Complex Numbers Questions with Solutions

A complex number has the form z = a + ib, where a and b are real numbers and i is the imaginary unit with i^2 = -1. This topic covers powers of i (which repeat in a cycle of four: i, -1, -i, 1), algebra of complex numbers, the conjugate, the modulus |z| = sqrt(a^2 + b^2), and the argument (amplitude). It also covers the polar and exponential forms of a complex number, De Moivre's theorem, the cube roots of unity (1, omega and omega^2, where 1 + omega + omega^2 = 0 and omega^3 = 1), square roots of complex numbers, and representing complex numbers in the Argand plane. Questions involving powers of omega are very common, and the two identities above solve most of them in a line or two. For the argument, always check the quadrant of the point before deciding the angle. Practise the complex numbers questions below, each with a step-by-step solution.