written in Nemeth Code. At the end of each description, you will find three additional links to examples in Nemeth Code. The first link takes you to a Braille Ready File (BRF) that includes examples using Nemeth Code in English Braille American Edition (EBAE). The second link takes you to a BRF file that includes examples using Nemeth Code within Unified English Braille (UEB) contexts. The third link takes you to a Microsoft Word document that includes examples in print and Simulated Braille (SimBraille), which adds shadow dots that can help sighted readers.

## List of Symbols and Math Terms

- Absolute value
- Alpha (lowercase)
- Angle
- Angle brackets
- Antiderivative
- Approximately equal to
- Arcs
- Barred brackets (used for ceiling, floor, and integer functions)
- Beta (lowercase)
- Biconditional statements, also known as if and only if, which use the two-way horizontal arrow
- Binomial coefficient
- Braces, also known as curly brackets
- Brackets
- Caret
- Cent sign
- Circle
- Circle with interior dot
- Combinations
- Complex fractions (linear format)
- Complex fractions (spatial format)
- Composite functions
- Conditional statements, also known as if-then statements, which use the right-pointing arrow
- Congruent to
- Conjunction (and)
- Corresponds to
- Cosecant
- Cosine
- Cotangent
- Curly d
- Decimal point
- Definite integral (using directly-under and directly-over)
- Definite integral (using subscripts and superscripts)
- Degrees (used in angle measure and temperature in Celsius/Centigrade or Fahrenheit)
- Delta (lowercase)
- Delta (uppercase)
- Density (lowercase rho)
- Derivative of f of x (using prime)
- Derivative of y with respect to x (using raised numbers after the first derivative)
- Directly-over indicator
- Directly-under indicator
- Directly-under indicator (used in place value)
- Disjunction (or)
- Divided by
- Divided into, also called a division bracket
- Dollar sign
- Ellipsis
- Empty set (also known as null set)
- Enlarged braces (used for systems of equations, compound functions, or piecewise functions)
- Enlarged brackets (used for matrices)
- Enlarged parentheses
- Enlarged vertical bars (used for determinants)
- Epsilon (lowercase)
- Equals, also known as is equal to
- Exponent, also known as superscript or power (which sometimes uses a baseline indicator)
- Factorial
- Five-Step Rule
- Function notation
- Gamma (lowercase)
- Greater than
- Greater than or equal to
- Greek letters (complete list)
- Greek letters (used for angle measure)
- Hollow dot
- Horizontal bar
- Hypercomplex fractions
- Indexed radical or root (also known as nth root)
- Infinity
- Integral
- Intersection (of sets)
- Interval notation
- Inverse of a function
- Inverse trigonometric functions
- Lambda (lowercase)
- Less than
- Less than or equal to
- Limits
- Lines
- Logarithms (including common and natural logarithms)
- Maps to, using the right-pointing arrow
- Mathematical comma
- Mean
- Minus sign
- Mixed number
- Negation, or not
- Negative sign
- Nemeth Code switch indicators
- Number systems
- Omega (lowercase used in angular velocity)
- Ordered pair
- Parallel to
- Parentheses (basic)
- Parentheses (used with combinations and the binomial coefficient)
- Partial derivative, also known as curly d
- Percent
- Permutations
- Perpendicular to
- Phi (lowercase)
- Pi (lowercase)
- Plus or minus
- Plus sign
- Power, also known as exponent or superscript (which sometimes uses a baseline indicator)
- Prime
- Proportion
- Ratio
- Rational numbers (as fractions, mixed numbers, and decimals)
- Rays
- Repeating decimals
- Secant
- Segments
- Shapes
- Sigma (lowercase)
- Sigma (uppercase)
- Signs of comparison
- Signs of omission, including general omission symbol, long dash, ellipsis, and shapes
- Similar to
- Simple fraction (linear format)
- Simple fraction (spatial arrangement)
- Since/Because
- Sine
- Square root
- Subscript (which sometimes uses a baseline indicator)
- Summation notation (using directly over and directly-under)
- Summation notation (using subscripts and superscripts)
- Superscript, also known as exponent or power (which sometimes uses a baseline indicator)
- Tally mark, also called hash mark
- Tangent
- Therefore
- Theta (lowercase)
- Tilde (for logical negation)
- Times sign (including multiplication cross, multiplication dot, and multiplication asterisk)
- Triangle
- Trigonometric functions
- Union (of sets)
- Vectors
- Vertical bar
- Wavelength

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## Definitions

### Absolute value

Absolute value is represented by a vertical bar (dots 1-2-5-6) on each side of a number or expression. The numeric indicator is not used inside grouping symbols such as the vertical bars representing absolute value. Note: The vertical bar is the same as the "ou" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Alpha (lowercase)

Alpha (dots 4-6, dot 1) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter a (dot 1). The lowercase form of this letter is often used to represent an angle measure. When writing this immediately after a trigonometric function such as sin, cos, tan, csc, sec, or cot, include a space before the Greek letter indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Angle

The angle symbol (dots 1-2-4-6, dots 2-4-5) is written with the shape indicator (dots 1-2-4-6) followed by dots 2-4-6. When naming an angle, insert a space before writing the capital letter(s) in the name of the angle. Any capital letter used to name an angle should have a capitalization indicator in front of each letter in its name. A double cap should not be used in the naming of an angle. Note: The shape indicator is the same as the "ed" contraction. Also, a letter m in front of an angle notation means the measure of. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Angle brackets

Angle brackets are a type of grouping symbol. The opening angle bracket (dots 4-6, dots 4-6, dots 1-2-3-5-6) and the closing angle bracket (dots 4-6, dots 4-6, dots 2-3-4-5-6) are commonly used when writing vector notation in component form. The numeric indicator is not used inside grouping symbols such as the angle brackets. See vectors for more Nemeth Code related to vectors. The example document includes vector notation as well a the angle brackets used in component form. Note: The "of" contraction is used after the two cells of dot 4-6 for the opening angle bracket and the "with" contraction is used after the two cells of dot 4-6 for the closing angle bracket. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Antiderivative

The antiderivative of a function is denoted by the capital letter of that function name, opening parenthesis (dots 1-2-3-5-6), x, closing parenthesis (dots 2-3-4-5-6). Note: The opening and closing parentheses are the same as the "of" and "with" contractions. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Approximately equal to

Approximately equal to (dot 4, dots 1-5-6, dot 4, dots 1-5-6) is represented in print as parallel wavy lines. Each wavy line is called a tilde (dot 4, dots 1-5-6). The entire approximately equal to symbol has a space before and after since it is a sign of comparison. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Arcs

Arcs are denoted using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified, which should be two capital letters for a minor arc or three capital letters for a major arc. The capitalization indicator must be placed before each capitalized letter since each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the arc symbol (dots 1-2-4-6, dot 1). This is the shape indicator followed by the letter a for arc. Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. Also, a letter m in front of an arc means the measure of. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Barred brackets (used for ceiling, floor, and integer functions)

The integer function is one of three types. The general integer function is written opening bracket (dot 4, dots 1-2-3-5-6), the letter x (dots 1-3-4-6), closing bracket (dot 4, dots 2-3-4-5-6). The greatest integer function, also known as the floor function, is represented with a special form of brackets that includes only the lower part of the bracket. The lower left bracket is dot 4, dots 5-6, dots 1-2-3-5-6 and the lower right bracket is dot 4, dots 5-6, dots 2-3-4-5-6. The least integer function, also known as the ceiling function, is represented with a special form of brackets that includes only the upper part of the bracket. The upper left bracket is dot 4, dots 4-5, dots 1-2-3-5-6 and the upper right bracket is dot 4, dots 4-5, dots 2-3-4-5-6. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Beta (lowercase)

Beta (dots 4-6, dots 1-2) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter b (dots 1-2). The lowercase form of this letter is often used to represent an angle measure. When writing this immediately after a trigonometric function such as sin, cos, tan, csc, sec, or cot, include a space before the Greek letter indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Biconditional statements, also known as if and only if, which use the two-way horizontal arrow

The two-way horizontal arrow (dots 1-2-4-6, dots 2-4-6, dots 2-5, dots 2-5, dots 1-3-5) is used for biconditional statements. It can also be written with a double arrow shaft, called a spear (dots 1-2-4-6, dots 2-4-6, dots 2-3-5-6, dots 2-3-5-6, dots 1-3-5). There should be a space before and after the double arrow when used as a biconditional statement since it is a sign of comparison. Note: The shape indicator is the same as the "ed" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Binomial coefficient

The binomial coefficient is written opening parenthesis, n or a number that represents n, directly-under indicator (dots 1-4-6), r or a number that represents r. It is used in probability and the binomial theorem and read "n choose r" where n and r are two numbers written vertically in print. Note: The opening and closing parentheses are the same as the "of" and "with" contractions. The directly-under indicator is the "sh" contraction. Think about the word shallow which has the "sh" contraction as being a way to put something under (or in a shallow position to) something else. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Braces, also known as curly brackets

Braces, sometimes called curly brackets, are a type of grouping symbol. The opening brace (dots 4-6, dots 1-2-3-5-6) and closing brace (dots 4-6, dots 2-3-4-5-6) are used as a third level of parentheses or for set notation. The numeric indicator is not used inside grouping symbols such as braces. Note: The opening and closing indicators after the dots 4-6 are the same as the "of" and "with" contractions. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Brackets

Brackets are a type of grouping symbol. The opening bracket (dot 4, dots 1-2-3-5-6) and closing bracket (dot 4, dots 2-3-4-5-6) are used as a second level of parentheses or for the integer function. Note that brackets are not always used in pairs. The numeric indicator is not used inside grouping symbols such as brackets. The closing right bracket is commonly used in the process of calculating definite integrals in calculus. Note: The opening and closing indicators after the dot 4 are the same as the "of" and "with" contractions. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Caret

The caret (dots 4-5-6, dots 1-2-6) can be used for representing an exponent or superscript. It is also commonly used in statistics and called a hat. When it is positioned over a variable, use the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified, which is the single variable. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the caret (dots 4-5-6, dots 1-2-6). Step 5: Write the termination indicator (dots 1-2-4-5-6). See examples Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Cent sign

The cent sign (dot 4, dots 1-4) is used to represent money and is placed right after the number representing the number of cents in both print and Braille. The word cent starts with a letter c, and the letter c is used in the Nemeth symbol. Also, the print cent sign includes a letter c with a slash through it. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Circle

The circle symbol (dots 1-2-4-6, dots 1-4). This is the shape indicator followed by the letter c for circle. A circle is always named by its center so when naming a circle, insert a space before writing the capital letter representing the center of the circle. Note: The shape indicator is the same as the "ed" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Circle with interior dot

The circle with interior dot (dots 1-2-4-6, dots 1-4, dots 4-5-6, dots 1-2-4-6, dots 1-6, dots 1-2-4-5-6) is written with the shape indicator (dots 1-2-4-6), the letter c (dots 1-4) for circle, the interior shape modification indicator (dots 4-5-6, dots 1-2-4-6), the interior dot (dots 1-6), and finally the termination indicator (dots 1-2-4-5-6). A circle is always named by its center so when naming a circle, insert a space before writing the capital letter representing the center of the circle. Note: The shape indicator is the same as the "ed" contraction and the termination indicator is the same as the "er" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Combinations

Combinations which are read n choose r can be written several different ways. Method 1: Write capital c, opening parenthesis (dots 1-2-3-5-6), n or a number that represent n, comma (dot 6), r or a number that represent r, closing parenthesis (dots 2-3-4-5-6). Method 2: Write capital c, subscript indicator (dots 5-6), n or a number that represents n, dots 2-4-6 which represents a print comma in a subscript, r or a number that represents r. Method 3: Write subscript indicator (dots 5-6), n or a number that represents n, baseline indicator (dot 5), capital c, r or a number that represent r. If r is a number, no subscript is needed before it, but if it is a variable, the subscript indicator is needed before it. Method 4: Write opening parenthesis (dots 1-2-3-5-6), n or a number that represents n, directly under indicator (dots 1-4-6), r or a number that represents r, closing parenthesis (dots 2-3-4-5-6). Note: The opening and closing parentheses are the same as the "of" and "with" contractions. The directly-under indicator is the same as the "sh" contraction. Think of the word shallow which has the "sh" contraction as being a way to put something under (or in a shallow position to) something else. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Complex fractions (linear format)

Complex fractions are fractions where the numerator or the denominator or both have at least one simple fraction. The opening complex fraction indicator (dot 6, dots 1-4-5-6) starts the outermost fraction and the closing complex fraction indicator (dot 6, dots 3-4-5-6) completes the outermost fraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Complex fractions (spatial format)

Complex fractions are fractions where the numerator or the denominator or both have at least one simple fraction. Spatial format for complex fractions are written on multiple lines. The lines above the fraction line include the outermost numerator centered. The line in between the numerator and denominator has the opening complex fraction indicator (dot 6, dots 1-4-5-6), fraction line as a series of dots 2-5 with a length corresponding to the length of either the numerator or denominator (whichever is longest), and the closing complex fraction indicator (dot 6, dots 3-4-5-6). The lines below the separation line include the outermost denominator centered. For additional information, see simple fractions (spatial format). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Composite functions

The hollow dot (dots 4-6, dots 1-6) is used for composite functions. It is represented by a small open circle in print and is generally written between the letters that represent the two functions. Since it is considered a sign of operation in this context, there is no space used on either side of the hollow dot. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Conditional statements, also known as if-then statements, which use the right-pointing arrow

The right-pointing arrow, used for conditional statements, has an uncontracted form (dots 1-2-4-6, dots 2-5, dots 2-5, dots 1-3-5) and a contracted form (dots 1-2-4-6, dots 1-3-5). The contracted form is used most frequently. It can also be written with a double arrow shaft, called a spear (dots 1-2-4-6, dots 2-3-5-6, dots 2-3-5-6, dots 1-3-5). There should be a space before and after the arrow when used as a conditional statement since it is a sign of comparison. Note: The shape indicator is the same as the "ed" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Congruent to

Congruent to (dot 4, dots 1-5-6, dots 4-6, dots 1-3) is represented in print as a wavy line over the equal sign. This wavy line is called a tilde (dot 4,dots 1-5-6). The congruent to symbol is the tilde followed by the equal sign (dots 4-6, dots 1-3). The entire congruent to symbol has a space before and after since it is a sign of comparison. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Conjunction (and)

Conjunction (dot 4, dots 1-4-6) is considered an operator that connects two statements with the word AND. There are no spaces before or after the symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Corresponds to

Corresponds to is represented by a two-way horizontal arrow (dots 1-2-4-6, dots 2-4-6, dots 2-5, dots 2-5, dots 1-3-5). There should be a space before and after the double arrow when used to represent the terms "corresponds to" since it is a sign of comparison. Note: The shape indicator is the same as the "ed" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Cosecant

Cosecant is a trigonometric function abbreviated as csc. Therefore, those three letters csc are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Cosine

Cosine is a trigonometric function abbreviated as cos. Therefore, those three letters cos are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Cotangent

Cotangent is a trigonometric function abbreviated as cot. Therefore, those three letters cot are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Curly d

The curly d (dot 4, dots 1-4-5) represents the partial derivative, often used in notation for calculus. Notice the letter d is used. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Decimal point

The decimal point (dots 4-6) in print is a small dot used to separate the whole number part and the decimal (or rational) part of a number. It is often just read as the word point when encountered in a number or problem. If a number includes a decimal point, but does not include a whole number part, it can be written after a zero or directly after a numeric indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Definite integral (using directly-under and directly-over)

The definite integral is an integral over a certain interval. It is occasionally written using directly under and directly over. In this case, it is written as a modified expression using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified, which is the integral sign (dots 2-3-4-6), Step 3: Write the directly-under indicator (dots 1-4-6). Step 4: Write the modifier, which is the starting value of the interval. Repeat Step 3: Write the directly-over indicator (dots 1-2-6). Repeat Step 4: Write the modifier, which is the ending value of the interval. Step 5: Write the termination indicator (dots 1-2-4-5-6). Then write the rest of the problem, which includes the function and then dx. Usually, the interval values are subscripts and superscripts instead of directly-under and directly-over. See Definite integral (using subscripts and superscripts). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Definite integral (using subscripts and superscripts)

The definite integral is an integral over a certain interval. It is most commonly written using the following steps. Step 1: Write the integral sign (dots 2-3-4-6). Step 2: Write a subscript indicator (dots 5-6) and the starting value of the interval. Step 3: Write the superscript indicator (dots 4-5) and the ending value of the interval. Step 4: Write the baseline indicator (dot 5) followed by the function and then dx. On occasion, the interval values are directly-over or directly-under instead of just raised and lowered. In this case, the Five-Step Rule is used. See Definite integral (using directly-under and directly-over). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Degrees (used in angle measure and temperature in Celsius/Centigrade or Fahrenheit)

Degrees (dots 4-5, dots 4-6, dots 1-6) can be used to specify an angle or arc measure, to write degrees Fahrenheit, and to write degrees Celsius or Centigrade. It begins with the superscript indicator (dots 4-5), since it is raised in print, followed by the two-cell hollow dot symbol (dots 4-6, dots 1-6). Also, the m in front of the angle notation means the measure of. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Delta (lowercase)

Delta (dots 4-6, dots 1-4-5) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter d (dots 1-4-5). The lowercase form of this letter is often used to represent a small change in the value of a variable in calculus. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Delta (uppercase)

Delta is a Greek letter. The uppercase form of Delta (dots 4-6, dot 6, dots 1-4-5) is most often used to represent the change in something. Begin with a Greek letter indicator (dots 4-6) and then write the capitalization indicator (dot 6) followed by the letter d (dots 1-4-5). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Density (lowercase rho)

Density is written using the lowercase Greek letter rho (dots 4-6, dots 1-2-3-5) which is actually the Greek letter indicator followed by the letter r. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Derivative of f of x (using prime)

The derivative of a function f of x or the derivative of y is written using prime which looks like an apostrophe in print and is written as a dot 3 which is the same dot configuration as an apostrophe in Braille. The second derivative repeats the dot 3 for the second prime. For subsequent derivatives use the superscript indicator (dots 4-5) followed by the number representing the level of derivative. If anything follows this number other than a space, insert the baseline indicator (dot 5). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Derivative of y with respect to x (using raised numbers after the first derivative)

The derivative of y with respect to x is written as a simple fraction which is written opening fraction indicator (dots 1-4-5-6), dy, horizontal fraction line (dots 3-4), dx, closing fraction indicator (dots 3-4-5-6). For the 2nd, 3rd, or other derivatives of y with respect to x, the superscript indicator (dots 4-5) is used to raise the 2, 3, etc. along with the baseline indicator (dot 5) after the 2, 3, etc. This superscript is placed between the d and the y in the numerator and after the dx in the denominator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Directly-over indicator

The directly-over indicator (dots 1-2-6) is used in a variety of contexts such as repeating decimals, the Five-Step Rule, summation notation, limits, and integrals for instance. To see any of these contexts by themselves, go to those definitions specifically. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Directly-under indicator

The directly-under indicator (dots 1-4-6) is used in a variety of contexts such as the binomial coefficient, combinations, limits, summation notation, and place value for instance. To see any of these contexts by themselves, go to those definitions specifically. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Directly-under indicator (used in place value)

The directly-under indicator (dots 1-4-6) is sometimes used in conjunction with certain digits when referring to place value. When a single digit in a number is underlined in print, the digit that is underlined is followed by this directly-under indicator (dots 1-4-6) and then the horizontal bar (dots 1-5-6). If more than one consecutive digit is underlined, this shortcut cannot be used. Instead, the Five-Step Rule needs to be applied. Step 1: Write the multipurpose indicator (dot 5) before the digit that is underlined. Step 2: Write the expression being modified, which should be the underlined digits. Step 3: Write the directly-under indicator (dots 1-4-6). Step 4: Write the modifier which is the horizontal bar (dots 1-5-6). Step 5: Write the termination indicator (dots 1-2-4-5-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Disjunction (or)

Disjunction (dot 4, dots 3-4-6) is considered an operator that connects two statements with the word OR. There are no spaces before or after the symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Divided by

The divided by sign (dots 4-6, dots 3-4), called the obelus, is a horizontal line with a dot above and below it in print. It is used between two numbers, variables, or other symbols to represent division. In a linear format, there should not be a space on either side of the divided by sign. Also, a numeric indicator is not used after the divided by sign if a number does follow it. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Divided into, also called a division bracket

The divided into sign (dots 1-3-5) is written between the divisor and the dividend. The divisor has a numeric indicator in front of it, but the dividend does not when the division arrangement contains only the divisor and dividend. This database does not cover spatial arrangements of division at this time. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Dollar sign

The dollar sign (dot 4, dots 2-3-4) is used to represent money and is placed in front of the number of dollars in both print and Braille. No numeric indicator is needed. Note: The letter s is used in the print symbol of a dollar sign, but with a vertical line(s) through it. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Ellipsis

The ellipsis (dot 3, dot 3, dot 3) is used as a sign of omission. The ellipsis usually has a space before and after the symbol unless punctuation comes after the ellipsis. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Empty set (also known as null set)

The empty set, or null set, can be written as a set of braces with a space between them or as a circle with a line through it. If writing it as a set of braces, write opening brace (dots 4-6, dots 1-2-3-5-6, space, closing brace (dots 4-6, dots 2-3-4-5-6). Note: The "of" and "with" contractions are used after the dots 4-6 for the opening and closing braces. If writing the empty set as a print zero with a line through it, write dots 4-5-6, dots 3-5-6. Note: the number zero is part of this second notation. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Enlarged braces (used for systems of equations, compound functions, or piecewise functions)

Enlarged braces are used when writing systems of equations, compound functions, or piecewise functions. To enlarge braces, just place a dot 6, as if it is capitalized, between the dots 4-6 and the opening or closing cell which are dots 1-2-3-5-6 and dots 2-3-4-5-6 respectively. This set of three cells is written at the beginning and/or end of each line it pertains to. Note: The "of" and "with" contractions are used after the dots 4-6, dot 6 for the opening and closing braces. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Enlarged brackets (used for matrices)

Enlarged brackets are used when writing matrices. To enlarge brackets, just place a dot 6, as if it is capitalized, between the dot 4 and the opening or closing cell which are dots 1-2-3-5-6 and dots 2-3-4-5-6 respectively. This set of three cells is written at the beginning and/or end of each line it pertains to. Note: The "of" and "with" contractions are used after the dot 4, dot 6 for the opening and closing braces. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Enlarged parentheses

Enlarged parentheses can be used in place of large brackets when writing matrices. To enlarge parentheses, just place a dot 6, as if it is capitalized, in front of both the opening parenthesis (dots 1-2-3-5-6) and closing parenthesis (dots 2-3-4-5-6). Note: The opening and closing parentheses are the same as the "of" and "with" contractions respectively. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Enlarged vertical bars (used for determinants)

Enlarged vertical bars are used when writing determinants. To enlarge vertical bars, just place a dot 6, as if it is capitalized, before the vertical bar (dots 1-2-5-6). This set of two cells is written at the beginning and end of each line it pertains to. Note: The vertical bar is the same as the "ou" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Epsilon (lowercase)

Epsilon (dots 4-6, dots 1-5) is a Greek letter. The lowercase form of this letter is often used to represent an arbitrarily small positive quantity in calculus. Begin with a Greek letter indicator (dots 4-6) and then write the letter e (dots 1-5). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Equals, also known as is equal to

The equals sign (dots 4-6, dots 1-3). There should be a space before and after the equals sign since it is a sign of comparison. Notice that you begin brailling using two fingers of the right hand followed by two fingers of the left hand - always right hand and then left hand. It's a pattern. Also, two fingers are equal to two fingers or two dots are equal to two dots. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Exponent, also known as superscript or power (which sometimes uses a baseline indicator)

The superscript indicator (dots 4-5) is used to represent exponents, powers, or raised characters such as degree. It is used after the base and before the actual raised character or exponent. If something comes after the exponent other than a space, a baseline indicator (dot 5) must be used after the exponent. If a space comes after the exponent, no baseline indicator is needed. If an exponent is raised to another exponent, the superscript indicator is used twice instead of once before the second exponent. The same indicator is used to represent a raised number such as in the notation for an inverse function which is the letter f (dots 1-2-4) followed by the superscript indicator (dots 4-5) and then a negative one (dots 3-6, dot 2). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Factorial

The factorial sign (dots 1-2-3-4-6) is used in probability to represent the number of permutations or possible arrangements of items. It is used directly after that number of items. In print, an exclamation point is used, but an exclamation point should not be used in Nemeth Code to represent factorial. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Five-Step Rule

The Five-Step Rule is as follows. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified. If these are capital letters representing points, the capitalization indicator must be placed before each capitalized letter since each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6) or the directly-under indicator (dots 1-4-6). Step 4: Write the modifier which is what is written directly-over or under. Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction or shallow which uses the "sh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Function notation

Function notation starts with a letter that represents the function name and then uses parentheses around the independent variable used in the function. The opening parentheses is dots 1-2-3-5-6 and the closing parentheses is dots 2-3-4-5-6. Note: The opening and closing parentheses are the same as the "of" and "with" contractions. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Gamma (lowercase)

Gamma (dots 4-6, dots 1-2-4-5) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter g (dots 1-2-4-5). The lowercase form of this letter is often used to represent an angle measure. When writing this immediately after a trigonometric function such as sin, cos, tan, csc, sec, or cot, include a space before the Greek letter indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greater than

The greater than sign (dots 4-6, dot 2) should be written with a space before and after it since it is a sign of comparison. Notice that you begin brailling using two fingers of the right hand followed by one finger of the left hand - always right hand and then left hand. It's a pattern. Also, two fingers are greater than one finger or two dots are greater than one dot. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greater than or equal to

The greater than or equal to sign (dots 4-6, dot 2, dots 1-5-6) is written with the greater than sign (dots 4-6, dot 2) followed by the horizontal bar (dots 1-5-6). There should be a space before and after the entire greater than or equal to sign since it is a sign of comparison. In print, the horizontal bar is written under the greater than symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greek letters (complete list)

Any letters of the Greek alphabet begin with dots 4-6 to identify it as a Greek letter. All lowercase letters begin with dots 4-6 and are followed by the letter(s) it represents. All uppercase letters will still begin with dots 4-6 followed by the capitalization indicator (dot 6) and then by the letter(s) it represents. The following Greek letters use a contraction to represent multiple letters: eta uses the contraction "wh" (dots 1-5-6), theta uses the contraction "th" (dots 1-4-5-6), and chi uses the contraction "and" (dots 1-2-3-4-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greek letters (used for angle measure)

The Greek letters most commonly used as variables that represent angle measure are alpha (dots 4-6, dot 1), beta (dots 4-6, dots 1-2), gamma (dots 4-6, dots 1-2-4-5), theta (dots 4-6, dots 1-4-5-6), and phi (dots 4-6, dots 1-2-4). Pi (dots 4-6, dots 1-2-3-4) is also used to represent a particular angle measure in radians. All of these letters are lowercase and have dots 4-6 in front of the letter(s) it represents. Alpha represents a, beta represents b, gamma represents g, theta represents th and uses the "th" contraction, phi represents f, and pi represents p. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Hollow dot

The hollow dot (dots 4-6, dots 1-6) can be used for composite functions or as part of the degree symbol (dots 4-5, dots 4-6, dots 1-6). It is represented by a small open circle in print. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Horizontal bar

The horizontal bar (dots 1-5-6) can be used in repeating decimals, the mean, segments, and to represent place value. To see any of these contexts by themselves, go to those definitions specifically. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Hypercomplex fractions

Hypercomplex fractions are fractions where the numerator or denominator or both contain at least one complex fractions. To denote a hypercomplex fraction use a dot 6, dot 6 before the fraction indicators at the beginning and end of the fraction line. Additional dot 6's are inserted for additional levels. For more information, see complex fractions (spatial format). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Indexed radical or root (also known as nth root)

The indexed root, nth root, or indexed radical are used to indicate roots other than the square root (which is the same as the 2nd root.) The index is assumed to be two for a square root, but is explicitly stated for all other roots. For example, the index of the third root, also known as the cube root, is three. This index is placed between the index-of-radical indicator (dots 1-2-6) and the radical symbol (dots 3-4-5). Then write what you are taking the root of without using any numeric indicators. Finally, finish or terminate the root with the termination indicator (dots 1-2-4-5-6). If the problem has a nested radical or a radical inside of another radical, then dots 4-6 is placed in front of the index-of-radical indicator and termination indicator for the inner radical. Two dots 4-6 cells are placed in front of a 3rd level radical, three dots 4-6 cells are placed in front of a 4th level radical, etc. This should always be placed in front of both the index-of-radical indicator and termination indicator corresponding to that particular square root. Note: The index-of-radical indicator is the same as the "gh" contraction, the radical indicator is the same as the "ar" contraction, and the termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Infinity

The infinity symbol (dot 6, dots 1-2-3-4-5-6) is used to represent that a list of numbers continues forever. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Integral

The integral, or indefinite integral, starts with the integral sign (dots 2-3-4-6). Then it is followed by the function and ends with dx. For the second integral or double integral, the integral sign is just repeated. The integral sign continues to be repeated for any subsequent integrals. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Intersection (of sets)

Intersection (dot 4, dots 1-4-6) is considered an operator, so there are no spaces before or after the symbol. Note: Since finding an intersection of two sets is the same as finding the elements that are the same or shared between two sets, think of dots 1-4-6 as the "sh" contraction used at the beginning of the word shared. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Interval notation

Interval notation uses grouping symbols such as parentheses and brackets. It is an opening parenthesis (dots 1-2-3-5-6) or opening bracket (dot 4, dots 1-2-3-5-6), number (with no numeric indicator) or negative infinity (dots 3-6, dot 6, dots 1-2-3-4-5-6), comma (dot 6), space, number (no numeric indicator) or infinity (dot 6, dots 1-2-3-4-5-6), and closing parenthesis (dots 2-3-4-5-6) or closing bracket (dot 4, dots 2-3-4-5-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Inverse of a function

The inverse of a function is written using a letter such as f, g, or h or a trig function such as sin, cos, or tan followed by the superscript indicator (dots 4-5), negative one (dots 3-6, dot 2). If a parenthesis follows, a baseline indicator (dot 5) must be used first. If trig functions are used, the baseline indicator is not needed since these are followed by a space. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Inverse trigonometric functions

Inverse trigonometric functions can be written two ways. Method 1: Use "arc" in front of the 3-letter abbreviation for the trigonometric function without using the "ar" contraction. A space can be inserted between "arc" and the trigonometric function. Method 2: Insert a superscripted negative one after the 3-letter abbreviation for the trigonometric function by writing the superscript indicator (dots 4-5) followed by the negative sign (dots 3-6) and the number one (dot 2). No numeric indicator is used in the superscript. Insert a space after the inverse trigonometric function if it is followed by a number or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Lambda (lowercase)

Lambda (dots 4-6, dots 1-2-3) is a Greek letter. The lowercase form of this letter is often used to represent wavelength. Begin with a Greek letter indicator (dots 4-6) and then write the letter l (dot 1-2-3). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Less than

The less than sign (dot 5, dots 1-3) should be written with a space before and after it since it is a sign of comparison. Notice that you begin brailling using one finger of the right hand followed by two fingers of the left hand - always right hand and then left hand. It's a pattern. Also, one finger is less than two fingers or one dot is less than two dots. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Less than or equal to

The less than or equal to sign (dot 5, dots 1-3, dots 1-5-6) is written with the less than sign (dot 5, dots 1-3) followed by the horizontal bar (dots 1-5-6). There should be a space before and after the entire less than or equal to sign since it is a sign of comparison. In print, the horizontal bar is written under the less than symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Limits

The less than or equal to sign (dot 5, dots 1-3, dots 1-5-6) is written with the less than sign (dot 5, dots 1-3) followed by the horizontal bar (dots 1-5-6). There should be a space before and after the entire less than or equal to sign since it is a sign of comparison. In print, the horizontal bar is written under the less than symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Lines

Lines are denoted using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified which should be two capital letters. The capitalization indicator must be placed before each capitalized letter since each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the two-way arrow symbol (dots 1-2-4-6, dots 2-4-6, dots 2-5, dots 2-5, dots 1-3-5). Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Logarithms (including common and natural logarithms)

There are four ways logarithms are written. 1) Natural logarithms are generally written as ln even though they mean log base e. 2) Common logarithms are generally written as log even though they mean log base 10. 3) Logarithms with a numeric base are written log immediately followed by the number indicating it is a subscript. No numeric indicator or subscript indicator is needed when functions are followed by a numeric subscript. 4) Logarithms with a letter or variable base are written log followed by the subscript indicator (dots 5-6) and the letter or variable in the subscript. Follow spacing rules as shown in the examples because logarithms are functions and therefore need to be followed by a space. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Maps to, using the right-pointing arrow

There are four ways logarithms are written. 1) Natural logarithms are generally written as ln even though they mean log base e. 2) Common logarithms are generally written as log even though they mean log base 10. 3) Logarithms with a numeric base are written log immediately followed by the number indicating it is a subscript. No numeric indicator or subscript indicator is needed when functions are followed by a numeric subscript. 4) Logarithms with a letter or variable base are written log followed by the subscript indicator (dots 5-6) and the letter or variable in the subscript. Follow spacing rules as shown in the examples because logarithms are functions and therefore need to be followed by a space. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Mathematical comma

The mathematical comma (dot 6), sometimes just called a comma, is used for numbers larger than three digits, lists, ordered pairs, etc. Unless the comma is within a multi-digit number, a space is written after the comma. See examples: Nemeth in EBAE (BRF), Nemeth within UEB co ntexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Mean

The mean can be written two ways. It can be written as the lowercase Greek letter mu (dots 4-6, dots 1-3-4) which is actually the Greek letter indicator followed by the letter m. The second way it can be written is called x bar. This is written x (dots 1-3-4-6) followed by the horizontal bar (dots 1-5-6). In print, this is written as x with a horizontal bar over it. Most statisticians use x bar to represent the mean of a sample and mu to represent the mean of a population. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Minus sign

The minus sign (dots 3-6), also known as a subtraction sign, is used between two numbers, variables, or other symbols to represent subtraction. In a linear format, there should not be a space on either side of the minus sign. Also, a numeric indicator is not used after the minus sign if a number does follow it. When placed in a spatial format, the minus sign is placed one cell to the left of the widest number in the problem and directly above the separation line. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Mixed number

A mixed number has a whole part followed by a fractional part. Mixed numbers begin with the numeric indicator with the whole number followed immediately by the fractional part. The fractional part of the mixed number includes the following in order: the opening mixed number indicator (dots 4-5-6, dots 1-4-5-6), numerator, horizontal fraction line (dots 3-4) or diagonal fraction line (dots 4-5-6, dots 3-4), denominator, and closing mixed number indicator (dots 4-5-6, dots 3-4-5-6). Note: The last three examples on the Braille documents should be read on an embossed copy instead of displayed on a single line Braille display. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Negation, or not

Negation (dots 3-4), or not, can be placed in front of many symbols such as equal to, greater than, less than, congruent to, parallel to, perpendicular to, element of, and subset of. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Negative sign

The negative sign (dots 3-6) is placed in front of numbers, variables, or other symbols. When placed in front of a number with a numeric indicator, it should be placed before the numeric indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Nemeth Code switch indicators

When Nemeth Code is to be used for mathematics and science, the actual math and technical notation is presented in Nemeth Code while the surrounding text is presented in UEB. Three symbols are used to support this switching. The opening Nemeth Code indicator (dots 4-5-6, dots 1-4-6) is used before beginning Nemeth Code. The Nemeth Code terminator (dots 4-5-6, dots 1-5-6) is used at the end of the Nemeth Code. Finally, the single-word switch indicator (dot 6, dot 3) is used to stay in Nemeth Code for a single word and is used right before that word. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Number systems

The different number systems which include natural, whole, integers, rational, real, complex, and imaginary are represented by uppercase letters. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Omega (lowercase used in angular velocity)

Omega (dots 4-6, dots 2-4-5-6), is a Greek letter. The lowercase form of this letter is often used to represent an angular velocity. Begin with a Greek letter indicator (dots 4-6) and then write the letter w (dots 2-4-5-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Ordered pair

Ordered pairs are grouped in an opening parenthesis (dots 1-2-3-5-6) and a closing parenthesis (dots 2-3-4-5-6) with a mathematical comma (dot 6) between the horizontal (x) coordinate and the vertical (y) coordinate. There is also a space after the comma. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Parallel to

Parallel to (dots 1-2-4-6, dots 1-2-3) is written using the shape indicator (dots 1-2-4-6) followed by the letter l (dots 1-2-3). Notice the number of times the letter l appears in the word parallel. In print, the symbol is represented by two vertical lines. Note: the shape indicator is the same as the "ed" contraction. Any capital letter used to name a segment, ray, or line should have a capitalization indicator in front of each letter in its name. A double cap should not be used in the naming of a segment, ray, or line. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Parentheses (basic)

Parentheses are grouping symbols that have a beginning and an ending often called an opening and a closing parenthesis. The opening parentheses (dots 1-2-3-5-6) is the beginning of the grouping while the closing parentheses (dots 2-3-4-5-6) is the end of the grouping. The numeric indicator is not used inside grouping symbols such as parentheses. For parentheses that occur over more than one line, see enlarged parentheses. Note: The opening and closing parentheses are the same as the "of" and "with" contractions. Parentheses are always used when writing function notation which is written f(x) and read f of x. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Parentheses (used with combinations and the binomial coefficient)

The binomial coefficient is written opening parenthesis, n or a number that represents n, directly-under indicator (dots 1-4-6), r or a number that represents r. It is used in probability and the binomial theorem and read "n choose r" where n and r are two numbers written vertically in print. Note: The opening and closing parentheses are the same as the "of" and "with" contractions. The directly-under indicator is the "sh" contraction. Think about the word shallow which has the "sh" contraction as being a way to put something under (or in a shallow position to) something else. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Partial derivative, also known as curly d

The partial derivative, sometimes called the curly d (dot 4, dots 1-4-5), is often used in notation for calculus. Notice how this uses the letter d (dots 1-4-5). See examples: Nemeth in EBAE (BRF), Nemeth within UEB co ntexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Percent

The percent sign (dot 4, dots 3-5-6) occurs immediately after a number or variable with no space between the number or variable and the percent sign. Note: Both the print and Braille percent signs use the number zero. Percents are parts per 100, or out of 100, which has two zeros in it. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Permutations

Permutations which are read n permutation r can be written several different ways. Method 1: Write capital p, opening parenthesis (dots 1-2-3-5-6), n or a number that represents n, comma (dot 6), r or a number that represents r, closing parenthesis (dots 2-3-4-5-6). Note: The opening and closing parentheses are the same as the "of" and "with" contractions. Method 2: Write capital p, subscript indicator (dots 5-6), n or a number that represents n, dots 2-4-6 which represents a print comma in a subscript, r or a number that represents r. Method 3: Write subscript indicator (dots 5-6), n or a number that represents n, baseline indicator (dot 5), capital p, r or a number that represents r. If r is a number, no subscript is needed before it, but if it is a variable, the subscript indicator is needed before it. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Perpendicular to

Perpendicular to (dots 1-2-4-6, dots 1-2-3-4) is written using the shape indicator (dots 1-2-4-6) followed by the letter p (dots 1-2-3-4). Notice that perpendicular has two p's including the beginning letter. In print, the symbol is represented as a short horizontal line centered at the bottom of a short vertical line. Note: The shape indicator is the same as the "ed" contraction. Any capital letter used to name a segment, ray, or line should have a capitalization indicator in front of each letter in its name. A double cap should not be used in the naming of a segment, ray, or line. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Phi (lowercase)

Phi (dots 4-6, dots 1-2-4) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter f (dot 1-2-4). The lowercase form of this letter is often used to represent an angle measure. When writing this immediately after a trigonometric function such as sin, cos, tan, csc, sec, or cot, include a space before the Greek letter indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Pi (lowercase)

Pi (dots 4-6, 1-2-3-4) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter p (dots 1-2-3-4). The lowercase form of this letter is often used to represent an angle measure in radians. It is also the ratio of the circumference of a circle to its diameter and is equal to approximately 3.14. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Plus or minus

The plus or minus sign, combined vertically in print, is simply represented as a plus sign (dots 3-4-6) followed directly by a minus sign (dots 3-6). No spaces should be used before or after this symbol when surrounded by other numbers, variables, or symbols. If the two signs are combined horizontally in print, a multipurpose indicator (dot 5) should be written between the two symbols to indicate the symbols are aligned horizontally. Again, no space should be used around these symbols when surrounded by other numbers, variables, or symbols. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Plus sign

The plus sign (dots 3-4-6), also known as an addition sign, is used between two numbers, variables, or other symbols to represent addition. In a linear format, there should not be a space on either side of the minus sign. Also, a numeric indicator is not used after the minus sign if a number does follow it. When placed in a spatial format, the plus sign is placed one cell to the left of the widest number in the problem and directly above the separation line. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Power, also known as exponent or superscript (which sometimes uses a baseline indicator)

The superscript indicator (dots 4-5) is used to represent exponents, powers, or raised characters such as degree. It is used after the base and before the actual raised character or exponent. If something comes after the exponent other than a space, a baseline indicator (dot 5) must be used after the exponent. If a space comes after the exponent, no baseline indicator is needed. If an exponent is raised to another exponent, the superscript indicator is used twice instead of once before the second exponent. The same indicator is used to represent a raised number such as in the notation for an inverse function which is the letter f (dots 1-2-4) followed by the superscript indicator (dots 4-5) and then a negative one (dots 3-6, dot 2). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Prime

Prime (dot 3) can be used to represent feet, minutes, transformations, complements for sets, the transpose of a matrix, and the first derivative of a function. Prime is the same dot configuration as an apostrophe and looks like an apostrophe in print. Double prime (dot 3, dot 3) can be used to represent inches, seconds, transformations, and the second derivative of a function. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Proportion

Proportions can be represented using the equals sign (dots 4-6, dots 1-3) or by using a proportion symbol (dots 5-6, dots 2-3). This symbol is read using the word as and relates to two ratios. There should be a space on each side of a proportion symbol. The print symbol looks like two colons, but colons should not be used in Nemeth Code to represent proportions. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Ratio

The ratio symbol (dot 5, dot 2) is read "is to" and relates two quantities. There should be a space on each side of the ratio symbol. The print symbol looks like a colon, but a colon should not be used in Nemeth Code to represent ratios. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Rational numbers (as fractions, mixed numbers, and decimals)

Rational numbers (as fractions, mixed numbers, and decimals) are any numbers that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Examples and non-examples are given in a variety of ways in the example documents. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Rays

Rays are denoted using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified which should be two capital letters. The capitalization indicator must be placed before each capitalized letter since each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the contracted right arrow (dots 1-2-4-6, dots 1-3-5), the uncontracted right arrow (dots 1-2-4-6, dots 2-5, dots 2-5, dots 1-3-5) or the left arrow (dots 1-2-4-6, dots 2-4-6, dots 2-5, dots 2-5). Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The contracted form of the right arrow is used most frequently. The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Repeating decimals

Repeating decimals are written differently based on whether one digit is repeated or multiple digits are repeated. If only one digit is repeated, that digit is followed by the horizontal bar (dots 1-5-6). If multiple digits are repeating, the Five-Step Rule is used. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified, which are the digits being repeated. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the horizontal bar (dots 1-5-6). Step 5: Write the termination indicator (dots 1-2-4-5-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Secant

Secant is a trigonometric function abbreviated as sec. Therefore, those three letters sec are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Segments

Segments are denoted using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified which should be two capital letters. The capitalization indicator must be placed before each capitalized letter since each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the horizontal bar (dots 1-5-6). Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Shapes

Shape symbols are represented in Braille using the shape indicator (dots 1-2-4-6) and then, depending on the shape, will be followed by a number, one or more letters, or a configuration of dots using one or more cells, which looks similar to the print shape. Numbers are used for regular polygons, such as a square. The most common shapes have been included as a list in the sample documents. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Sigma (lowercase)

Sigma (dots 4-6, dots 2-3-4) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the letter s (dots 2-3-4). The lowercase form of this letter is often used to represent the standard deviation in statistics. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Sigma (uppercase)

Sigma is a Greek letter. The uppercase form of this letter (dots 4-6, dot 6, dots 2-3-4) is often used to represent the summation operator for a series. Begin with a Greek letter indicator (dots 4-6) followed by the capitalization indicator (dot 6) and then by the letter s (dots 2-3-4). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Signs of comparison

Signs of comparison are always written with a space before and after the sign. The most common signs of comparison have been included as a list in the example documents. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Signs of omission, including general omission symbol, long dash, ellipsis, and shapes

Several different signs of omission are used in print. The Braille versions include the general omission symbol (dots 1-2-3-4-5-6) represented by a question mark or physical blank space in print, the long dash (dots 3-6, dots 3-6, dots 3-6, dots 3-6) represented by a blank line in print, and the ellipsis (dot 3, dot 3, dot 3) represented by three dots in print. Other omissions in print can be represented by shapes such as a square or a circle, which can also be represented in Braille. These begin with a shape indicator (dots 1-2-4-6). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Similar to

Similar to (dot 4, dots 1-5-6) is represented in print as a wavy line called a tilde. The similar to symbol has a space before and after since it is a sign of comparison. Therefore, the symbol is written space, dot 4, dots 1-5-6, space. Any capital letter used to name a triangle should have a capitalization indicator in front of each letter in its name. A double cap should not be used in the naming of a triangle. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Simple fraction (linear format)

Fractions in a linear format are usually written as follows: opening fraction indicator (dots 1-4-5-6), numerator, horizontal fraction line (dots 3-4) or diagonal fraction line (dots 4-5-6, dots 3-4), denominator, and closing fraction indicator (dots 3-4-5-6). The numerator and denominator do not use the numeric indicator in front of them. Sometimes, fractions (which have a diagonal fraction line with the numerator and denominator on the same level) are written as follows: numeric indicator (dots 3-4-5-6), numerator, slash (dots 4-5-6, dots 3-4), denominator. Note: The last two examples on the Braille documents should be read on an embossed copy instead of displayed on a single line Braille display. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Simple fraction (spatial arrangement)

Spatial arrangement for fractions are written on three lines. The first line includes the numerator. The second line has the opening fraction indicator (dots 1-4-5-6), separation line as a series of dots 2-5 with a length corresponding to the length of either the numerator or denominator (whichever is longest), and the closing fraction indicator (dots 3-4-5-6). The third line includes the denominator. Note: Sample Braille documents should be embossed instead of displayed on a single line Braille display. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Since/Because

The symbol for since (dot 4, dots 3-4), also read as because, is sometimes used as a shortcut notation for the word. In print and Braille, it looks like three dots in the shape of a downward facing triangle. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Sine

Sine is a trigonometric function abbreviated as sin. Therefore, those three letters sin are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. Note: The contraction for "in" must not be used. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Square root

Square root uses the radical symbol (dots 3-4-5) to start the square root and uses the termination indicator (dots 1-2-4-5-6) to end or terminate the square root. If the problem has a nested square root or a square root inside of another square root, then a dots 4-6 is placed in front of the "ar" and "er" signs for the inner square root. Two dots 4-6 cells are placed in front of a 3rd level radical, three dots 4-6 cells are placed in front of a 4th level radical, etc. This should always be placed in front of both the "ar" and the "er" corresponding to that particular square root. Note: The radical symbol is the same as the "ar" contraction which is used in the word square and the termination indicator is the same as the "er" contraction (dots 1-2-4-5-6) which is used in the word terminate. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Subscript (which sometimes uses a baseline indicator)

The subscript indicator (dots 5-6) is not always used to represent a subscript. It is not used when it follows a variable (letter) or a function name such as log and the subscript is a number. In these special cases, just write the letter immediately followed by the number with no numeric indicator. When this special case does not apply, use the subscript indicator before the subscript and the baseline indicator (dot 5) after the subscript if something other than a space follows the subscript. There is still no numeric indicator used within the subscript itself. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Summation notation (using directly over and directly-under)

Summation notation is most commonly written as a modified expression using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified, which is uppercase sigma (dots 4-6, dot 6, dots 2-3-4). Step 3: Write the directly-under indicator (dots 1-4-6). Step 4: Write the modifier, which is what is written directly-under. This is usually a variable or letter equals a number. Repeat Step 3: Write the directly-over indicator (dots 1-2-6). Repeat Step 4: Write the modifier, which is what is written directly-over. This is usually a number or infinity (dot 6, dots 1-2-3-4-5-6). Step 5: Write the termination indicator (dots 1-2-4-5-6). Then just write the rest of the problem. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Summation notation (using subscripts and superscripts)

Summation notation is occasionally written using subscripts and superscripts. In this case, use the following steps. Step 1: Write uppercase sigma (dots 4-6, dot 6, dots 2-3-4). Step 2: Write the subscript indicator (dots 5-6), only if the subscript is not a number. Step 3: Write the expression in the subscript. Step 4: Write the superscript indicator (dots 4-5). Step 5: Write the expression in the superscript, which is usually a number or infinity (dot 6, dots 1-2-3-4-5-6). Step 6: Write the baseline indicator (dot 5), and then the rest of the problem. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Superscript, also known as exponent or power (which sometimes uses a baseline indicator)

The superscript indicator (dots 4-5) is used to represent exponents, powers, or raised characters such as degree. It is used after the base and before the actual raised character or exponent. If something comes after the exponent other than a space, a baseline indicator (dot 5) must be used after the exponent. If a space comes after the exponent, no baseline indicator is needed. If an exponent is raised to another exponent, the superscript indicator is used twice instead of once before the second exponent. The same indicator is used to represent a raised number such as in the notation for an inverse function which is the letter f (dots 1-2-4) followed by the superscript indicator (dots 4-5) and then a negative one (dots 3-6, dot 2). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Tally mark, also called hash mark

Tally marks (dots 4-5-6), also called hash marks, are used for counting or tallying results. A tally mark is written one after another in groups of five with a space between the groups. In print, one vertical line is made for each of the first four tally marks; the fifth tally mark is represented by a diagonal line across the previous four. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Tangent

Tangent is a trigonometric function abbreviated as tan. Therefore, those three letters tan are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Therefore

The symbol for therefore (dot 6, dots 1-6) is sometimes used as a shortcut notation for the word. In print and Braille, it looks like three dots in the shape of an upward facing triangle. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Theta (lowercase)

Theta (dots 4-6, dots 1-4-5-6) is a Greek letter. Begin with a Greek letter indicator (dots 4-6) and then write the contraction "th" (dot 1-4-5-6). The lowercase form of this letter is often used to represent an angle measure. When writing this immediately after a trigonometric function such as sin, cos, tan, csc, sec, or cot, include a space before the Greek letter indicator. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Tilde (for logical negation)

Tilde (dot 4, dots 1-5-6), used to represent negation of a statement, is represented in print as a wavy line. Since this is considered an operator, there is no space before or after the symbol in an expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Times sign (including multiplication cross, multiplication dot, and multiplication asterisk)

The times sign can be written three different ways: multiplication cross (dot 4, dots 1-6), multiplication dot (dots 1-6), or multiplication asterisk (dot 4, dots 3-4-5-6). It is used between two numbers, variables, or other symbols to represent multiplication. In a linear format, there should not be a space on either side of the times sign. No numeric indicator is used after a multiplication cross or multiplication dot. A numeric indicator is used after the multiplication asterisk. When placed in a spatial format, the times sign is placed one cell to the left of the last number in the problem and directly above the separation line and that sign is usually the multiplication cross. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Triangle

The triangle symbol (dots 1-2-4-6, dots 2-3-4-5) is written with the shape indicator (dots 1-2-4-6) followed by the letter t (dots 2-3-4-5). When naming a triangle, insert a space before writing the three capital letters that represent the vertices of the triangle. Any capital letter used to name a triangle should have a capitalization indicator in front of each letter in its name. A double cap should not be used in the naming of a triangle. Note: The shape indicator is the same as the "ed" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Trigonometric functions

Trigonometric functions are written using 3-letter abbreviations. Sine is abbreviated sin, cosine is abbreviated cos, tangent is abbreviated tan, cosecant is abbreviated csc, secant is abbreviated sec, and cotangent is abbreviated cot. Therefore, those three letters are used to represent the function. Insert a space after the function if it is followed by an angle name, angle measure, or expression. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Union (of sets)

Union (dot 4, dots 3-4-6) is considered an operator, so there are no spaces before or after the symbol. Note: Since finding a union of two sets is the same as combining or adding the two sets together without repeating any elements, think of dots 3-4-6 as the addition symbol. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Vectors

Vectors are denoted using the Five-Step Rule. Step 1: Write the multipurpose indicator (dot 5). Step 2: Write the expression being modified which should be two capital letters or one lowercase letter. If capital letters are used, the capitalization indicator must be placed before each capitalized letter since, in this case, each letter represents a separate point. Step 3: Write the directly-over indicator (dots 1-2-6). Step 4: Write the modifier which is the vector symbol (dots 1-2-4-6, dots 2-5, dots 2-5, dot 4, dots 1-3-5). Step 5: Write the termination indicator (dots 1-2-4-5-6) Note: The directly-over indicator is the same as the "gh" contraction. Think about the word high which also uses the "gh" contraction. The termination indicator is the same as the "er" contraction which is used in the word termination. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Vertical bar

The vertical bar (dots 1-2-5-6) can be used for absolute value, such that, divides, and magnitude, length, or norm of vectors. Absolute value is represented by a vertical bar on each side of a number or expression. Double or single vertical lines on each side of a vector are used to represent magnitude, length, or norm of vectors. A single vertical line is used to represent the word divides used in number theory and the words such that used in set notation. The numeric indicator is not used inside grouping symbols such as the vertical bars representing absolute value. Note: The vertical bar is the same as the "ou" contraction. See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Wavelength

Wavelength is often represented by the lowercase Greek letter Lambda (dots 4-6, dots 1-2-3). Begin with a Greek letter indicator (dots 4-6) and then write the letter l (dot 1-2-3). See examples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)