# 3.1 The Complex Numbers

Perform computations involving complex numbers.

Some functions have zeros that are not real numbers. In order to find the zeros of such functions, we must consider the **complex-number system**.

# The Complex-Number System

We know that the square root of a negative number is not a real number. For example, $\sqrt{-1}$ is not a real number because there is no real number `x` such that ${x}^{2}=-1$. This means that certain equations, like ${x}^{2}=-1$ or ${x}^{2}+1=0$, do not have real-number solutions, and certain functions, like $f(x)={x}^{2}+1$, do not have real-number zeros. Consider the graph of $f(x)={x}^{2}+1$.

We see that the graph does not cross ...

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