Glossary — Pure Mathematics

Amplitude — the distance from the middle line of a wave graph to a peak.

Arithmetic sequence — a sequence that rises or falls by the same amount at every step.

Assumption — in a statement of the form if A then B, the part A that is taken as true.

Asymptote — a line that a curve approaches but never reaches.

Chain rule — the rule for differentiating a function of a function.

Common ratio — the fixed multiplier between consecutive terms of a geometric sequence, written $r$.

Conclusion — in a statement of the form if A then B, the part B that follows from the assumption.

Counter example — a single value that makes a "for all" statement false, and so disproves it.

Definite integral — an integral worked out between two limits, which gives a number.

Derivative — the gradient of a curve at a point, or the rate of change of one quantity with respect to another.

Differential equation — an equation that links a derivative to the quantities it describes.

Discriminant — the value $b^2 - 4ac$, which tells you how many real roots the equation $ax^2 + bx + c = 0$ has.

Exponential function — a function with the variable in the index, such as $y = 2^x$ or $y = e^x$.

Factor theorem — the result that $(x - a)$ is a factor of a polynomial $f(x)$ exactly when $f(a) = 0$.

Factorial — the product of all the whole numbers from $1$ up to a given whole number, written $n!$.

Geometric sequence — a sequence made by multiplying by the same amount at every step.

Gradient — how steeply a line or curve rises; the change in the vertical coordinate divided by the change in the horizontal coordinate.

Iterative method — a method that finds a root by applying the same rule again and again, using each answer as the next input.

Logarithm — the power to which a base must be raised to give a number; $\log_a x$ asks what power of $a$ gives $x$.

Magnitude — the length or size of a vector, written $|\mathbf{a}|$.

Modulus — the distance of a number from zero, so it is never negative.

Natural logarithm — the logarithm to base $e$, written $\ln x$.

Newton–Raphson — an iterative method for finding a root that uses the tangent to the curve at each step.

Parametric equations — a pair of equations that give $x$ and $y$ separately in terms of a third variable, called the parameter.

Partial fractions — the simpler fractions that a rational expression is split into.

Period — the length of the interval over which a repeating graph repeats.

Point of inflection — a point where a curve changes from convex to concave, or from concave to convex.

Polynomial — a sum of terms of the form $ax^n$, where $n$ is a whole number.

Position vector — the vector from the origin to a point.

Product rule — the rule for differentiating a product of two functions.

Proof by contradiction — a proof that assumes the statement is false and works until it reaches an impossibility.

Proof by deduction — a proof that works from the assumption with algebra to reach the conclusion.

Proof by exhaustion — a proof that checks every case in a finite set.

Quotient rule — the rule for differentiating a quotient of two functions.

Radian — the angle at the centre of a circle that cuts off an arc equal in length to the radius.

Recurrence relation — a rule that links each term of a sequence to the one before it.

Remainder theorem — the result that dividing a polynomial $f(x)$ by $(x - a)$ leaves the remainder $f(a)$.

Resultant — a single vector that has the same effect as two or more vectors together.

Root — a value of the variable that makes the two sides of an equation equal.

Scalar — a quantity that has a size but no direction.

Sequence — an ordered list of numbers.

Series — the total of the terms of a sequence.

Sigma notation — a short way of writing a sum, using the symbol $\sum$.

Statement — a sentence that is either true or false.

Stationary point — a point on a curve where the gradient is zero.

Surd — a root that cannot be written as an exact fraction.

Tangent — a straight line that touches a curve at a point and has the same gradient as the curve there.

Trapezium rule — a formula that estimates a definite integral by adding the areas of trapezium strips.

Vector — a quantity that has both a size and a direction.