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Real part/Practice Problem 1

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Problem

Find the conditions on $w$ and $z$ so that $\mathrm{Re}(w\cdot z) = \mathrm{Re}(w) \cdot \mathrm{Re}(z)$ .

Solution

Let $w = a + bi$ and $z = c + di$ . Then $w\cdot z = (a + bi)\cdot(c + di) = (ac - bd) + (ad + bc)i$ . So $\mathrm{Re}(w\cdot z) = ac - bd$ . $\mathrm{Re}(w)\cdot \mathrm{Re}(z) = ac$ . Now $ac = ac - bd$ if and only if $bd = 0$ , so at least one of $b$ and $d$ must equal 0. Thus $\mathrm{Re}(w\cdot z) = \mathrm{Re}(w) \cdot \mathrm{Re}(z)$ if and only if at least one of $w$ and $z$ is real.

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