CAIE A-Level Mathematics 3.9.1 Complex-Number Terminology and Forms

Practise using real and imaginary parts, modulus, argument and conjugates and converting complex numbers between Cartesian, modulus–argument and exponential forms.

Exam points

equate real and imaginary parts separately after writing each number as x + iy
calculate |z| and arg z with the correct quadrant and specified argument interval
use conjugates and argument laws to express quotients or products in exact exponential form

[Maximum number: 2]

The complex numbers u and v are defined by u=-4+2 i and v=3+i.

Hence express $\frac{u}{v}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ , where r and $\theta$ are exact.

State or imply $r=\sqrt{2}$
B1 FT
State or imply that $\theta=\frac{3}{4} \pi$
B1 FT

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3.9.1—Complex numbers question 1