The Complex Number System

A complex number is a number that can be put in the form $a+bi$ where $a$ and $b$ are real numbers and $i$ is called the imaginary unit, where $i^2=-1$. In this expression, $a$ is called the real part and $b$ the imaginary part of the complex number. We will write $\text{Re}\{a+bi\}=a$ to indicate the real part of the complex number, and $\text{Im}\{a+bi\}=b$ to indicate the imaginary part. 

For example, to indicate that the real part of the number $2+3i$ is $2$, we would write $\text{Re}\{2+3i\}=2$. To indicate that the imaginary part of $4-5i$ is $-5$, we would write $\text{Im}\{4-5i\} = -5$.

Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number $a+bi$ can be identified with the point $(a,b)$. Thus, for example, complex number $-2+3i$ would be associated with the point $(-2,3)$ and would be plotted in the complex plane as shown below. 

The complex point $-2+3i$

The complex number $-2+3i$ is plotted in the complex plane, $2$ to the left on the real axis, and $3$ up on the imaginary axis.

A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the set of ordinary real numbers can be thought of as a subset of the set of complex numbers. It is beneficial to think of the set of complex numbers as an extension of the set of real numbers. This extension makes it possible to solve certain problems that can't be solved within the realm of the set of real numbers. 

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. 

Complex numbers allow for solutions to certain equations that have no real number solutions. For example, the equation:

$(x + 1)^2 = -9$

has no solution if we restrict ourselves to the real numbers, since the square of a real number is never negative. However, we can see that the complex numbers $1+3i$ and $1-3i$ are solutions, since

$\begin{aligned}\left(1+3i-1\right)^2&=(3i)^2\\&=9i^2\\&=-9\end{aligned}$ 

and

 $\begin{aligned} \left(1-3i-1\right)^2 &=(-3i)^2 \\& =9i^2\\&=-9 \end{aligned}$

It turns out that if we allow $x$ to be a complex number, then any polynomial equation in $x$ of degree $n$ will have $n$ (not necessarily unique) solutions.