A-Level Further Mathematics · Topic 1

Complex numbers, matrices and series: every key term you need (+ practice quiz)

16 flashcard terms for A-Level Further Mathematics Topic 1, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 8-question quiz — free, no account needed.

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Imaginary unit: i is defined so that i squared equals -1. It extends the reals to the complex numbers, where every polynomial has a full set of roots.
Argand diagram: Plots a complex number with the real part on the horizontal axis and the imaginary part on the vertical, turning algebra into geometry.
Modulus and argument: The modulus is the distance from the origin; the argument is the angle from the positive real axis. Together they give the polar form.
Complex conjugate: Changes the sign of the imaginary part. For polynomials with real coefficients, non-real roots always occur in conjugate pairs.
De Moivre's theorem: Raising a complex number in polar form to a power multiplies the argument and raises the modulus to that power.
Roots of unity: The n solutions of z to the n equals 1, evenly spaced round the unit circle and summing to zero.
Matrix multiplication: Row times column. It is associative but not commutative, so the order in which transformations are applied matters.
Determinant: The scale factor by which a transformation multiplies area or volume. A determinant of zero means the transformation is singular and non-invertible.
Inverse matrix: Undoes the transformation. Exists only when the determinant is non-zero, and solves simultaneous equations written in matrix form.
Eigenvector: A non-zero vector whose direction is unchanged by a transformation; the eigenvalue is the factor by which it is scaled.
Method of differences: Writes each term as a difference so that intermediate terms cancel in a sum, leaving only the first and last — a telescoping series.
Maclaurin series: Expresses a function as an infinite polynomial using its derivatives at zero. The general case about any point is the Taylor series.
Hyperbolic functions: cosh and sinh are defined from exponentials and satisfy identities parallel to the trigonometric ones, with some signs reversed.
Polar curve: Defines r as a function of the angle theta rather than y as a function of x. Areas are found by integrating half r squared with respect to theta.
Proof by induction: Prove the base case, assume the statement for n = k, then prove it for n = k + 1. Establishes results for all positive integers.
Simple harmonic motion: Acceleration is proportional to displacement and directed towards equilibrium, giving a second-order differential equation with sinusoidal solutions.

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