Maths Pure 3 · Complex numbers. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The number i is defined by i2 = −1. A complex number z = x + iy has a real part x and an imaginary part y. You add, subtract and multiply like ordinary algebra, replacing i2 by −1 each time. The conjugate z* = x − iy is used for division.
On an Argand diagram z is the point (x, y). Its modulus |z| is the distance from the origin, and its argument is the angle measured anticlockwise from the positive real axis, with −π < arg z ≤ π. These give the polar form r(cos θ + i sin θ).
Equations in z describe loci, such as circles and lines. A polynomial equation with real coefficients has its non-real roots in conjugate pairs.
Students read the centre of a circle with the wrong signs. |z − 3 + 2i| = 4 must be written as |z − (3 − 2i)| = 4, so the centre is 3 − 2i, the point (3, −2), not (−3, 2). For arg z, always sketch the point first to find the quadrant.
Answer: 2 − i, with modulus √5
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