Math glossary

The distance of a number from zero on the number line, so it is never negative. Written with vertical bars.

Example: $|-7| = 7$ and $|7| = 7$.

Absolute value calculator

An angle greater than $0^\circ$ and less than $90^\circ$.

Example: Each angle of an equilateral triangle, $60^\circ$, is acute.

Types of triangles

A combination of numbers, letters (variables) and operations, with no equals sign.

Example: $3x^2 - 5x + 2$

Simplifying expressions

The amount of turn between two rays that meet at a point (the vertex), measured in degrees or radians.

Degrees and radians converter

The amount of flat surface a shape covers, measured in square units.

Example: A rectangle 5 cm by 3 cm has area $5 \times 3 = 15\text{ cm}^2$.

Area of common shapes · Area calculator

A list of numbers where the same amount (the common difference) is added each time. The $n$th term is $a + (n-1)d$.

Example: $3, 7, 11, 15, \ldots$ has common difference $d = 4$.

A line that a graph gets closer and closer to but does not reach as $x$ or $y$ grows large.

Example: $y = \frac{1}{x}$ has asymptotes $x = 0$ and $y = 0$.

Rational expressions

A single value that represents a set of data. Usually means the mean, but the median and mode are averages too.

Averages and spread

A line that divides a shape or graph into two mirror-image halves. For $y = ax^2 + bx + c$ it is $x = -\frac{b}{2a}$.

Quadratic equations

In a power $a^n$, the number $a$ that is multiplied by itself. Also the bottom side of a shape used for area.

Example: In $2^5$, the base is 2.

Exponent calculator

A polynomial with exactly two terms.

Example: $x + 3$ and $2a^2 - 5b$ are binomials.

Polynomials

The distance around a circle: $C = 2\pi r$, where $r$ is the radius.

Example: A circle of radius 5 cm has $C = 10\pi \approx 31.4$ cm.

Circle calculator

The number multiplying a variable in a term.

Example: In $7x^2$, the coefficient is 7.

Polynomials

A selection of items where order does not matter: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

Example: Choosing 2 of 4 people: $\binom{4}{2} = 6$ ways.

Permutation and combination calculator

A shared bottom number that lets fractions be added or compared. The smallest one is the LCM of the denominators.

Example: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Fractions

Two angles that add up to $90^\circ$.

Example: $35^\circ$ and $55^\circ$ are complementary.

A function applied to the output of another: $f(g(x))$ means do $g$ first, then $f$.

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $f(g(x)) = (x+1)^2$.

Functions

A whole number greater than 1 that has a factor other than 1 and itself.

Example: 12 is composite: $12 = 2^2 \times 3$.

Prime factorization calculator

A value that does not change, such as a number on its own in an expression.

Example: In $4x + 9$, the constant is 9.

A pair of numbers $(x, y)$ giving the position of a point: across first, then up.

Distance and midpoint calculator

In a right triangle, the adjacent side divided by the hypotenuse: $\cos\theta = \frac{\text{adj}}{\text{hyp}}$.

Trigonometry basics

The number that, multiplied by itself three times, gives the original number.

Example: $\sqrt[3]{64} = 4$ because $4^3 = 64$.

A number written with a decimal point, where digits after the point are tenths, hundredths and so on.

Example: $0.75 = \frac{3}{4}$

Decimal to fraction converter

The highest power of the variable in a polynomial.

Example: $4x^3 - x + 1$ has degree 3.

Polynomials

The bottom number of a fraction: how many equal parts the whole is split into.

Example: In $\frac{3}{8}$, the denominator is 8.

Fractions

The rate of change of a function, or the slope of its graph at a point. For $x^n$ it is $nx^{n-1}$.

Example: The derivative of $x^3$ is $3x^2$.

Derivatives and the power rule · Derivative calculator

A number calculated from a square matrix. For $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ it is $ad - bc$; the matrix has an inverse only if it is not 0.

How to find the determinant · Determinant calculator

A straight line through the centre of a circle from one side to the other; twice the radius.

Circle calculator

For $ax^2 + bx + c = 0$, the value $b^2 - 4ac$. Positive: two real roots; zero: one; negative: none that are real.

Quadratic equations · Quadratic equation solver

All the input values a function is allowed to take.

Example: The domain of $\sqrt{x}$ is $x \ge 0$.

Functions

A statement that two expressions are equal, joined by an equals sign.

Example: $2x + 1 = 9$

Equations

A triangle with all three sides equal and all three angles $60^\circ$.

Types of triangles

The small raised number in a power: it says how many times the base is multiplied by itself. Also called an index or power.

Example: $3^4 = 3 \times 3 \times 3 \times 3 = 81$

Exponent calculator

Numbers, variables and operations put together, without an equals sign.

Simplifying expressions

A number or expression that divides another exactly.

Example: The factors of 12 are 1, 2, 3, 4, 6 and 12.

Factors calculator

The product of all whole numbers from 1 up to $n$, written $n!$. By definition $0! = 1$.

Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Permutation and combination calculator

To write an expression as a product of simpler expressions.

Example: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Factoring polynomials · Factoring calculator

A part of a whole, written as one number over another: $\frac{a}{b}$.

Fractions · Fraction calculator

A rule that gives exactly one output for each input.

Example: $f(x) = 2x + 3$ gives $f(4) = 11$.

Functions · Function calculator

A list of numbers where each term is the previous one multiplied by the same number (the common ratio). The $n$th term is $ar^{n-1}$.

Example: $2, 6, 18, 54, \ldots$ has common ratio $r = 3$.

Another word for slope: how steep a line is.

Slope and the equation of a line

The largest whole number that divides two or more numbers exactly. Also called the highest common factor.

Example: $\gcd(12, 18) = 6$

Prime factors, GCD and LCM · GCD and LCM calculator

The longest side of a right triangle, opposite the right angle.

The Pythagorean theorem · Pythagorean theorem calculator

An equation that is true for every value of the variable.

Example: $\sin^2\theta + \cos^2\theta = 1$

Trig identities calculator

A fraction whose numerator is greater than or equal to its denominator.

Example: $\frac{7}{3} = 2\frac{1}{3}$

Fractions

A statement that one value is less than ($<$), greater than ($>$), at most ($\le$) or at least ($\ge$) another.

Example: $2x - 1 > 5$ gives $x > 3$.

Solving inequalities · Inequality calculator

A whole number that can be positive, negative or zero.

Example: $-3, 0, 12$

The reverse of a derivative, or the area under a curve. $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$.

Integral calculator

Where a graph crosses an axis. The $y$-intercept of $y = mx + c$ is $c$.

Slope and the equation of a line

Money paid for borrowing or earned for saving. Simple interest is $I = Prt$; compound interest adds interest to the balance each period.

Interest calculator

The matrix $A^{-1}$ that gives the identity matrix when multiplied by $A$. It exists only when $\det A \ne 0$.

Inverse matrix calculator

A number that cannot be written as a fraction of two integers; its decimal never ends or repeats.

Example: $\sqrt{2}$ and $\pi$

Square roots

A triangle with two equal sides, and the two angles opposite them equal.

Types of triangles

The smallest positive number that is a multiple of two or more numbers.

Example: $\operatorname{lcm}(4, 6) = 12$

Prime factors, GCD and LCM · GCD and LCM calculator

Terms with exactly the same variables and powers, so they can be added together.

Example: $3x^2$ and $-5x^2$ are like terms: together they make $-2x^2$.

Simplifying expressions

An equation where the variable has power 1. Its graph is a straight line.

Example: $5x - 2 = 13$ gives $x = 3$.

Solving linear equations · Linear equation solver

The power a base must be raised to in order to give a number: $\log_b a = c$ means $b^c = a$.

Example: $\log_2 8 = 3$ because $2^3 = 8$.

Logarithm calculator

A rectangular grid of numbers arranged in rows and columns.

Determinant calculator

The sum of the values divided by how many values there are.

Example: The mean of 2, 4 and 9 is $\frac{15}{3} = 5$.

Averages and spread · Mean, median and mode calculator

The middle value when the data is in order; with an even count, the mean of the two middle values.

Example: The median of 3, 8, 1, 6 is $\frac{3 + 6}{2} = 4.5$.

Mean, median and mode calculator

The point halfway between two points: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Distance and midpoint calculator

A whole number and a proper fraction written together.

Example: $3\frac{1}{2} = \frac{7}{2}$

Fractions

The value that appears most often in a data set. There can be more than one.

Mean, median and mode calculator

A polynomial with one term.

Example: $-4x^3y$

The result of multiplying a number by a whole number.

Example: Multiples of 6: 6, 12, 18, 24, …

A counting number: 1, 2, 3, … (some books include 0).

The top number of a fraction: how many parts are taken.

Example: In $\frac{3}{8}$, the numerator is 3.

Fractions

An angle greater than $90^\circ$ and less than $180^\circ$.

The agreed order for working out a calculation: brackets, powers, multiplication and division (left to right), then addition and subtraction (left to right). Known as BIDMAS, BODMAS or PEMDAS.

Example: $2 + 3 \times 4 = 14$, not 20.

Basic calculator

The U-shaped graph of a quadratic function $y = ax^2 + bx + c$.

How to graph an equation

Lines that never meet. Parallel lines have the same slope.

Slope and the equation of a line

A number out of 100. $p\%$ means $\frac{p}{100}$.

Example: $20\%$ of 50 is $0.2 \times 50 = 10$.

How to calculate percentages · Percentage calculator

The total distance around the outside of a shape.

Example: A 5 by 3 rectangle has perimeter $2(5 + 3) = 16$.

An arrangement of items where order matters: $P(n, r) = \frac{n!}{(n-r)!}$.

Example: Arranging 2 of 4 people in order: $P(4, 2) = 12$.

Permutation and combination calculator

Lines that meet at a right angle. Their slopes multiply to $-1$ (unless one is vertical).

Slope and the equation of a line

The ratio of any circle’s circumference to its diameter, about 3.14159. It is irrational.

Circle calculator

A closed flat shape made of straight sides.

Example: Triangles, quadrilaterals and hexagons are polygons; a circle is not.

Polygons and quadrilaterals

An expression made of terms with whole-number powers of the variable, added or subtracted.

Example: $2x^3 - x + 7$

Polynomials · Polynomial calculator

Writing a number as a product of prime numbers.

Example: $60 = 2^2 \times 3 \times 5$

Prime factorization calculator

A whole number greater than 1 whose only factors are 1 and itself.

Example: 2, 3, 5, 7, 11, 13, …

Prime factors, GCD and LCM

How likely an event is, from 0 (impossible) to 1 (certain). With equally likely outcomes it is $\frac{\text{favourable}}{\text{total}}$.

Example: Rolling a 6 on a fair die: $\frac{1}{6}$.

Probability calculator

A statement that two ratios are equal.

Example: $\frac{2}{3} = \frac{8}{12}$

Ratios and proportions

In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Example: Legs 3 and 4 give hypotenuse $\sqrt{9 + 16} = 5$.

The Pythagorean theorem · Pythagorean theorem calculator

One of the four regions the $x$- and $y$-axes split the plane into, numbered I to IV anticlockwise from the top right.

An equation of the form $ax^2 + bx + c = 0$ with $a \ne 0$. Solutions: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Quadratic equations · Quadratic equation solver

A polygon with four sides. Its angles add up to $360^\circ$.

Polygons and quadrilaterals

Values that split ordered data into four equal parts: $Q_1$ (lower), $Q_2$ (median) and $Q_3$ (upper).

Quartile calculator

The result of a division.

Example: In $20 \div 4 = 5$, the quotient is 5.

A unit of angle: the angle made by an arc as long as the radius. $\pi$ radians $= 180^\circ$.

Degrees and radians converter

The distance from the centre of a circle to its edge; half the diameter.

Circle calculator

All the output values a function can produce.

Example: The range of $x^2$ is $y \ge 0$.

Functions

The largest value minus the smallest value in a data set.

Example: Data 4, 9, 2, 7: range $= 9 - 2 = 7$.

Averages and spread

A comparison of two quantities, written $a : b$.

Example: 6 boys and 4 girls is $6 : 4 = 3 : 2$.

Ratios and proportions · Ratio calculator

A fraction whose top and bottom are polynomials.

Example: $\frac{x + 1}{x^2 - 4}$

Rational expressions · Rational expression calculator

A number that can be written as $\frac{a}{b}$ with integers $a$ and $b \ne 0$.

Example: $0.25 = \frac{1}{4}$ and $-3 = \frac{-3}{1}$

1 divided by a number; flip a fraction to get it.

Example: The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.

Fractions

An angle of exactly $90^\circ$, often marked with a small square.

A value of the variable that makes an equation true; for $f(x) = 0$, where the graph crosses the $x$-axis.

Example: The roots of $x^2 - 9 = 0$ are $x = 3$ and $x = -3$.

What is a solution in math?

Writing a number as $a \times 10^n$ with $1 \le |a| < 10$ and $n$ an integer.

Example: $45\,000 = 4.5 \times 10^4$

Scientific notation converter

Shapes with the same angles whose sides are all in the same ratio: one is an enlargement of the other.

To write an expression in a shorter, equivalent form: collect like terms, cancel common factors and so on.

Simplifying expressions

Two or more equations that must be true at the same time; also called a system of equations.

Example: $x + y = 5$ and $x - y = 1$ give $x = 3$, $y = 2$.

Systems of equations · System of equations solver

In a right triangle, the opposite side divided by the hypotenuse: $\sin\theta = \frac{\text{opp}}{\text{hyp}}$.

Trigonometry basics · Trig calculator

How steep a line is: rise over run, $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Example: From $(1, 2)$ to $(3, 8)$: $m = \frac{6}{2} = 3$.

Slope and the equation of a line · Slope calculator

The equation of a line written as $y = mx + b$: slope $m$, $y$-intercept $b$.

Slope and the equation of a line

A number that, multiplied by itself, gives the original number. $\sqrt{a}$ means the non-negative one.

Example: $\sqrt{49} = 7$

Square roots · Square root calculator

A measure of how spread out data is around the mean: the square root of the variance.

Standard deviation calculator

Two angles that add up to $180^\circ$.

Example: $110^\circ$ and $70^\circ$ are supplementary.

A root that cannot be simplified to a whole number or fraction.

Example: $\sqrt{12} = 2\sqrt{3}$

How to simplify radicals

The total area of all the faces or curved surfaces of a 3D shape.

Example: A cube with side 2 has surface area $6 \times 2^2 = 24$.

Surface area calculator

In a right triangle, the opposite side divided by the adjacent side: $\tan\theta = \frac{\text{opp}}{\text{adj}}$.

Trigonometry basics

A single part of an expression, separated from the others by $+$ or $-$ signs. Also one number in a sequence.

Example: $4x^2 - 3x + 1$ has three terms.

A quadrilateral with one pair of parallel sides. Area $= \frac{1}{2}(a + b)h$.

Area of a trapezium

A vector of length 1, found by dividing a vector by its length.

Unit vector calculator

A letter that stands for a number that can change or is unknown.

Example: In $y = 2x + 1$, $x$ and $y$ are variables.

Find x

A quantity with both size (magnitude) and direction.

Example: $\vec{v} = (3, 4)$ has magnitude $5$.

Vector calculator

A corner point where lines or edges meet; for a parabola, its turning point.

Example: The vertex of $y = (x - 2)^2 + 1$ is $(2, 1)$.

The amount of space a 3D shape takes up, measured in cubic units.

Example: A cylinder: $V = \pi r^2 h$.

3D shapes · Volume calculator

A number with no fractional part: 0, 1, 2, 3, …

How many standard deviations a value is from the mean: $z = \frac{x - \mu}{\sigma}$.

Statistics basics