# Welcome to the Nemeth Braille Searchable Database!

## Introduction

Sara Larkin, Susan Osterhaus, and Tina Herzberg

## 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Alpha (lowercase)

Alpha is a Greek letter. The lowercase form of this letter is often used to represent an angle measure. To write Alpha in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “a” (dot 1). 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Angle

The angle symbol 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, an “m” in front of an angle notation means “the measure of”. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF). See samples: 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. They are commonly used when writing vector notation in component form. The opening angle bracket is dots 4-6 written twice in front of dots 1-2-3-5-6. The closing angle bracket is dots 4-6 written twice in front of dots 2-3-4-5-6. The numeric indicator is not used inside grouping symbols such as the angle brackets. See vectors for more Nemeth code related to vectors. The sample document includes vector notation as well as 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 samples: 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 followed by the letter x in parentheses. The opening parenthesis is dots 1-2-3-5-6 and the closing parenthesis is dots 2-3-4-5-6. Note: the opening and closing parentheses are the same as the “of” and “with” contractions. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Approximately equal to

“Is approximately equal to” is represented in print as parallel wavy lines. Each wavy line is called a tilde. Each tilde is represented using dot 4 followed by dots 1-5-6. The entire “is approximately equal 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, dot 4, dots 1-5-6, space. See samples: 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: Begin with 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 which is dots 1-2-6. Step 4: Write the modifier which is the arc symbol. It is the shape indicator (dots 1-2-4-6) followed by the letter a (dot 1) for arc. Step 5: Write the termination indicator which is dots 1-2-4-5-6. Note: the directly-over indicator is the same as the “gh” contraction. Think “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, an “m” in front of an arc means “the measure of”. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Beta (lowercase)

Beta is a Greek letter. The lowercase form of this letter is often used to represent an angle measure. To write Beta in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “b” (dots 1-2). 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 samples: 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” that use the two-way horizontal arrow

The two-way horizontal arrow, used for biconditional statements, is written using the shape indicator (dots 1-2-4-6) followed by dots 2-4-6, dots 2-5, dots 2-5, dots 1-3-5. It can also be written with a double arrow shaft, called a spear, which is written using the shape indicator (dots 1-2-4-6) followed by 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Binomial coefficient

The binomial coefficient which is used in probability and the binomial theorem is written with parentheses and read “n choose r” where n and r are two numbers written vertically in print. Remember the open parenthesis is dots 1-2-3-5-6 and the closing parenthesis is dots 2-3-4-5-6. The directly-under indicator (dots 1-4-6) is also used in this notation. This is used between the numbers that appear on the first line and the second line 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 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 samples: 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. They are used as a third level of parentheses or for set notation. The opening brace is dots 4-6 in front of dots 1-2-3-5-6. The closing brace is dots 4-6 in front of dots 2-3-4-5-6. 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Brackets

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

### Circle

The circle symbol is written with the shape indicator (dots 1-2-4-6) followed by the letter “c” (dots 1-4) 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 samples: 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 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 samples: 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: capital c followed by n comma r inside a set of parentheses which are dots 1-2-3-5-6 and dots 2-3-4-5-6. Method 2: capital c followed by the subscript indicator (dots 5-6) n comma r. Method 3: subscript indicator (dots 5-6) n capital c 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: parentheses with r below n. Remember the opening and closing parentheses are dots 1-2-3-5-6 and dots 2-3-4-5-6 respectively. The directly-under indicator (dots 1-4-6) is also used in this notation. This is used between the first line which represents n and the second line which represents r. 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Conditional statements, also known as “if-then” statements that use the right-pointing arrow

The right-pointing arrow, used for conditional statements, has an uncontracted form and a contracted form. The uncontracted form is written using the shape indicator (dots 1-2-4-6) followed by dots 2-5, dots 2-5, dots 1-3-5. The contracted form which consists of the shape indicator (dots 1-2-4-6) followed by dots 1-3-5 is often used. It can also be written with a double arrow shaft, called a spear, which is written using the shape indicator (dots 1-2-4-6) followed by 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Congruent to

“Is congruent to” is represented in print as a wavy line over the equal sign. This wavy line is called a tilde. The tilde is represented using dot 4 followed by dots 1-5-6. The “congruent to” symbol is the tilde followed by the equal sign. The entire “congruent to” symbol has a space before and after since it is a sign of comparison. Therefore, the “is congruent to” symbol is written space, dot 4, dots 1-5-6, dots 4-6, dots 1-3, space. Note: Any capital letter used to name an angle, segment, triangle, or other polygon 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, segment, or triangle. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Conjunction (and)

Conjunction (and) is written as dot 4 followed by dots 1-4-6. Since this is considered an operator, there are no spaces before or after the symbol. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Corresponds to

The two-way horizontal arrow, used to represent the terms “corresponds to”, is written using the shape indicator (dots 1-2-4-6) followed by 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 samples: 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 samples: 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 samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Curly d

The curly d which represents the partial derivative is often used in notation for calculus. To write this in Nemeth code, write dot 4 followed by the letter d (dots 1-4-5). See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Definite integral

The definite integral is an integral over a certain interval. It starts with the integral sign which is dots 2-3-4-6. Then it is followed by a subscript indicator (dots 5-6) and the starting value of the interval. Next is the superscript indicator (dots 4-5) and the ending value of the interval. Finally 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 which starts with the multipurpose indicator (dot 5) and ends with the termination indicator (dots 1-2-4-5-6) instead of the baseline indicator. Also the directly-under indicator (dots 1-4-6) is used instead of the subscript indicator and the directly-over indicator (dots 1-2-6) is used instead of the superscript indicator. See samples: 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 can be used to specify an angle or arc measure, to write degrees Fahrenheit, and to write degrees Celsius or Centigrade. The degree symbol is actually three cells long. It begins with the superscript indicator (dots 4-5), since it is raised, followed by the two-cell hollow dot symbol (dots 4-6 and then dots 1-6). Also, the m in front of the angle notation means “the measure of”. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Delta (lowercase)

Delta is a Greek letter. The lowercase form of this letter is often used to represent a small change in the value of a variable in calculus. To write Delta in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “d” (dots 1-4-5). See samples: 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 is most often used to represent the change in something. To write Delta in Nemeth code, begin with a Greek letter indicator (dots 4-6) followed by the capitalization indicator (dot 6) and then by the letter “d” (dots 1-4-5). See samples: 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 starts with an opening fraction indicator (dots 1-4-5-6) followed by “dy”, then uses the slash (dots 3-4) followed by “dx”, and finishes with the 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 samples: 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 and is written as a dot 3 which is the same as an apostrophe. 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Disjunction (or)

Disjunction (or) is written as dot 4 followed by dots 3-4-6. Since this is considered an operator, there are no spaces before or after the symbol. See samples: 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, the opening brace is dots 4-6 in front of dots 1-2-3-5-6. The closing brace is dots 4-6 in front of 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, use a dot 4-5-6 in front of a zero (dots 3-5-6). See samples: 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 samples: 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 samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Enlarged vertical lines (used for determinants)

Enlarged vertical lines are used when writing determinants. To enlarge vertical lines, just place a dot 6, as if it is capitalized, before the vertical line (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 line is the same as the “ou” contraction. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Epsilon (lowercase)

Epsilon is a Greek letter. The lowercase form of this letter is often used to represent an arbitrarily small positive quantity in calculus. To write Epsilon in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “e” (dots 1-5). See samples: 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 is written dots 4-6 followed by dot 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 samples: 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 (which is 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 followed by the superscript indicator and then a negative one. See samples: 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: Begin with 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 “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 samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Gamma (lowercase)

Gamma is a Greek letter. The lowercase form of this letter is often used to represent an angle measure. To write Gamma in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “g” (dots 1-2-4-5). 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greater than

”Is greater than” is written dots 4-6 followed by dot 2. There should be a space before and after the ”is greater than” sign 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Greater than or equal to

”Is greater than or equal to” is written with the ”greater than” sign followed by the horizontal bar (dots 1-5-6) so it is dots 4-6, dot 2, dots 1-5-6. There should be a space before and after the entire ”is 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 samples: 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, beta, gamma, theta, and phi. Pi 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 samples: 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 the dots 4-6 and are followed by the letter(s) it represents. All uppercase letters will still begin with the 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Infinity

The infinity symbol is written using dot 6 followed by the full cell (dots 1-2-3-4-5-6). See samples: 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 uses the opening bracket (dot 4, dots 1-2-3-5-6) followed by x and then a 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 samples: 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 which is 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Intersection (of sets)

Intersection is written as dot 4 followed by dots 1-4-6. Since this is considered an operator, 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 samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Lambda (lowercase)

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

### Less than

”Is less than” is written dot 5 followed by dots 1-3. There should be a space before and after the ”is less than” sign 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Less than or equal to

”Is less than or equal to” is written with the ”less than” sign followed by the horizontal bar (dots 1-5-6) so it is dot 5, dots 1-3, dots 1-5-6. There should be a space before and after the entire ”is 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 samples: 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: Begin with 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. It is the shape indicator (dots 1-2-4-6) followed by 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 “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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

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

The right-pointing arrow, used to represent the terms “maps to”, has an uncontracted form and a contracted form. The uncontracted form is written using the shape indicator (dots 1-2-4-6) followed by dots 2-5, dots 2-5, dots 1-3-5. The contracted form which consists of the shape indicator (dots 1-2-4-6) followed by dots 1-3-5 is often used. There should be a space before and after the arrow when used to represent the terms “maps to” since it is a sign of comparison. Note: the shape indicator is the same as the “ed” contraction. See samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Omega (lowercase)

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

### Is parallel to

“Is parallel to” 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 samples: 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 closed parentheses. 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 samples: 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, which is used in probability and the binomial theorem, is written with parentheses and read “n choose r” where n and r are two numbers written vertically in print. Remember the open parenthesis is dots 1-2-3-5-6 and the closing parenthesis is dots 2-3-4-5-6. The directly-under indicator (dots 1-4-6) is also used in this notation. This is used between the numbers that appear on the first line and the second line 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 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Partial derivative

The partial derivative, sometimes called the curly d, is often used in notation for calculus. To write this in Nemeth code, write dot 4 followed by the letter d (dots 1-4-5). See samples: 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: capital p followed by n comma r inside a set of parentheses which are dots 1-2-3-5-6 for the opening and dots 2-3-4-5-6 for the closing. Note: the opening and closing parentheses are the same as the “of” and “with” contractions. Method 2: capital p followed by the subscript indicator (dots 5-6) n followed by dots 2-4-6 which represents a print comma in a subscript and then an r. Method 3: subscript indicator (dots 5-6) n baseline indicator (dot 5) capital p 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Is perpendicular to

“Is perpendicular to” 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Phi (lowercase)

Phi is a Greek letter. The lowercase form of this letter is often used to represent an angle measure. To write Phi in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “f” (dot 1-2-4). 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Pi (lowercase)

Pi is a Greek letter. 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. To write Pi in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “p” (dots 1-2-3-4). See samples: 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 (which is 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 followed by the superscript indicator and then a negative one. See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Prime

Prime can be used to represent feet, transformations, or to represent the derivative of a function. Either way it is written using dot 3 which is the same as an apostrophe since it looks like an apostrophe in print. See samples: 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 sample documents. See samples: 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: Begin with 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 right or left arrow symbol. It is the shape indicator (dots 1-2-4-6) followed by dots 2-5, dots 2-5, dots 1-3-5 for the right arrow or dots 2-4-6, dots 2-5, dots 2-5 for the left arrow. Step 5: Write the termination indicator (dots 1-2-4-5-6). Note: The contracted form of the right arrow which consists of the shape indicator (dots 1-2-4-6) followed by dots 1-3-5 is often used. The directly-over indicator is the same as the “gh” contraction. Think “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 samples: 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 samples: 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: Begin with 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 “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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Sigma (lowercase)

Sigma is a Greek letter. The lowercase form of this letter is often used to represent the standard deviation in statistics. To write Sigma in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter “s” (dots 2-3-4). See samples: 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 is often used to represent the summation operator for a series. To write Sigma in Nemeth code, 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Similar to

“Is similar to” is represented in print as a wavy line called a tilde. The tilde is represented using dot 4 followed by dots 1-5-6. The “is 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 samples: 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 samples: 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 samples: 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 samples: 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 (which is 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 followed by the superscript indicator and then a negative one. See samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Theta (lowercase)

Theta is a Greek letter. The lowercase form of this letter is often used to represent an angle measure. To write Theta in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the contraction “th” (dot 1-4-5-6). 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Tilde (for logical negation)

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

### Triangle

The triangle symbol 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 samples: 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 samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)

### Union

Union is written as dot 4 followed by dots 3-4-6. Since this is considered an operator, 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 samples. See samples: 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: Begin with 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. It is the shape indicator (dots 1-2-4-6) followed by 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 “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 samples: 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, 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 words “such that,” for example 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 samples: 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. To write Lambda in Nemeth code, begin with a Greek letter indicator (dots 4-6) and then write the letter "l" (dot 1-2-3). See samples: Nemeth in EBAE (BRF), Nemeth within UEB contexts (BRF), or Nemeth in Print and SimBraille (PDF)