Real part/Practice Problem 1


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


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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