Math B30 Glossary An annuity is a series of equal payments or deposits made at regular intervals. An arithmetic sequence is a sequence in which each successive term is formed by adding the same number. In an arithmetic sequence, the difference between consecutive terms is constant. An arithmetic series is the sum of the terms of an arithmetic sequence. The formulas for finding the sum are: or . A column matrix is a matrix with dimensions nx1. In an arithmetic sequence, the difference between consecutive terms is constant. This constant is called the common difference and denoted with the letter, d. In an geometric sequence, the ratio between consecutive terms is constant. This constant is called the common ration and is denoted with the letter, r. Common logarithms are logarithms of positive numbers to a base of 10. Compound interest is an interest which is earned on principal plus the last interest earned. Conditional probability is the probability that an event will occur, given that another event has already occured. The probability that Event A will occur on the condition that Event B has already occured, denoted P(A/B), can be determined using the following formula: The degree of a polynomial function defined as is n. Dependent events are two events in which the outcome of one event is affected by the outcome of another event. In other words, if the happening of one event can increase or decrease the chances of another event, the the two events are dependent. If two events A and B are dependent, then P(A and B) = P(A) P(B/A) where P(B/A) represents the probability of B after event A has occured. The determinant of a matrix A where is defined as ad - bc. The dimension of a matrix is the number of rows x number of columns. The discriminant of a quadratic equation of the form, , is the expression . The discriminant is often denoted with a capital D; thus, . The doubling period is the amount of time that it takes for cells to duplicate themselves. Each integer in a matrix is referred to as an element or entry. An element is a member of a set. When listing the elements contained within a set, set braces, { }, are used to enclose the elements belonging to a set. For example, the set of prime numbers from 1 to 20, is written as {3, 5, 7, 11, 13, 17, 19}. Exponential decay is decay which can be related to an exponential function. An exponential equation is an equation in which the variable is located in an exponent position. An exponential function is a function in which a variable occurs as an exponent. An exponential function with a base 'a' is defined as . Exponential growth is growth which can be related to an exponential function. The factor theorem states that if x = a is substituted into a polynomial and the resulting value is 0, then x - a is a factor of the polynomial. The factor property states that when attempting to determine a factor of a polynomial, test values of "a" which are factors of the constant term of the polynomial. The future value of an investment is the sum of all deposits plus the interest earned. The general term of an arithmetic sequence is given by the formula . The general term of a geometric sequence is given by the formula . A geometric sequence is a sequence in which each successive term is formed by multiplying the same number. In an geometric sequence, the ratio between consecutive terms is constant. A geometric series is the sum of the terms of an geometric sequence. The formula for finding the sum is: The half life is the amount of time that it takes for a given element to decrease to half of its original amount. A horizontal asymptote is a horizontal line which the graph approaches as x grows infinitely large or small. If the limit of infinity of a function is k, then the line y = k will be a horizontal asymptote of the graph of the function. An identity matrix is a square matrix with 1's on the main diagonal from the upper left to the lower right and 0's off the main diagonal. An identity matrix is denoted as I. Independent events are two events in which the occurrence of one event does not affect the occurence of another event. If two events A and B are independent, then P(A and B) = . An infinite geometric series is a geometric series which contains an infinite number of terms. The intersection of two sets A and B, written as , consists of those elements that are common to both set A and set B. For example, if A ={3, 6, 9, 12, 15, 18, 21, 24, 27, 30} and B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}, then = {6, 12, 18}. Two 2X2 matrices are inverses of each other if their products, in both orders, is a 2X2 identity matrix. The inverse of a matrix A where is defined as . Irrational Numbers are numbers that cannot be expressed as fractions. The decimal expansion of an irrational number neither terminates or repeats. Therefore, irrational numbers include all non-perfect roots, all non-terminating, non-repeating decimals and =3.1415.... Examples: The leading coefficient of a polynomial function defined as is . The limit of infinity of a function, f(x), is the value that the function approaches as x grows infinitely large or small. The notation means the limit of f(x) as x approaches infinity. A logarithm is an exponent. A logarithm of the form means the exponent to which 'a' is raised to produce x. A logarithmic equation is an equation which contains a logarithm. The logarithmic function is the inverse of the exponential function. Thus, since the exponential function is defined by the equation , the logarithmic function is defined by the equation . A matrix is a rectangular array of numbers. The plural of matrix is matrices. The mean or average of a sample of n data values is the sum of the data values divided by n. The mean is determined by taking the sum of all the data values and dividing by the number of data values.If a data set contains n data values, , the mean, denoted , can be determined using the formula . The median is the middle entry in a set of data arranged in either increasing or decreasing order. If the number of data values, n, is even, the median can be found by finding the average of the two centre entries. Thus, with an even number of data values, the median is not always equal to one of the data values. The mode is the most frequently occurring data value. The multiplicity of a zero is defined as the number of times in which that zero occurs. A normal curve is a frequency curve used to represent a normal distribution. All normal curves are bell-shaped and symmetric about a vertical line passing through the peak or the mean of the data. A normal distribution is a distribution in which the data is concentrated around the mean. The frequency curve of a normal distribution takes on a symmetrical bell-shape. The order of a matrix is the number of rows x number of columns. An ordinary annuity is an annuity in which the frequency of the equal payments is the same as the frequency of compounding. A polynomial function is a function of the form where n is a natural number (1, 2, 3, ...) and , the coefficients of the powers of x, are real numbers. The present value, PV, of an investment is the amount of money that must be invested now to give the required amount at the end of a certain time period. The product of the roots of a quadratic equation of the form, , is . A quadratic equation is a second degree equation that can be written in the form and a, b and c are any real numbers. The quadratic formula is the formula, , which can be used to solve any quadratic equation of the form . A quadratic inequality is an inequality of the form ax +bx+c < 0 (or > or or ) The range is a measure of how widely the data values are spread apart. It is calculated by finding the difference between the greatest and smallest data values. A rational function is a function that is expressed as the ratio of two polynomials. The general form of a rational function is where p(x) and q(x) are polynomials and because division by zero is an impossible operation. Rational Numbers are numbers that can be written in the form where a and b are integers and b 0. Rational Numbers include all integers, fractions, perfect roots and all terminating and repeating decimals. (All terminating and repeating decimals can be expressed as fractions.) Examples:  Real Numbers ( ) are all numbers that can be expressed as decimals. Real numbers correspond to every point on the number line and include all rational and irrational numbers. The roots of an equation are the values of the variable that satisfy the equation. For instance, in solving the equation, , we find that x = 5 and x = -3; thus, 5 and -3 are referred to as the roots of the equation. A row matrix is a matrix with dimensions 1xn. A series is a sum of the terms of a sequence. A sequence is a set of numbers written in a definite order. A set is a well-defined collection of objects. Sigma Notation is a notation used to abbreviate long summations of a given expression. A skewed distribution is a distribution in which the data is not concentrated around the mean. A square matrix is a matrix with dimensions nxn. Standard deviation (s) is the measure of the amount of dispersion in a set of data. The formula below is used to determine the standard deviation of a sample, = A standard normal curve is representative of a normal distribution with mean of 0 and a standard deviation of 1. The area under a standard normal curve is one unit squared. The sum of the roots of a quadratic equation of the form, , is . A system of inequalities is a set of a least two inequalities. The union of two sets A and B, written as , consists of those elements that are common to set A or set B or both. For example, if A = {cat, dog, hamster, goldfish} and B = {rat, mouse, beaver, hamster}, then = {cat, dog, hamster, golfish, rat, mouse, beaver}. Venn diagrams are one useful way of understanding the relations between sets. A typical Venn diagram uses overlapping circles to represent groups of items or ideas that share common properties. A vertical asymptote is a vertical line which the graph approaches in an infinitely close manner but never crosses. Vertical asymptotes will always exist at excluded values of x. Thus, if a number, k, makes the denominator zero, then the line x = k will be a vertical asymptote of the graph of the function. A zero of a polynomial function is any value of x that causes y to be equal to zero. A z-score represents the number of standard deviations that the data value is from the mean. If a data value has a positive z-score, this means that the data value is located a number of standard deviations above (to the right of) the mean and if a data value has a negative z-score,this means that it is located a number of standard deviations below (to the left of) the mean.