Basic Properties of Complex Numbers Review

# Basic Properties of Complex Numbers Review

We will now review some of the recent material regarding complex numbers.

• Recall from the The Set of Complex Numbers page that the Imaginary Unit is defined to be $i = \sqrt{-1}$. An Imaginary Number is a number of the form $bi$ where $b \in \mathbb{R}$. For example, $2i$ is an imaginary number. A Complex Number is a number of the form $a + bi$ where $a, b \in \mathbb{R}$. For example, $3 + 2i$ is a complex number. The set of complex numbers is denoted by $\mathbb{C}$.
• If $z = a + bi \in \mathbb{C}$ then we defined the Real Part of $z$ as $\mathrm{Re} (z) = a$, and we defined the Imaginary Part of $z$ as $\mathrm{Im} (z) = b$. Furthermore, $z$ can be represented as an ordered pair or vector $(a, b)$ in the complex plane where the complex plane is $\mathbb{R}^2$ with the $x$-axis labelled as the Real Axis and the $y$-axis labelled as the Imaginary Axis.
(1)
\begin{align} \quad z + w = (a + c) + (b + d)i \end{align}
• We defined Multiplication between $z = a + bi, w = c + di \in \mathbb{C}$ to obtain the Product $z \cdot w$ by:
(2)
\begin{align} \quad z \cdot w = (ac - bd) + (ad + bc)i \end{align}
• On the Division of Complex Numbers page we defined Division of two complex numbers. First we noted that if $w = c + di \in \mathbb{C}$ and $w \neq 0$ then the inverse of $w$ is:
(3)
\begin{align} \quad w^{-1} = \frac{1}{w} = \frac{c}{c^2 + d^2} - \frac{d}{c^2 + d^2}i \end{align}
• So if $z = a + bi, w = c + di \in \mathbb{C}$ and $w \neq 0$ then we can define the Quotient $\displaystyle{\frac{z}{w}}$ as:
(4)
\begin{align} \quad zw^{-1} = \frac{z}{w} = \frac{ac + bd}{c^2 + d^2} - \frac{ad - bc}{c^2 + d^2}i \end{align}
• On the The Set of Complex Numbers is a Field page we then noted that the set of complex numbers $\mathbb{C}$ with the operations of addition $+$ and multiplication $\cdot$ defined above make $(\mathbb{C}, +, \cdot)$ an algebraic field (similarly to that of the real numbers with the usually defined addition and multiplication).
• On the The Conjugate of a Complex Number page we then defined the Complex Conjugate of $z = a + bi$ to be the complex number $\overline{z} = a - bi$. We then proved some basic properties regarding the complex conjugate. We saw that $z = \overline{z}$ if and only if $z \in \mathbb{R}$, $z + \overline{z} = 2a$, $z - \overline{z} = 2bi$, and that $z \cdot \overline{z} = a^2 + b^2$.
(5)
\begin{align} \quad \mid z \mid = \sqrt{a^2 + b^2} \end{align}
• Geometrically, the absolute value of a complex number $z = a + bi$ gives us the length of the position vector $(a, b)$ in the complex plane.
• On the Square Roots of Complex Numbers page we proved a very important result which said that if $z \in \mathbb{C}$ then there exists a $w \in \mathbb{C}$ such that $z = w^2$. In other words, every complex number has a square root. More generally, every complex number has precisely two square roots (allowing repeated roots).
• We saw that the square roots of a complex number $z$ are both real if and only if $z$ is real and positive. We saw that the square roots of $z$ are both imaginary if and only if $z$ is real and negative. We saw that the square roots of $z$ are the same if and only if $z = 0$.