3.9 Complex numbers

Understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument and conjugate, and use the fact that two complex numbers are equal if and only if both their real parts and their imaginary parts are equal.

The notations Re z, Im z, |z|, arg z and z* should be known. The argument of a complex number usually means an angle θ with −π < θ ≤ π, but in some cases the interval 0 ≤ θ < 2π may be more convenient; answers may use either interval unless the question specifies otherwise.

Taught in: Complex numbers: what i is, and the parts of a complex number · 4 questions

Carry out addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy.

For calculations involving multiplication or division, full details of the working should be shown.

Taught in: Complex arithmetic: add, subtract, multiply and divide · 4 questions

Use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs.

For example, in solving a cubic or quartic equation where one complex root is given.

Taught in: Conjugate pairs: non-real roots arrive in twos · 5 questions

Represent complex numbers geometrically by means of an Argand diagram.

Taught in: Argand diagrams — complex numbers as geometry · 4 questions

Carry out multiplication and division of two complex numbers expressed in polar form r(cos θ + i sin θ) ≡ re^(iθ).

Includes the results |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂), and the corresponding results for division.

Taught in: Polar form: multiply the lengths, add the angles · 5 questions