Basic Properties of Complex Numbers Review

Table of Contents

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.

We then began to define some operations on the set of complex numbers. On the Addition and Multiplication of Complex Numbers page we defined Addition between $z = a + bi, w = c + di \in \mathbb{C}$ to obtain the sum $$z + w$$ by:

(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 $z \cdot \overline{z} = a^2 + b^2$ .

On the The Absolute Value/Modulus of a Complex Number page we define the Absolute Value or Modulus of a complex number $z = a + bi \in \mathbb{C}$ to be:

(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 $$z = 0$$ .

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