Express $-1+i$ in exponential form.

Question

Express $-1+i$ in exponential form.

My attempt so far

Let $z=-1+i$

$$r=|z|=\sqrt2$$

$$\theta=\tan^{-1}(-1)=-\frac{\pi}{4}$$

Now, this is where I go wrong (I don't know why it's wrong!):

So in exponential form: $-1+i=\sqrt2 e^{-i\pi/4}$

According to the solutions, $\theta=3\pi/4$.

Thanks in advance

Answer 1

If $\tan \theta = -1$, then

• $\cos\theta > 0, \text{ and } \sin\theta <0 \implies \theta = \frac {3\pi}4 $,

  or else

• $\cos\theta < 0 ,\text{ and } \sin \theta > 1\implies \theta = \frac {7\pi}4 = -\frac{\pi}{4}$.

Since we are working with $-1 + i$, $\cos \theta \lt 0$, $\sin \theta > 0$, and hence, $\theta = \frac{3\pi}{4}$.

Hence, $-1 + i = \sqrt 2e^{3\pi/4}.$

Answer 2

You almost got it! $$ -1+i=\sqrt{2}\,\mathrm{e}^{3\pi i/4}=\mathrm{e}^{\ln 2/2}\,\mathrm{e}^{3\pi i/4}=\ldots $$