e^{i\theta}=\cos\theta+i\sin\theta (with \theta in radians).

Exponential form of a complex number z?

z=re^{i\theta} with r=|z| and \theta=\arg z.

Modulus of e^{i\theta}?

1, since \cos^2\theta+\sin^2\theta=1.

\dfrac{r1e^{i\theta1}}{r2e^{i\theta2}}?

\frac{r1}{r2}e^{i(\theta1-\theta2)}.

When is re^{i\theta} real?

When \theta is a multiple of \pi (positive if a multiple of 2\pi).

Exponential form of a complex numberAQA A-Level Further Maths: Flashcards

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Define $\mathrm{e}^{\mathrm{i}\theta}$.

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Define $\mathrm{e}^{\mathrm{i}\theta}$ .: $\mathrm{e}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$ (with $\theta$ in radians).
Exponential form of a complex number $z$ ?: $z=r\mathrm{e}^{\mathrm{i}\theta}$ with $r=|z|$ and $\theta=\arg z$ .
Modulus of $\mathrm{e}^{\mathrm{i}\theta}$ ?: $1$ , since $\cos^2\theta+\sin^2\theta=1$ .
Value of $\mathrm{e}^{\mathrm{i}\pi}$ ?: $-1$ .
Value of $\mathrm{e}^{\mathrm{i}\pi/2}$ ?: $\mathrm{i}$ .
Conjugate of $r\mathrm{e}^{\mathrm{i}\theta}$ ?: $r\mathrm{e}^{-\mathrm{i}\theta}$ .
$r_1\mathrm{e}^{\mathrm{i}\theta_1}\times r_2\mathrm{e}^{\mathrm{i}\theta_2}$ ?: $r_1r_2\mathrm{e}^{\mathrm{i}(\theta_1+\theta_2)}$ .
$\dfrac{r_1\mathrm{e}^{\mathrm{i}\theta_1}}{r_2\mathrm{e}^{\mathrm{i}\theta_2}}$ ?: $\frac{r_1}{r_2}\mathrm{e}^{\mathrm{i}(\theta_1-\theta_2)}$ .
$\left(r\mathrm{e}^{\mathrm{i}\theta}\right)^n$ ?: $r^n\mathrm{e}^{\mathrm{i}n\theta}$ .
When is $r\mathrm{e}^{\mathrm{i}\theta}$ real?: When $\theta$ is a multiple of $\pi$ (positive if a multiple of $2\pi$ ).
$z=\mathrm{e}^{\mathrm{i}\theta}$ : $z+\frac1z$ ?: $2\cos\theta$ .
$z=\mathrm{e}^{\mathrm{i}\theta}$ : $z-\frac1z$ ?: $2\mathrm{i}\sin\theta$ .
Exponential form of $1-\sqrt3\,\mathrm{i}$ ?: $2\mathrm{e}^{-\mathrm{i}\pi/3}$ .

Exam questions on Exponential form of a complex number

The complex number $z=2\mathrm{e}^{\mathrm{i}\pi/3}$ .
Find $z^3$ in the form $a+b\mathrm{i}$ .2 marks
The complex numbers $w_1=3\mathrm{e}^{\mathrm{i}\pi/4}$ and $w_2=2\mathrm{e}^{-\mathrm{i}\pi/4}$ .
Express $w_1^2w_2$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$ .2 marks
The complex number $z=1-\sqrt3\,\mathrm{i}$ .
Express $z$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).