Complex Numbers

We have just seen that $x^2+1=0$ has no solution among the real numbers. More generally, there are polynomial equations with real coefficients that have no real solution at all. To be able to solve every such polynomial equation, we enlarge the real number system into a bigger one — the complex number system — that is guaranteed to contain a solution.

This section defines three things needed to work with complex numbers:

1. Complex numbers in rectangular form;
2. The Argand plane (the geometric picture of a complex number);
3. Algebraic operations (addition, subtraction, multiplication) on complex numbers.

The complex number system is built by extending the real numbers with the imaginary unit $i$, where $i^2=-1$. Combining $i$ with two real numbers $x$ and $y$, by the processes of addition and multiplication, produces a complex number $x+iy$. The symbol "$+$" here is treated the same way as vector addition (the notation was introduced by Carl Friedrich Gauss, 1777–1855). …