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I don't know where to start... It's a multiple-choice question: I can choose from $\sqrt{2}, 0, 2, 1$

Thank you!

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    You need to know the facts used in the answers. But for multiple choice, the norm of $1-i$ is $\sqrt{2}$, and the norm of $e^{ix}$ is $1$ for every real $x$. so $0$, $2$, and $1$ are impossible.2012-10-20

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Using this or this, $e^{\frac{i\pi}4}=\cos \frac{\pi}4 +i\sin\frac{\pi}4=\frac{1+i}{\sqrt 2}$

$(1-i)\cdot e^{\frac{i\pi}4}=(1-i)\cdot \frac{(1+i)}{\sqrt 2}=\frac{1-i^2}{\sqrt 2}=\sqrt 2$

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    @hkproj, may look into the proof using calculus in the 2nd link.2012-10-20
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$e^{i\pi/4}(1-i)=(1-i)(1+i)\frac{\sqrt{2}}{2}=2\frac{\sqrt{2}}{2}=\sqrt{2}$