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Intro to complex numbers and conjugates

Thus far we have focused on real numbers. However, we mentioned at the beginning that we can fully factor any polynomial of degree $n$ into $n$ linear factors if we used complex valued numbers. This means that we need to explore what complex valued numbers actually are. We begin with a motivating example, showing what problem complex numbers are designed to solve.

At first glance it may seem like only defining $i$ so that $i^2 = -1$ won’t be sufficient to solve all our problems with these ‘non-real roots’. For example; how about solving $x^2 = -4$ or $x^2 = -9$? Actually, we can still evaluate these non-real roots by using $i$ to remove the negative on its own, and then evaluating the positive square root afterward. Let’s see an example.

You may notice that the zeros in the above example are incredibly similar. In fact, purely by how they were found you can see that the only difference is in the sign of the imaginary part (the term with $i$). An astute student may even notice that, because of how the $\pm$ sign came to be (by square rooting) and how $i$ must be brought into the answer (inside a square root), that in fact these things will always come together. This phenomena is actually true and is (one of) the reason(s) we give this relationship between these two complex values a special name; they are called ‘conjugate pairs’ or ‘complex conjugates’.

In general, if some complex-valued number is a zero of a polynomial, then the complex conjugate must also be a zero of the polynomial.