OAE Mathematics key terms

The number that, when added to a given number, produces zero. For any real number a, the additive inverse is −a, since a + (−a) = 0.

The number that, when multiplied by a nonzero number, yields 1. For a nonzero real number a, the multiplicative inverse is 1/a, also called the reciprocal.

Any number expressible as a ratio p/q of two integers where q is not zero. The set includes integers, terminating decimals, and repeating decimals, and is closed under the four basic operations except division by zero.

A real number that cannot be written as a ratio of two integers. Its decimal expansion is infinite and non-repeating, as with the square root of 2 or pi.

The property that between any two distinct rational numbers there is another rational number. This shows the rationals cannot be listed in order without gaps, even though they are countable.

The conventional sequence for evaluating expressions: grouping symbols first, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right.

The rule that multiplication distributes over addition and subtraction: a(b + c) = ab + ac. It underlies expanding expressions and combining like terms in algebra.

The distance of a number from zero on the number line, always nonnegative. Written |x|, it equals x when x is nonnegative and −x when x is negative.

A way to write numbers as a product of a factor between 1 and 10 and a power of ten, such as 3.2 × 10^5. Useful for very large or very small quantities and for estimating magnitude.

Reasoning with relationships of the form y = kx, where the ratio of two quantities stays constant. It supports solving scale, rate, and unit-conversion problems using equivalent ratios.

A rate expressed per single unit of the second quantity, found by dividing. For example, 150 miles in 3 hours gives a unit rate of 50 miles per hour.

The ratio of an amount of increase or decrease to the original amount, expressed as a percent. Computed as (new − old) ÷ old × 100.

A sequence in which each term after the first differs from the previous term by a fixed constant called the common difference. Its nth term is a₁ + (n − 1)d.

A sequence in which each term after the first is found by multiplying the previous term by a fixed nonzero constant called the common ratio. Its nth term is a₁r^(n−1).

A definition of a sequence that states one or more initial terms and expresses each later term in terms of earlier ones, such as aₙ = aₙ₋₁ + 3.

A closed rule that gives the nth term of a sequence directly as a function of n, allowing any term to be computed without finding preceding terms.

A relation that assigns to each input exactly one output. The set of inputs is the domain and the set of outputs is the range; a graph fails the vertical line test if it is not a function.

The domain of a function is the set of all valid inputs; the range is the set of all resulting outputs. Restrictions often arise from division by zero or even roots.

A function whose graph is a straight line, writable as f(x) = mx + b where m is the slope and b the y-intercept. It has a constant rate of change.

The steepness of a line, equal to the change in y divided by the change in x between two points. It is the constant rate of change of a linear function.

A function of the form f(x) = ax² + bx + c with a not zero. Its graph is a parabola opening upward when a is positive, with a vertex giving the maximum or minimum.

The turning point of a parabola, where the function attains its minimum or maximum value. Its x-coordinate is −b/(2a) for a quadratic in standard form.

The vertical line through a parabola's vertex that divides the graph into mirror-image halves. For f(x) = ax² + bx + c it is the line x = −b/(2a).

The quantity b² − 4ac in the quadratic formula. Its sign reveals whether a quadratic equation has two real solutions, one repeated real solution, or two complex solutions.

A function of the form f(x) = a·b^x with b positive and not equal to 1. It models constant percent growth or decay, with b greater than 1 for growth.

The exponent to which a base must be raised to produce a given number: log_b(x) = y means b^y = x. Logarithms are the inverses of exponential functions.

A function that reverses another function's action, so that composing the two returns the original input. It exists only when the original function is one-to-one.

An input value for which the function equals zero. Zeros correspond to x-intercepts of the graph and to factors of a polynomial expression.

How one quantity changes relative to another, computed as the change in output divided by the change in input over an interval. Average rate of change uses endpoints of the interval.

Figures with the same shape but possibly different size, having congruent corresponding angles and proportional corresponding side lengths. The scale factor relates all corresponding lengths.

The constant ratio between corresponding lengths of two similar figures. Areas scale by the square of the factor and volumes by its cube.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Its converse tests whether a triangle is right.

The distance between points (x₁, y₁) and (x₂, y₂) is the square root of (x₂ − x₁)² + (y₂ − y₁)². It follows from the Pythagorean theorem on the coordinate plane.

The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is the average of the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Motions or resizings of figures on the coordinate plane: translations slide, rotations turn, reflections flip, and dilations resize. Congruence is preserved by all but dilation.

The total area of all faces or curved surfaces of a three-dimensional solid, measured in square units. Formulas depend on the solid, such as 6s² for a cube.

The amount of space a three-dimensional solid occupies, measured in cubic units. Common formulas include lwh for prisms and (4/3)πr³ for spheres.

Summary statistics describing a typical value in a data set: mean, median, and mode. The median resists extreme values while the mean is pulled toward outliers.

Statistics describing how dispersed data are, including range, interquartile range, variance, and standard deviation. Larger values indicate more variability around the center.

A measure of typical distance of data values from the mean, equal to the square root of the variance. It is expressed in the same units as the data.

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation. Roughly 68 percent of values lie within one standard deviation of the mean and 95 percent within two.

A data value far from the rest of the distribution, often defined as lying more than 1.5 interquartile ranges below the first quartile or above the third quartile.

An assignment of probabilities to each possible value of a random variable, with all probabilities between 0 and 1 summing to 1. It may be discrete or continuous.

The long-run average outcome of a random variable, computed by multiplying each value by its probability and summing the products. Also called the mean of the distribution.

The probability that event A occurs given that event B has occurred, written P(A|B) and equal to P(A and B) divided by P(B) when P(B) is nonzero.

Two events whose joint probability equals the product of their individual probabilities. Knowing one occurred does not change the probability of the other.

Events that cannot occur at the same time, so their joint probability is zero. For such events, the probability of either is the sum of their probabilities.

A selection of items from a group in which order does not matter, counted by n! divided by r!(n − r)!. Contrasted with permutations, which count ordered arrangements.

An arrangement of items in which order matters, counted by n! divided by (n − r)! when choosing r items from n distinct items.

If one choice can be made in m ways and a second in n ways, the two together can be made in m × n ways. It extends to any number of sequential choices.