Exercise 426Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Exponential form

#exponential form#Euler#polar

Statement

Write each of the following in exponential form:
(a) $z_1=1+i$ ; (b) $z_2=-1+i\sqrt3$ ; (c) $z_3=-1$ .

Hint

Compute the modulus $r$ and argument $\theta$ , then write $z=re^{i\theta}$ .

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(a) $|z_1|=\sqrt2$ , $\arg=\pi/4$ , so $z_1=\sqrt2 e^{i\pi/4}$ .

(b) $|z_2|=\sqrt{1+3}=2$ , $\arg=2\pi/3$ (in the second quadrant). So $z_2=2e^{2i\pi/3}$ .

(c) $|z_3|=1$ , $\arg=\pi$ , so $z_3=e^{i\pi}$ .

Conclusion. $z_1=\sqrt2 e^{i\pi/4}$ , $z_2=2e^{2i\pi/3}$ , $z_3=e^{i\pi}$ .