Some Facts About Complex Numbers

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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.

Similarly, we can compute that \Im (z^2) = 2(\Re z)(\Im z). Check these facts to make sure you understand.

Next time, we’ll use them for something!