Some Facts About Complex Numbers « Calculus: Problems and Solutions

Some Facts About Complex Numbers

July 9, 2009 · 1 Comment

Let’s look at some properties of complex numbers, numbers of the form $a+ib$ where $a$ and $b$ are real numbers and $i^2=-1$ . We’ll use these together with Euler’s formula to deduce some trig identities.

First, if $z = a+ib$ , I’m going to call $a$ the real part of $z$ and $b$ the imaginary part. We write $a=\Re z$ and $b=\Im z$ ; the big goofy symbols are for historical reasons.

Now what if we start squaring $z$ ? What is $\Re (z^2)$ ? Computing, we get $z^2 = a^2+2iab + i^2b = a^2-b^2+2iab$ , so
$\Re (z^2) = a^2-b^2 = (\Re z)^2-(\Im z)^2$ .