I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes into calculating the amplitude but I can't be sure.
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The absolute value of the numerator is $\sqrt{2^2+2^2} = 2\sqrt2$, not $2$. – Théophile Mar 19 '14 at 02:17
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Try multiplying the numerator and denominator by $1+i$. This will give you $\frac{(2i+2)(1+i)}{1^2+1^2}$. Then, FOIL the numerator and note $i^2=-1$.
Batman
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Ahhhhh so the easiest way to go about these problems is just eliminating the denominator? Makes sense thanks! – Bourezg Mar 19 '14 at 02:16
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Yes, in general, if the denominator is $a+bi$, multiply the top and bottom by the conjugate, i.e., $a-bi$. – Théophile Mar 19 '14 at 02:18
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The secret is to multiply and divide by the conjugate of the denominator:
$$\frac{2i+2}{1-i} =\frac{2i+2}{1-i}\frac{1+i}{1+i}=\frac{2(1+i)(1+i)}{1-i^2}=\frac{(1+i^2)}{1+1}=1+2i+i^2=1+2-1=2i.$$
ZHN
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What about $$\frac{1+i}{1-i}=\frac{2\sqrt{2}(\sqrt{2}/2+i\sqrt{2}/2)}{\sqrt{2}(\sqrt{2}/2-i\sqrt{2}/2)} =2\frac{e^{i\frac{\pi}{4}}}{e^{-i\frac{\pi}{4}}}=2e^{i\frac{\pi}{2}}=2i?$$
Michael Hoppe
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