FY13CE Pure Mathematics key terms

A proof technique that starts by assuming the statement to be proved is false, then derives a logical impossibility. The contradiction shows the original assumption cannot hold, so the statement must be true.

A proof that splits a claim into finitely many exhaustive cases and verifies each one separately. Valid only when the cases genuinely cover every possibility, such as all residue classes modulo some integer.

A single specific case in which a general statement fails. One counterexample is enough to disprove a universal claim, making it the standard tool for refuting conjectures about numbers or functions.

A statement believed to be true on the basis of evidence but not yet proved or disproved. In this exam, a conjecture must be either justified by proof or destroyed by a counterexample.

A proof method for statements indexed by natural numbers: establish a base case, then show that truth for an arbitrary value k forces truth for k+1. Together these give truth for all integers from the base onwards.

A condition that is both required for and enough to guarantee a statement. "P if and only if Q" asserts that Q is necessary and sufficient for P, so the two statements are logically equivalent.

A logical statement of the form "if P then Q", written P implies Q. It is false only when P is true and Q is false; when P is false the implication is automatically true, a point often tested.

The statement obtained by swapping hypothesis and conclusion of an implication, so "if P then Q" becomes "if Q then P". A statement and its converse need not share the same truth value.

A function expressible as one polynomial divided by another, defined wherever the denominator is nonzero. Its graph may have vertical asymptotes at denominator zeros and a horizontal or oblique asymptote determined by degrees.

A decomposition of a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator. It makes integration and series expansion of rational functions tractable.

The statement that a polynomial f has factor (x - a) exactly when f(a) = 0. It converts root-finding into factorisation and underpins the solution of polynomial equations and inequalities.

The result that dividing a polynomial f(x) by (x - a) leaves remainder f(a). It allows remainders to be computed by substitution instead of long division, and generalises to divisors of higher degree.

An expression formed from a variable and constants using only addition, subtraction and non-negative whole-number powers. Its degree is the highest power present, and it is defined and continuous for all real inputs.

The function giving the non-negative distance of a number from zero, written |x| and defined as x when x is non-negative and -x otherwise. Its graph is V-shaped and equations involving it often split into cases.

A line that a curve approaches ever more closely as the variable tends to infinity or to a value where the function is undefined. Vertical, horizontal and oblique asymptotes are all examined for rational and exponential graphs.

A sequence in which consecutive terms differ by a fixed common difference. Its nth term is a + (n-1)d and the sum of the first n terms is n/2 times the first term plus the last term.

A sequence in which each term is a fixed multiple, the common ratio, of the previous one. Sums converge only when the ratio lies strictly between -1 and 1, giving the sum to infinity a/(1-r).

The property of a series whose partial sums approach a finite limiting value as more terms are added. For a geometric series this happens exactly when the common ratio has modulus less than one.

Compact notation using the Greek capital sigma to denote a sum, with an index, lower and upper limits written beneath and above. It is used to state series sums and to manipulate standard results algebraically.

A summation technique in which each term of a series is written as a difference of consecutive values of some expression, so most terms cancel when the sum is expanded, leaving only a few boundary terms.

An equation defining each term of a sequence using one or more earlier terms together with an initial value. Solving it means finding a closed formula, often verified afterwards by induction.

The expression of a power of a sum, such as (a + b)^n, as a sum of terms involving binomial coefficients. For non-integer or negative powers it becomes an infinite series valid over a restricted range.

A relation between x and y that describes a curve in the plane, as opposed to a parametric or polar description. Converting between forms is a routine skill in coordinate geometry questions.

A pair of equations expressing x and y separately in terms of a third variable, the parameter. They describe curves such as circles and parabolas and allow gradients to be found by differentiating with respect to the parameter.

The measure of steepness of a line or curve, equal to the change in y divided by the change in x for a line, and to the derivative at a point for a curve. Parallel lines share it; perpendicular lines multiply to -1.

A straight line touching a curve at a point and having the same gradient as the curve there. Its equation follows from the point and the derivative evaluated at that point.

The straight line through a point on a curve perpendicular to the tangent there. Its gradient is the negative reciprocal of the curve's gradient at that point.

An angular measure in which one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn is 2 pi radians, and calculus formulas for trigonometric functions require this measure.

The distance along a section of a curve. For a circle of radius r subtending angle theta in radians it equals r times theta, and for a general curve it is found by integrating the square root of one plus the squared derivative.

An equation true for all values of the variables, such as the Pythagorean identity linking sine and cosine, or double-angle formulas. Identities are used to simplify expressions and to transform integrals.

The replacement of sine of a small angle by the angle itself in radians, with cosine approximated by one minus half the angle squared. These approximations underpin the derivative of sine and simple harmonic motion.

The value a function or sequence approaches as its input approaches a specified point or grows without bound. Limits define continuity, derivatives and convergence, and can be evaluated by algebra or numerically.

The instantaneous rate of change of a function, defined as the limit of a difference quotient. Geometrically it is the gradient of the tangent, and it measures sensitivity of the output to small changes in input.

The rule for differentiating a composite function: differentiate the outer function at the inner function and multiply by the derivative of the inner function. It is the most frequently used rule in this exam.

A technique for differentiating a relation not given in the form y equals f(x), by differentiating every term with respect to x and treating y as a function of x. It produces derivatives involving both variables.

A point where the derivative of a function is zero, so the tangent is horizontal. It may be a maximum, a minimum or a point of inflection, distinguished by the second derivative or a sign change.

The derivative of the derivative, measuring how the gradient itself changes. Its sign at a stationary point usually classifies it as a maximum when negative or a minimum when positive.

An integration technique derived from the product rule, rewriting an integral of a product as a boundary term minus another integral. It suits products of polynomials with exponentials, logarithms or trigonometric functions.

A technique that reverses the chain rule by changing the integration variable to simplify an integrand, adjusting the differential and, for definite integrals, the limits accordingly.

The integral of a function between two limits, representing the signed area under its graph. It is evaluated by finding an antiderivative and subtracting its values at the limits.

The volume generated when a region is rotated about an axis, computed by integrating pi times the square of the radius with respect to the axis of rotation. The limits are the coordinates bounding the region.

A vector giving the location of a point relative to a chosen origin, written as a column or in unit-vector notation. Differences of position vectors give displacement vectors between points.

The product of two vectors giving a number: the product of their magnitudes and the cosine of the angle between them. It tests perpendicularity, since it is zero exactly when the vectors are perpendicular.

A description of a line as a fixed position vector plus a scalar multiple of a direction vector. It encodes every point on the line as the parameter varies over all real values.

The complex number obtained by changing the sign of the imaginary part, so a + bi becomes a - bi. Multiplying a number by its conjugate gives a real value, used to divide complex numbers and find polynomial roots.

The modulus is the distance of a complex number from the origin; the argument is the angle its position vector makes with the positive real axis. Together they give the polar form used for multiplication and powers.

The result that raising a complex number in polar form to an integer power multiplies its argument by that power and raises its modulus to that power. It simplifies powers and roots of complex numbers.

Solutions of polynomial equations that are not real. For polynomials with real coefficients they occur in conjugate pairs, and every polynomial of degree n has exactly n roots counted with multiplicity over the complex numbers.

A rectangular array of numbers arranged in rows and columns, used to represent linear transformations and to solve simultaneous equations. Its order states the number of rows then columns.

A number computed from a square matrix that indicates whether the matrix is invertible and how it scales areas or volumes. A zero determinant means the transformation collapses space and no inverse exists.

The matrix that undoes the effect of a given square matrix, existing only when the determinant is nonzero. For a two-by-two matrix it is formed by swapping diagonal entries, negating the others and dividing by the determinant.

A mapping of the plane or space that preserves vector addition and scalar multiplication, representable by a matrix acting on column vectors. Rotations, reflections, stretches and shears are the standard examples.

An iterative numerical method for solving f(x) = 0 that uses the tangent at the current estimate to produce a better one. It converges rapidly when the starting value is close to the root.

Repeated application of a formula to generate a sequence of approximations to a solution. Convergence depends on the behaviour of the derivative of the iteration function near the fixed point.

A numerical integration method that approximates the area under a curve by summing the areas of trapezia joining successive points on the curve. Accuracy improves with more, narrower strips.