1.8: Complex Numbers

Media

Videos

Title	Topics	Type	Length
Introduction to the Complex Numbers as a Result of Solving Quadratic Equations	DEFINITIONS Imaginary unit (\(i\)) Complex numbers (\(a + b i\)) Complex conjugates THEOREMS Conjugate Pairs Theorem EXAMPLES Solve \(x^2 + x + 5 = 0\) and \(2x^2 + 3x + 12 = 0\) Given that \(2 - 3i\) is a solution of a quadratic equation, state another solution and write the original quadratic equation	Lecture	21:02
An Introduction to the Complex Numbers	College Algebra	Lecture	11:45
The Complex Plane and Plotting Complex Numbers	College Algebra	Lecture	5:39

Definitions and Theorems 

Definition: Unit Imaginary Number

The unit imaginary number, denoted as \(i\), is defined as the square root of negative \( 1 \). That is,\[\sqrt{ −1 } =i. \nonumber \]

Theorem: The Square of the Imaginary Unit

\[ i^2 = -1 \nonumber \]

Proof

\[ i^2 = (i)^2 =  \left( \sqrt{-1} \right)^2 = -1 \nonumber \]

Definition: Pure Imaginary Number

Any number of the form \( b i \), where \( b \neq 0 \), is called a pure imaginary number (or simply an imaginary number).

Definition: Complex Number

A complex number is a number that can be written in the form \(a+bi\) where

• \(a\) is called the real part, and
• \(b\) is called the imaginary part of the complex number.

If \(b=0\), then \(a+bi\) is a real number. If \(a=0\) and \(b\) is not equal to \( 0 \), the complex number is called an imaginary number. The form, \( a + bi \) is called the standard form of the complex number.

Definition: Complex Number System

The set of all numbers of the form \( a + bi \) is called the complex number system (symbolically denoted as \( \mathbb{C} \)).

Definition: Complex Plane

The complex plane is a coordinate system in which the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part. In this sense, we define the horizontal axis to be the real axis and the vertical axis to be the complex axis. Complex numbers are the points on the plane, expressed as ordered pairs \((a,b)\), where \(a\) represents the coordinate for the horizontal axis and \(b\) represents the coordinate for the vertical axis.

Definition: Equality of Complex Numbers

Two complex numbers \(z_1=a+b i\) and \(z_2=c+d i\) are said to be equal if and only if\[a=c \text { and } c=d.\nonumber \]

Definition: Addition and Subtraction of Complex Numbers

Given two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), we define addition and subtraction as follows:

Addition:\[ z_1 + z_2 = ( a+bi )+( c+di )=( a+c )+( b+d )i. \nonumber \]Subtraction:\[z_1 - z_2 = ( a+bi )−( c+di )=( a−c )+( b−d )i. \nonumber \]

Definition: Multiplication of Complex Numbers

Given two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), we define multiplication as follows:\[ \begin{array}{rcl}
z_1 \cdot z_2 & = & (z_1)(z_2) \\[6pt]
& = & ( a+bi )( c+di ) \\[6pt]
& = & ac + ad i + bc i + bd i^2 \\[6pt]
& = & ac +(ad + bc)i + bc (-1) \\[6pt]
& = & (ac - bd) + (ad + bc)i \\[6pt]
\end{array} \nonumber \]

Definition: Complex Conjugate

The complex conjugate of \( z = a + bi \) is defined to be \( \bar{z} = a - bi \). Likewise, the complex conjugate of \( z = a - bi \) is defined to be \( \bar{z} = a + bi \).

Theorem: Products of Complex Conjugates

Let \( z = a + bi \), so that its complex conjugate is \( \bar{z} = a - bi \). Then\[ z \cdot \bar{z} = a^2 + b^2. \nonumber \]

Proof

Let \( z = a + bi \). Then\[ \begin{array}{rcl} z \cdot \bar{z}& = & (a + bi)(a - bi) \\[6pt]  & = & a^2 - abi + abi - b^2 i^2 \\[6pt]  & = & a^2 - \cancel{abi} + \cancel{abi} - b^2 (-1) \\[6pt]  & = & a^2 + b^2 \\[6pt] \end{array} \nonumber \]

Definition: Division of Complex Numbers

Given two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), we define division as follows:\[ \begin{array}{rcl}
z_1 \div z_2 & = & \dfrac{z_1}{z_2} \\[6pt]
& = & \dfrac{a+bi}{c+di} \\[6pt]
& = & \dfrac{a+bi}{c+di} \cdot \dfrac{c - di}{c - di} \\[6pt]
& = & \dfrac{(a+bi)(c - di)}{(c+di)(c - di)} \\[6pt]
& = & \dfrac{ac -ad i + bc i - bd i^2}{c^2+ d^2} \\[6pt]
& = & \dfrac{ac + (-ad + bc) i - bd (-1)}{c^2+ d^2} \\[6pt]
& = & \dfrac{(ac + bd) + (bc - ad) i}{c^2+ d^2} \\[6pt]
& = & \dfrac{ac + bd}{c^2 + d^2} + \dfrac{bc - ad}{c^2 + d^2} i \\[6pt]
& = & r + si \\[6pt]
\end{array} \nonumber \]where \( r \) and \( s \) are real numbers.

Theorem: Powers of the Imaginary Unit

Let \( p \in \mathbb{W}\) (the Whole Number system) and \( r \) be the remainder when \( p \) is divided by 4. Then\[ i^p = i^r. \nonumber \]

Proof

Let \( p \in \mathbb{W} \) and \( r \) be the remainder when \( p \) is divided by 4. Then \( r \) is either 0, 1, 2, or 3. The Divison Algorithm tells us that\[ \dfrac{p}{4} = q + \dfrac{r}{4}, \nonumber \]where \( q \) is the quotient and \( r \) is the remainder. Multiplying both sides by 4, we get\[ p = 4q + r. \nonumber \]Hence,\[ \begin{array}{rclcl}
i^p & = & i^{4q + r} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & i^{4q} \cdot i^r & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \left(i^4\right)^q \cdot i^r & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \left(1\right)^q \cdot i^r & \quad & \left( i^4 = 1 \right) \\[6pt]
& = & 1 \cdot i^r &  & \\[6pt]
& = & i^r &  & \\[6pt]
\end{array} \nonumber \]Therefore, all whole-numbered powers of \( i \) can be simplified to\[ \begin{array}{rcl}
i^0 & = & 1 \\[6pt]
i^1 & = i \\[6pt]
i^2 & = & -1 \\[6pt]
i^3 & = & i^2 \cdot i = (-1) \cdot i = -i \\[6pt]
\end{array} \nonumber \]

Definition: Modulus (of a complex number)

The modulus of a complex number \(z=a+b i\) is given by\[|z|=\sqrt{a^2+b^2}.\nonumber \]