Glossary

The words the formula pages use, in plain language. Each entry links to the formulas where it matters.

ABCDEFGHILMNPRSTVZ

Amplitude

Half the distance between the highest and lowest values of a wave. For $y=A\sin (Bx)$ it is $|A|$.

Sine wave (amplitude, period and phase)

Annuity

A series of equal payments made at regular intervals, such as monthly savings deposits or loan repayments.

Future value of an annuityPresent value of an annuityLoan repayment formula

Antiderivative

A function whose derivative is the given function. $x^3/3$ is an antiderivative of $x^2$; adding any constant gives another.

Power rule for integralsDefinite integral (area under a curve)

Arc

A piece of the circumference of a circle between two points.

Arc lengthArea of a sector

Arithmetic sequence

A sequence in which each term is the previous one plus a fixed amount, the common difference d: 3, 7, 11, 15, …

Arithmetic sequence (n-th term)Arithmetic series (sum)

Asymptote

A line that a curve gets closer and closer to without reaching it, like the x-axis for $y=2^{-x}$.

Exponential growth and decayDefinition of a logarithm

Axis of symmetry

The line that cuts a parabola into two mirror-image halves; for $y=a{x}^2+bx+c$ it is $x=-b/2a$.

Vertex of a parabola

Base

In $b^x$, the number being raised to a power; in ${\log }_{b}x$, the number whose powers the logarithm counts.

Laws of exponentsDefinition of a logarithmChange of base formula

Binomial coefficient

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, read "n choose k": the number of ways to choose k items from n when order does not matter, $\frac{n!}{k!(n-k)!}$.

CombinationsBinomial theoremBinomial distribution

Chord

A straight line joining two points on a circle. The longest chord is a diameter.

Equation of a circle

Circumference

The distance all the way round a circle, $2\pi r$.

Circumference of a circle

Coefficient

The number multiplying a variable: in $5x^2$ the coefficient is 5.

Quadratic formulaBinomial theorem

Common difference

In an arithmetic sequence, the fixed amount d added to get from one term to the next. It can be negative.

Arithmetic sequence (n-th term)

Common ratio

In a geometric sequence, the fixed number r each term is multiplied by to get the next.

Geometric sequence (n-th term)Geometric series (sum)Infinite geometric series (sum to infinity)

Complex number

A number $a+bi$ where $i^2=-1$. Quadratics with a negative discriminant have two complex roots.

Quadratic formulaDiscriminant

Compound interest

Interest calculated on the original amount plus the interest already added, so the balance grows exponentially.

Compound interestContinuous compoundingEffective annual rate

Conditional probability

The probability of an event given that another has happened, written $P(A\mid B)$.

Conditional probabilityBayes' theorem

Cosine

In a right-angled triangle, adjacent side ÷ hypotenuse; on the unit circle, the x-coordinate of the point at angle θ.

SOH CAH TOA (right-triangle trigonometry)Law of cosinesPythagorean identity

Definite integral

${\int }_a^{b}f(x)dx$: the signed area between the curve and the x-axis from $x=a$ to $x=b$.

Definite integral (area under a curve)Average value of a functionPower rule for integrals

Degree

A unit of angle: a full turn is 360°. One degree is $\pi /180$ radians.

Degrees and radians

Derivative

The rate of change of a function, and the slope of its tangent line at each point; written $f\prime (x)$ or $\frac{dy}{dx}$.

Power rule for derivativesDifference quotientEquation of a tangent line

Diameter

A chord through the centre of a circle; twice the radius.

Circumference of a circleArea of a circle

Discriminant

$b^2-4ac$ for a quadratic $a{x}^2+bx+c$; its sign tells how many real roots there are.

DiscriminantQuadratic formula

Displacement

Distance in a particular direction from a starting point. It can be negative, unlike distance travelled.

Displacement: s = ut + ½at²Velocity and displacement: v² = u² + 2asDisplacement from average velocity

Euler's number

e ≈ 2.718 28, the base of natural logarithms; $e^x$ is the function that is its own derivative.

Exponential growth and decayContinuous compoundingDerivatives of eˣ and ln x

Expected value

The long-run average outcome of a random quantity: each value times its probability, all added up.

Expected value

Exponent

The small raised number in $b^n$ that says how many times the base is multiplied; also called a power or index.

Laws of exponentsLogarithm power rule

Factorial

$n!=n\times (n-1)\times \cdots \times 2\times 1$, with $0!=1$. It counts the orderings of n different items.

PermutationsCombinations

Geometric sequence

A sequence in which each term is the previous one times a fixed ratio r: 3, 6, 12, 24, …

Geometric sequence (n-th term)Geometric series (sum)

Gradient

Another word for slope: rise over run.

Slope formulaSlope-intercept form

Half-life

The time for half of a decaying quantity to disappear. It stays the same however much is left.

Half-life formulaExponential growth and decay

Hypotenuse

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

Pythagorean theoremSOH CAH TOA (right-triangle trigonometry)

Identity

An equation that is true for every value of the variable, such as ${\sin }^2\theta +{\cos }^2\theta =1$.

Pythagorean identityDouble-angle formulasAngle sum and difference formulas

Independent events

Events where one happening does not change the probability of the other, so $P(A\cap B)=P(A)P(B)$.

Multiplication rule for independent events (and)

Logarithm

The exponent you need: ${\log }_{b}x=y$ means $b^y=x$.

Definition of a logarithmChange of base formulaLogarithm product rule

Mean

The sum of the values divided by how many there are; the usual "average".

Mean (average)Standard deviationZ-score (standard score)

Mutually exclusive

Events that cannot both happen at once, so $P(A\cap B)=0$ and $P(A\cup B)=P(A)+P(B)$.

Addition rule (or)

Normal distribution

The symmetric bell-shaped distribution described by its mean μ and standard deviation σ.

Normal distribution probabilityZ-score (standard score)

Parabola

The U-shaped graph of a quadratic function.

Quadratic formulaVertex of a parabola

Period

The length of one complete cycle of a repeating function; for $\sin (Bx)$ it is $2\pi /|B|$.

Sine wave (amplitude, period and phase)

Perpendicular

At right angles. Two non-vertical lines are perpendicular when their slopes multiply to −1.

Slopes of perpendicular linesDistance from a point to a line

Present value

What a future amount of money is worth today, given an interest rate.

Present valuePresent value of an annuity

Principal

The amount of money originally invested or borrowed.

Simple interestCompound interestLoan repayment formula

Radian

The angle that cuts off an arc equal in length to the radius; a full turn is $2\pi$ radians.

Degrees and radiansArc length

Radius

The distance from the centre of a circle or sphere to its edge.

Area of a circleVolume of a sphereEquation of a circle

Root

A solution of an equation; for $f(x)=0$, an x-value where the graph crosses or touches the x-axis.

Quadratic formulaDiscriminant

Sample standard deviation

The standard deviation of a sample used to estimate a population's: divide by $n-1$ instead of n.

Standard deviationMargin of error (proportion)

Sector

A slice of a circle between two radii, like a slice of pizza.

Area of a sectorArc length

Series

The sum of the terms of a sequence.

Arithmetic series (sum)Geometric series (sum)Infinite geometric series (sum to infinity)

Sine

In a right-angled triangle, opposite side ÷ hypotenuse; on the unit circle, the y-coordinate of the point at angle θ.

SOH CAH TOA (right-triangle trigonometry)Law of sinesSine wave (amplitude, period and phase)

Slant height

On a cone or pyramid, the distance from the apex down the sloping face to the edge of the base.

Surface area of a cone

Slope

Rise over run: how much y changes for each step of 1 in x.

Slope formulaSlope-intercept formPoint-slope form

Standard deviation

A measure of spread: roughly the typical distance of the values from their mean.

Standard deviationZ-score (standard score)Normal distribution probability

Tangent

In trigonometry, opposite ÷ adjacent, or $\sin \theta /\cos \theta$. In calculus, the straight line that just touches a curve at a point.

SOH CAH TOA (right-triangle trigonometry)Equation of a tangent line

Term

One of the numbers in a sequence, or one of the parts added together in an expression.

Arithmetic sequence (n-th term)Geometric sequence (n-th term)Binomial theorem

Variance

The square of the standard deviation: the average squared distance from the mean.

Standard deviation

Vertex

A corner of a shape; for a parabola, its turning point.

Vertex of a parabola

Z-score

How many standard deviations a value is above (+) or below (−) the mean.

Z-score (standard score)Normal distribution probability