Mathematics Glossary A-M

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Term
Definition
A
Absolute Maximum
The maximum value of the function over its entire domain. (See Global Maximum)
Absolute Minimum
The minumum value of the function over its entire domain. (See Global Minimum)
Absolute Value
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The distance that a number is from zero on the number line. This must always be either zero or positive.
Addition of Functions
. The domain for this combination function is , where A is the domain of function f, and B is the domain of function g. Example if:  and , then 
Additive Identity
The number which when added to any other number, results in the other number. Zero(0) is the Additive Identity. (See Identity Element) 
Additive Inverse
The number which when added to a certain number, results in the Identity Element for Addition. (0)
Algebraic Expression
An expression which has at least one variable in it.
Algebraic Function
A function which can be built using any algebraic operations. Rational functions all qualify, and in addition any roots can be included in the numerator and/or denominator.
Antiderivative
An antiderivative of f is a function F such that F' = f.
Asymptote
A linear boundary which a graph approaches, but never touches. In the function , both the x-axis and y-axis are asymptotes of the function.
Associative Property
a+(b+c)=(a+b)+c. How you group items makes no difference in the outcome.
B
Bezier Curves
Special parametric curves that are often used in manufacturing, and in describing the shape of characters sent to laser printers. Bezier curves "smoothly" connect a set of points.
Biconditional (Statement)
An "If-Then" Statement that is true in both directions. 
Sample conditional statement: pq. This be thought of as If p, then q AND If q then p. It is often presented as p "if and only if" q (sometimes shown as iff). The biconditional statement is only true when both p and q have the same truth values.
C  
Cardinality (of a set)
The number of elements in a set. 
Example: A={4,6,-9,12} 
n(A)=4 (because there are 4 elements in set A)
Closed Interval
An Interval which includes both endpoints  . Another way to write this interval above is: [a,b]. The "squared" brackets indicate that a and b are to be included.
Closed Interval Method

To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]:

1. Find the values of f at the critical numbers of f in (a,b)

2. Find the values of f at the endpoints of the interval.

3. The largest of the values from Steps 1 and 2 is the absolute maximum value; and the smallest of these values is the aboslute minimum value.
Closure Property
If two elements of a set can be combined using an operation and a third number from that same set always results, then the set is said to be closed under that operation.
Coefficient
The numerical parts of an expression. They are usually thought of as the numbers multiplied by the variables. But constants can also be coefficients because they can be thought of as being multiplied by some variable to the zero power.
Collinear
Points which all lie in a straight line.
Column Matrix
A matrix with only 1 column.
Common Binomial (Factoring)
Common Monomial (Factoring)
Commutative Property
ab=ba. What order you use in your calculation makes no difference in the outcome.
Complement (of set A)
The set of elements in the universal set that are not in set A.
Completing the Square
Solving an unfactorable quadratic equation by creating a perfect square trinomial, so that the method of taking the square root of both sides of the equation can be used.
Complex Numbers
C 
Composition of Functions
The composition of functions has the second function substituted into the first function.. The domain for this combination function is , where A is the domain of function f, and B is the domain of function g. Example if:  and , then 
Conditional (Statement)
An "If-Then" Statement. The "If" section has the condition which has to be met, and the "Then" section has what results if the condition is true. If the condition is false, there is no action. 
Sample conditional statement: pq. This is read If p, then q. A conditional statement will only be false when p is true and q is false. Otherwise, it is true.
Conjugate

A binomial factor which together with the original binomial factor have a product which is the difference of two squares.

(a-b) is the conjugate of (a+b). Complex conjugates have a product which is the sum of two squares. .

(a-bi) is the conjugate of (a+bi)
Constant
a,b 
Monomials that contain no variables.
Constant Function
A function for which the entire range has a constant value. , where c is a constant.
Continuous
A function is continuous at a number a if 
Three things must take place for this to be true:
1) a must be in the domain of f,
2)  exists, and 
3)
Continuous from the left
A function is continuous from the left if  (This is called a left-hand limit.
Continuous from the right
A function is continuous from the right if (This is called a right-hand limit.
Continous on an interval
A function is continuous on an interval if it is continuous at every point of the interval. At the endpoints of the interval, the function must be continuous from right (at the right end) and continuous from the left (at the left end).
Coordinates
Each point in the coordinate plane corresponds to an ordered pair of numbers called its coordinates.
Coordinate Plane
The plane determined by two perpendicular axes. 
Cramer's Rule
The solution to the system:  is: 
Critical Number
A critical number of a function f is a number c in the domain of f such that either f'(c)=0 or f'(c) does not exist.
Cubic Function
A polynomial of degree 3 is of the form: 
Cycloid
A curve traced out by a point on the circumference of a circle as the circle rolls along a straight line. 
D
Degree of Monomial
The sum of the exponents of its variables. The degree of a non-zero constant is zero. The constant zero has no degree.
Degree of Polynomial
The degree of the term (monomial) which has the largest degree of each of the individual terms (monomials).
Derivative
The slope of the tangent line to a curve at any instant a. This is calculated by taking the slope of a secant line through the curve, and then taking the limit of that secant slope as x gets closer and closer to a. This is expressed as: . Another way to express this same concept uses the letter h to symbolize the gap between the point where the derivative is desired and a nearby point. Then the derivative is expressed as: .
Determinant
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An operation perfomed on square matrices. A second order determinant: . A third order determinant: 
Difference of Two Cubes
Difference of Two Squares
Difference of Two nth Powers
Differentiable
A function is differentiable at c if f '(c) exists. To be differentiable over an interval, it must be differentiable at every point in the interval. Differentiate is the verb form of derivative. For f '(c) to exist, f (c) must be defined and .
Dimensions
The number of rows and columns in the matrix. We always list the number or rows first. A 4x7 matrix is a matrix with 4 rows and 7 columns.
Discontinuous
A function which cannot be drawn with a writing device without a gap or picking up of the writing device. There are either separate parts of the graph, or a hole(s) in the graph.
Disjoint Sets
Sets with no elements in common. If sets A and B are disjoint, then AB=.
Distance Formula
Given the two points  and , the distance, d is calculated using the following formula:  This formula is based on the Pythagorean Theorem.
Distributive Property
The multiplication "is distributed" over the addition.
Division of Functions
. The domain for this combination function is , where A is the domain of function f, and B is the domain of function g. Example if:  and , then 
Domain
The set of all x-coordinates in a relation.
closed interval
E
Elements
The individual items or objects that are in sets
Elements
The individual values in a matrix.
Empty Set
{} 
A set with no contents. (See Null Set)
Equal Matrices
Two matrices that have the same dimensions and their corresponding elements are equal.
Equal Rights Amendment for Algebra 
Whatever you do to one side of an equation, you must do the same to the other side also.
Equal Rights Amendment for Algebra (Inequalities)
For inequalities, the Equal Rights Amendment for Algebra works the same except when multiplying or dividing both sides by a negative. Then the inequality sign must be reversed.
Equation
Two equal mathematical expressions.
Equivalent (statements)
Two symbolic statements are equivalent to each other when their truth values are identical. 
Exclusive OR
Either of two statements can be true, but NOT BOTH. (See Inclusive OR)
Exponential Function
A function of the form , where the base a is a positive constant. 
Example: 
Expression
A mathematical statement using numbers, variables, and operations.
Extraneous Roots
Roots (or solutions) to a quadratic equation which have been obtained by correct algebraic steps, but which do not check in the original equation. Extraneous roots are not solutions.
Extreme Values
The absolute maximum and absolute minumum of a function.
Extreme Value Theorem
If f is continuous over a , then it must attain an absolute maximum and an absolute minimum in that interval.
F
Factorial
The factorial of a number is the number multiplied by every integer less than the number down to 1. The symbol used for factorial is "!". For example, 
Fermat's Theorem
If a function as a local maximum or local minimum at c, and if f '(c) exists, then f '(c)=0.
Finite Set
A set whose cardinality is a finite number. (A set whose number of elements is countable)
Formula
A mathematical sentence that expresses the relationship between certain quantities.
Function
F(x) 
A relation where each value of the domain has exactly one corresponding element in the range.
G
Global Maximum
The maximum value of the function over its entire domain. (See Absolute Maximum)
Global Minimum

The minumum value of the function over its entire domain. (See Absolute Minimum)

H
Half-Open Interval
An Interval which includes exactly one endpoint Another way to write this interval is: [a,b). The "squared" bracket indicates that in the interval above a is to be included and the rounded parenthesis indicates that b is the boundary, but is not to be inlcuded.
Handshaking
Method of multiplying two polynomials. Each term of the first polynomial must be multiplied by each term of the second polynomial.
Horizontal Line Test
A test for whether a relation is one-to-one. If the relation never has a horizontal line intersect the graph in more than one point, it passes the test and is one-to-one.
I
Identity Element
The number which when operated on with any other number, results in the other number. Zero(0) is the Identity Element for Addition. (See Additive Identity) One (1) is the Identity Element for Multiplication.
Identity Matrix
The identity matrix I for multiplication is a square matrix with a 1 for every element of the principal diagonal (top left to bottom right) and a 0 in all other positions.
Imaginary Numbers
IM 
Square roots of negative numbers. By definition, 
Inclusive OR
Either of two statements can be true, or BOTH of them. (See Exclusive OR)
Indeterminate Form
In calculus when an expression is in a form where the limit cannot be determined.
Inductive Reasoning
The process of arriving at a general conclusion based on repeated observations of specific examples.
Infinite
That which has no limit or end.
Infinite Discontinuity
A discontinuity where the jump is infinite.
Infinite Set
A set which is equivalent to a proper subset of itself. (Equivalence means that each element of the one set can be put into a one-to-one correspondence with the other set)
Initial Point
The initial point of a parametric curve is the point which represents the x and y values when the parameter takes on the lowest value in its domain.
Integers
IN 
{...,-1,0,1,...}
Intersection
What is common between two sets
Interval
A mathematical expression of "betweeness"  Another way to write this interval above is: [a,b]. The "squared" brackets indicate that a and b are to be included.
Inverse of a Function
The inverse of a function literally undoes the action of a function. Every function has an inverse, but not every inverse will be a function.
Inverse Matrix
The matrix which when multiplied by the original matrix gives the identity matrix as the solution.
Irrational Numbers 
IR
 Decimal Form: non-repeating, non-terminating 
Iteration
Iteration is the repeated application of a function or process in which the output of each step is used as the input for the next iteration. Iteration is an important tool for solving problems (e.g., Newton's method, the logistic equation, fractals) as well as a subject of investigation (e.g., Julia sets). Any function that has the same type of mathematical object for both its argument and result can be iterated.
J
Jump Discontinuity
A discontinuity where there the function "jumps" from one value to another.
L
Left-Hand Limit
A function is continuous from the left if  (This is called a left-hand limit.
Less Than
< 
A number which is to the left of the other number on a number line.
Like Terms
Two monomials (terms) that are the same or differ only in their coefficients.
Line
In geometry one of the three "undefined terms" that geometry is based on. Informally, it is infinite in length, and is straight.
Linear Equation
An equation whose graph is a line.
Linear Function
A linear equation that is also a function.
Local Maximum
The maximum value of the function near a specified value in the domain. (See Relative Maximum)
Local Minimum
The minimum value of the function near a specified value in the domain. (See Relative Minimum)
Logarithmic Function
A function in the form , where the base a is a positive constant. 
Example: 
M
Mapping
A pictoral matching up of domain elements with their corresponding range elements.
Matrix
[] 
A rectangular array of numbers with columns and rows.
Matrix Addition
Corresponding elements of the two matrices are added.
Matrix Multiplication
Corresponding elements in that row of the first matrix are multiplied with the corresponding element of that column of the second matrix, and then added. 
Minor
The determinant formed when the row and column containing that element are deleted.
Monomial
An expression that is a number, a variable, or the product of a number and one or more variables. Also called a term.
Multiplicative Identity
The number which when multiplied by any other number (except for 0), results in the other number. One (1) is the Multiplicative Identity. (See Reciprocal), (See Identity Element) 
Multiplicative Inverse
The number which when multiplied by a given number results in the Multiplicative Identity (1). 
(See Reciprocal)
Multiplication of Functions
. The domain for this combination function is , where A is the domain of function f, and B is the domain of function g. Example if:  and , then