JCE Mathematics key terms

A whole number with no fractional or decimal part, including negative numbers, zero and positive numbers. It is closed under addition, subtraction and multiplication but not division.

A number that can be written as one integer divided by another non-zero integer. Its decimal form either terminates or repeats a fixed pattern forever.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion never terminates and never repeats, as with the square root of two or pi.

A square or higher root of a number that cannot be simplified to a rational value, such as the square root of five. Surds are often left in exact form or simplified by removing square factors.

A number or algebraic expression that divides another exactly, leaving no remainder. Listing factors of a term helps simplify fractions and solve equations.

The result of multiplying a given number by any integer. Multiples of six include six, twelve and eighteen, and the lowest shared multiple of two numbers is their least common multiple.

An integer greater than one whose only positive factors are one and itself. Primes are the building blocks of all integers through prime factorisation.

The largest number that divides two or more given numbers exactly. It is found by comparing prime factorisations and taking the lowest power of each shared prime.

A way of writing very large or very small numbers as a value between one and ten multiplied by a power of ten. Also called scientific notation.

The size of an increase or decrease expressed as a fraction of the original amount, multiplied by one hundred. It uses the starting value, not the final value, as the denominator.

Interest calculated on both the original amount and on interest already added. The balance grows by a fixed multiplier each period, giving exponential rather than linear growth.

A comparison of two or more quantities by division, often written with a colon. Ratios can be simplified like fractions and used to share an amount in given proportions.

A relationship in which one quantity is a constant multiple of another, so doubling one doubles the other. Its graph is a straight line through the origin.

A relationship in which one quantity increases as the other decreases so that their product stays constant. Its graph is a curve that never crosses either axis.

Rules for combining powers with the same base, such as adding indices when multiplying and subtracting them when dividing. They also cover zero, negative and fractional indices.

The power to which a base must be raised to produce a given number. Logarithms turn multiplication into addition and are the inverse of exponential functions.

An expression made from variables and constants using only addition, subtraction and non-negative whole-number powers. Its degree is the highest power of the variable present.

A polynomial of degree two, written in the form ax squared plus bx plus c where a is not zero. Its graph is a parabola and it has at most two real roots.

The expression b squared minus four ac taken from a quadratic equation. Its sign reveals whether the equation has two, one or no real roots.

Rewriting a quadratic in the form a times a squared bracket plus a constant. This reveals the turning point of the parabola and allows exact solution of the equation.

A statement that a polynomial has a factor of x minus k exactly when substituting k gives zero. It links factors of an expression to roots of the corresponding equation.

Two or more equations that must be satisfied by the same values of the variables. They are solved by elimination or substitution, and may be linear or non-linear.

An equation in which each variable appears only to the first power, so its graph is a straight line. Solving it usually means isolating the unknown by inverse operations.

A statement comparing two expressions that are not necessarily equal, using symbols such as less than or greater than or equal to. Multiplying or dividing by a negative reverses the symbol.

A rule that assigns exactly one output value to each permitted input value. The set of allowed inputs is the domain and the set of outputs is the range.

The function that reverses the effect of another, so applying both returns the original input. Its graph is the reflection of the original graph in the line y equals x.

A function formed by applying one function to the output of another, written as f of g of x. The order of application matters and is not generally reversible.

A function in which the variable appears in the exponent, such as a to the power x with a positive base. It models growth or decay at a constant relative rate.

A function written as one polynomial divided by another. Its graph often has vertical asymptotes where the denominator is zero and may have horizontal asymptotes.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Asymptotes may be vertical, horizontal or oblique.

A measure of how steep a line is, calculated as the change in vertical distance divided by the change in horizontal distance between two points.

A point where two or more line segments or edges meet, such as a corner of a polygon. For a parabola it names the maximum or minimum turning point.

A straight line that touches a curve at a single point and has the same direction as the curve there. Its gradient equals the derivative at that point.

A relationship stating that each side of a triangle divided by the sine of its opposite angle is constant. It is used to find unknown sides or angles in non-right triangles.

A generalisation of Pythagoras' theorem relating the three sides of a triangle to one angle. It finds a side from two sides and the included angle, or an angle from three sides.

A rule stating that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. It is used to find unknown lengths.

A ratio of two sides of a right-angled triangle defined by an acute angle, namely sine, cosine or tangent. These ratios extend to all angles through the unit circle.

A unit of angle measure defined by the arc length on a circle of radius one. One full turn equals two pi radians, so one radian is about 57.3 degrees.

The distance measured along a curved section of a circle or other curve. For a circle it equals the radius multiplied by the angle in radians.

A region of a circle bounded by two radii and the arc between them, resembling a slice of pie. Its area is a fraction of the circle's area set by the central angle.

The arithmetic average of a set of values, found by adding them and dividing by how many there are. It is sensitive to extreme values.

The middle value of a data set placed in order, or the average of the two middle values when the count is even. It is resistant to outliers.

The difference between the upper and lower quartiles of a data set, covering the middle half of the values. It measures spread without being distorted by extreme values.

A measure of how far values typically lie from the mean, found from the square root of the variance. A larger value indicates more spread-out data.

A description giving the probability of each possible outcome of a random variable. For a discrete variable the probabilities must sum to one.

The probability distribution of the number of successes in a fixed number of independent trials, each with the same success probability. It is defined by the trial count and that probability.

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

The probability that an event occurs given that another event has already happened. It is calculated by dividing the probability of both events by the probability of the given event.

Events whose outcomes do not affect each other, so the probability of both happening equals the product of their separate probabilities. Sampling with replacement usually produces independence.

A simplified mathematical description of a real situation, built from assumptions and variables. Its predictions are only as good as the assumptions behind it.

A condition accepted as true when building a model, often to make the mathematics manageable. Stating assumptions clearly allows the model's limitations to be judged.

The process of finding the input that gives the largest or smallest possible value of a quantity. It typically involves differentiating, setting the derivative to zero and checking the result.

How quickly one quantity changes with respect to another, often found as the gradient of a graph or the derivative of a function. Average and instantaneous rates differ.