Complex numbers

Complex numbers have only ever appeared in Section A. However, this does not mean that a section B question is impossible. Questions generally include polar form and the use of De Moivre’s theorem.

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Be able to work with complex numbers in rectangular form. It’s important to be able to plot numbers on the Argand diagram. Example: If z z

1=1+iand 2=–1+2 i, plot the following: ●

Be able to calculate the modulus |z| and the conjugate z̅ of a complex number z.

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Be able to solve equations with real or imaginary coefficients

A quadratic equation with real coefficients either has two real roots OR two complex roots that are conjugates of each other. A cubic equation with real coefficients either has three real roots OR one real root and two complex roots that are conjugates of each other.

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Be able to put complex numbers into polar form and use De Moivre’s theorem. To find the argument: Understand the difference in the questions below.

Example 1: Evaluate z4 where For solving an equation, we need general polar form As there are going to be four different solutions, sub in four consecutive values for n to find the solutions. ●

Be able to prove De Moivre’s Theorem by

∈ induction for n N (this proof has never been examined)