Mathematics Basic key terms

A whole number that can be positive, negative, or zero, with no fractional or decimal part, such as -7, 0, or 42. Integers are closed under addition, subtraction, and multiplication.

A number that can be written as a fraction with an integer numerator and a nonzero integer denominator. Its decimal form either terminates or repeats, and it can always be located exactly on a number line.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion never terminates and never repeats, so values such as the square root of 2 or pi can only be approximated.

The distance of a number from zero on the number line, always zero or positive. It is written with two vertical bars, so both 5 and -5 have an absolute value of 5.

A straight line where every point corresponds to one real number, with equal spacing between consecutive whole numbers. It visually represents order, distance, and the relative size of values.

The value a digit carries because of its position within a written number. Moving one place to the left multiplies the digit's value by ten in base-ten notation, which governs regrouping and rounding.

Replacing a number with a nearby value that has fewer significant digits, chosen by inspecting the next digit. If that digit is five or more, the retained digit increases by one.

Producing an approximate answer by rounding numbers or using compatible values before computing. It is used to judge whether an exact result is reasonable and to solve problems quickly.

The agreed sequence for evaluating expressions: grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. Following it ensures everyone gets the same value.

A small raised number showing how many times a base is used as a factor. In 3 to the fourth power, 3 is the base, 4 is the exponent, and the value is 81.

A number that, when multiplied by itself, produces a given value. The positive square root of 49 is 7, since 7 times 7 equals 49; perfect squares have whole-number roots.

A compact way to write very large or very small numbers as a number between 1 and 10 multiplied by a power of ten. For example, 4,500 is written as 4.5 times 10 cubed.

A comparison of two quantities by division, often written with a colon or as a fraction. It states how many units of one quantity correspond to units of another.

An equation stating that two ratios are equal, such as 3/4 equals 9/12. Solving a proportion usually means finding the missing value by cross-multiplying.

A rate expressed per single unit of the second quantity, found by dividing. A price of 12 dollars for 4 items gives a unit rate of 3 dollars per item.

A ratio comparing a number to 100, indicated by the percent sign. Twenty-five percent means 25 out of every 100, equal to the fraction one fourth or the decimal 0.25.

A letter or symbol that stands for an unknown or changeable number in an expression or equation. The same variable used twice in one expression represents the same value both times.

A mathematical phrase combining numbers, variables, and operation signs without an equals sign. It can be simplified or evaluated, but it cannot be solved because it makes no claim of equality.

A statement that two expressions have equal value, joined by an equals sign. Solving it means finding every value of the variable that makes the statement true.

An equation in which every variable appears to the first power and no variables are multiplied together. Its solutions form a straight line when graphed, and it typically has exactly one solution.

A statement comparing two expressions with symbols such as less than, greater than, or not equal. Its solution is usually a range of values rather than a single number.

The numerical factor multiplying a variable in a term. In the term 7x, the coefficient is 7; a term written as x alone has an implied coefficient of 1.

Terms that have identical variable parts raised to the same powers, so only their coefficients differ. They can be combined by adding or subtracting those coefficients, as 4x and 9x combine to 13x.

A rule allowing a factor outside parentheses to multiply each term inside: a times the sum of b and c equals ab plus ac. It is the main tool for removing parentheses when simplifying.

Two or more equations considered together, whose solution must satisfy every equation at once. For two linear equations, the solution is the point where their graphs intersect.

A solving method in which one variable is replaced by an equivalent expression from another equation. It reduces a system to a single equation in one unknown.

A rule assigning exactly one output value to each allowed input value. Repeated inputs must give the same output, which is why a vertical line may cross its graph only once.

The complete set of input values for which a function is defined. For a function given by a fraction, any input making the denominator zero must be excluded.

The set of all output values a function actually produces from its domain. It may be described as an interval, a list, or an inequality on the output variable.

A measure of how steeply a line rises or falls, computed as the change in output divided by the change in input between two points. Horizontal lines have slope zero.

The point where a graph crosses the vertical axis, occurring where the input value equals zero. In the form y equals mx plus b, the letter b gives this value.

How much one quantity changes relative to another over an interval, found by dividing the change in output by the change in input. For a linear function it equals the slope everywhere.

A function whose highest variable power is two, graphing as a parabola. Its graph opens upward when the leading coefficient is positive and has a single highest or lowest point called the vertex.

The turning point of a parabola, where the curve reaches its minimum or maximum output. The vertical line through it, the axis of symmetry, splits the parabola into matching halves.

The total distance around the outside of a closed two-dimensional figure, found by adding all side lengths. For a rectangle it equals twice the length plus twice the width.

The amount of flat surface enclosed by a two-dimensional figure, measured in square units. A rectangle's area equals its length multiplied by its width.

The amount of three-dimensional space occupied by a solid, measured in cubic units. A rectangular box's volume equals length times width times height.

The distance around a circle, equal to pi multiplied by the diameter, or twice pi times the radius. It plays the same role for circles that perimeter plays for polygons.

A relationship for right triangles stating that the squares of the two legs add to the square of the hypotenuse. It lets you find a missing side when the other two are known.

A triangle containing one ninety-degree angle. The side opposite that angle is the longest side, called the hypotenuse, and the other two sides are the legs.

A figure formed by two rays sharing a common endpoint called the vertex, measured in degrees. Angles are classified as acute, right, obtuse, or straight by their size.

Figures with identical shapes whose corresponding angles are equal and whose corresponding side lengths share a constant ratio. Their areas compare by the square of that ratio.

The constant multiplier relating corresponding lengths of two similar figures. A scale factor of 3 triples every length but multiplies area by 9.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. Extreme values can pull the mean noticeably toward them.

The middle value of a data set arranged in order; with an even count, it is the mean of the two middle values. It is less affected by extreme values than the mean.

The value or values appearing most often in a data set. A set may have no mode or several modes, and the mode can be used with non-numerical categories.

A data value far from the rest of the set, often caused by measurement error or a genuinely unusual case. Outliers can strongly shift the mean while barely affecting the median.

A number from zero to one expressing how likely an event is, where zero means impossible and one means certain. For equally likely outcomes it equals favorable outcomes divided by total outcomes.

The set of all possible outcomes of a random experiment. Listing it completely is the first step in computing probabilities for single or combined events.

The event consisting of all outcomes in the sample space that are not in the given event. An event's probability plus its complement's probability always equals one.

Events whose outcomes do not influence each other, so the probability of both occurring equals the product of their individual probabilities. Repeated coin flips are independent.