Complex Numbers in College Algebra: Definitions, Operations, and Applications

Section 1.3: Complex Numbers

Introduction

This section introduces complex numbers, a fundamental concept in College Algebra. Students will learn how to simplify powers of the imaginary unit i, perform arithmetic operations with complex numbers, and apply these skills to equations and applications involving radicals with negative radicands.

Simplifying Powers of i

The Imaginary Unit

The imaginary unit, denoted by i, is defined as:

• $i = \\sqrt{-1}$

• Equivalently, $i^2 = -1$

Powers of i

Powers of i repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = i^2 \\cdot i = -1 \\cdot i = -i$

• $i^4 = (i^2)^2 = (-1)^2 = 1$

• $i^5 = i^4 \\cdot i = 1 \\cdot i = i$

• $i^6 = i^4 \\cdot i^2 = 1 \\cdot -1 = -1$

• $i^7 = i^4 \\cdot i^3 = 1 \\cdot -i = -i$

• $i^8 = i^4 \\cdot i^4 = 1 \\cdot 1 = 1$

Example:

• Simplify $i^{10} \\cdot i^3$: $i^{10} \\cdot i^3 = (i^4)^{2} \\cdot i^2 \\cdot i^3 = 1 \\cdot -1 \\cdot i^3 = -i$

• Simplify $i^{-21}$: $i^{-21} = \\frac{1}{i^{21}} = \\frac{1}{(i^4)^5 \\cdot i} = \\frac{1}{1 \\cdot i} = \\frac{1}{i}$

Definition of a Complex Number

Standard Form

A complex number is any number that can be written in the form $a + bi$, where a and b are real numbers. The variable z is often used to denote a complex number.

• a is called the real part

• b is called the imaginary part

Adding and Subtracting Complex Numbers

Procedure

To add or subtract complex numbers, combine the real parts and combine the imaginary parts separately.

Example:

• $(7 - 5i) + (-2 + i) = (7 + -2) + (-5i + i) = 5 - 4i$

• $(7 - 5i) - (-2 + i) = (7 - (-2)) + (-5i - i) = 9 - 6i$

Multiplying Complex Numbers

Procedure

Multiply complex numbers as you would binomials, using the distributive property (FOIL method). Remember that $i^2 = -1$.

Example:

• $(4 - 3i)(7 + 5i) = 4 \\cdot 7 + 4 \\cdot 5i - 3i \\cdot 7 - 3i \\cdot 5i = 28 + 20i - 21i - 15i^2 = 28 - i + 15 = 43 - i$

• $(\\sqrt{3} - 5i)^2 = (\\sqrt{3} - 5i)(\\sqrt{3} - 5i) = 3 - 10i\\sqrt{3} + 25(-1) = -22 - 10i\\sqrt{3}$

Complex Conjugate

Definition

The complex conjugate of $z = a + bi$ is denoted as $\\overline{z} = a - bi$.

Example:

• If $z = -2 - 7i$, then $\\overline{z} = -2 + 7i$

Multiplying by the Conjugate

Multiplying a complex number by its conjugate yields a real number:

• $(a + bi)(a - bi) = a^2 + b^2$

Example:

• $(-2 - 7i)(-2 + 7i) = 4 + 49 = 53$

Theorem: Product of a Complex Number and Its Conjugate

The product of a complex number $z = a + bi$ and its conjugate $\\overline{z} = a - bi$ is the real number $a^2 + b^2$.

Finding the Quotient of Complex Numbers

Procedure

• Goal: Eliminate the imaginary part from the denominator.

• Express in standard form: $a + bi$

• Method: Multiply the numerator and denominator by the complex conjugate of the denominator.

Example:

• To write $\\frac{5 - 13i}{5 + 2i}$ in the form $a + bi$, multiply numerator and denominator by the conjugate of the denominator: $\\frac{5 - 13i}{5 + 2i} \\cdot \\frac{5 - 2i}{5 - 2i}$ (Full expansion omitted for brevity; see textbook for detailed steps.)

Simplifying Radicals with Negative Radicands

Procedure

When simplifying radicals with negative radicands, use the property $\\sqrt{-m} = i\\sqrt{m}$ for $m > 0$.

Example:

• $\\sqrt{-108} = i\\sqrt{108} = 6i\\sqrt{3}$

Summary Table: Powers of i

Power	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$
$i^5$	$i$
$i^6$	$-1$
$i^7$	$-i$
$i^8$	$1$

Key Concepts and Applications

• Complex numbers extend the real numbers and are essential for solving equations with no real solutions.

• Operations with complex numbers follow algebraic rules, with special attention to the property $i^2 = -1$.

• Complex conjugates are used to rationalize denominators and simplify quotients.

• Radicals with negative radicands are expressed in terms of i.

Additional info: The examples and procedures above are foundational for further study in algebra, including quadratic equations and polynomial functions.