The Set of Complex Numbers

# The Set of Complex Numbers

By now you should be relatively familiar with the set of real numbers denoted $\mathbb{R}$ which includes numbers such as $2$, $-4$, $\displaystyle{\frac{6}{13}}$, $\pi$, $\sqrt{3}$, …

We will now introduce the set of complex numbers.

 Definition: The number $i = \sqrt{-1}$ is called the Imaginary Unit. An Imaginary Number is a number of the form $bi$ where $b \in \mathbb{R}$, and a Complex Number is a number of the form $a + bi$ where $a, b \in \mathbb{R}$. The set of all complex numbers is denoted $\mathbb{C}$. If $z = a + bi \in \mathbb{C}$ then the Real Part of $z$ is $\mathrm{Re}(z) = a$, while the Imaginary Part of $z$ is $\mathrm{Im}(z) = b$.

Some people implicitly define $i$ such that $i^2 = -1$.

In set-builded notation we can denote the set of complex numbers as:

(1)
\begin{align} \quad \mathbb{C} = \{ a + bi : a, b \in \mathbb{R} \: \mathrm{and} \: i = \sqrt{-i} \} \end{align}

Furthermore, we can express any complex number $z = a + bi$ in terms of its real and imaginary parts as:

(2)
\begin{align} \quad z = \mathrm{Re} (z) + \mathrm{Im}(z) i \end{align}

Another notation for the real part of a complex number $z$ is $\Re (z)$, and another notation for the imaginary part of a complex number $z$ is $\Im (z)$.

 Definition: The Complex Plane consists of a horizontal axis known as the Real Axis and a vertical axis known as the Imaginary Axis, which contains every complex number $z = a + bi \in \mathbb{C}$ as a point $(a, b)$ on this plane.

in a sense, the complex plane $\mathbb{C}$ can often be viewed as $\mathbb{R}^2$.

For example, consider the complex number $z = 2 + 3i$. We can think of this number as the point $(2, 3)$ in the complex plane and represent it as such or even as a vector whose initial point is the origin and whose terminal point is $(2, 3)$ as illustrated below:

Note that the set of points $z \in \mathbb{C}$ for which $\mathrm{Im}(z) = 0$ is simply the real line, while the set of points for which $\mathrm{Re}(z) = 0$ is simply the imaginary line. Thus, the real and imaginary lines are embedded in the complex plane.