Complex Numbers in Precalculus: Operations and Applications

Chapter 2: Polynomial and Rational Functions

Section 2.1: Complex Numbers

This section introduces the concept of complex numbers, their algebraic properties, and operations. Complex numbers are essential in solving equations that do not have real solutions and are foundational for further study in mathematics.

Objectives

• Add and subtract complex numbers

• Multiply complex numbers

• Divide complex numbers

• Perform operations with square roots of negative numbers

• Solve quadratic equations with complex imaginary solutions

Complex Numbers and Imaginary Numbers

Definition and Structure

Complex numbers extend the real number system by introducing the imaginary unit i, defined as:

$i = \sqrt{-1}$, where $i^2 = -1$

• The set of all complex numbers is written as $a + bi$, where a and b are real numbers, and i is the imaginary unit.

• Real part: The number a in $a + bi$.

• Imaginary part: The number b in $a + bi$.

• If $b \neq 0$, the complex number is called an imaginary number.

• If the complex number is in the form $bi$, it is called a pure imaginary number.

Operations on Complex Numbers

General Principles

The form $a + bi$ is similar to a binomial. Operations such as addition, subtraction, and multiplication follow the same rules as binomial algebra.

Adding and Subtracting Complex Numbers

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$

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

Example:

• $(5 - 2i) + (3 + 3i) = (5 + 3) + (-2 + 3)i = 8 + i$

• $(2 + 6i) - (12 - i) = (2 - 12) + (6 - (-1))i = -10 + 7i$

Multiplying Complex Numbers

• Use distributive property (FOIL method) as with binomials.

• Remember that $i^2 = -1$.

Example 1:

• $(2 - 9i)$: $2 \times -9i = 14i - 63i^2 = 14i - 63(-1) = 14i + 63 = 63 + 14i$

Example 2:

• $(5 + 4i)(6 - 7i) = 5 \times 6 + 5 \times (-7i) + 4i \times 6 + 4i \times (-7i)$

• $= 30 - 35i + 24i - 28i^2$

• $= 30 - 11i - 28(-1)$

• $= 30 - 11i + 28 = 58 - 11i$

Conjugate of a Complex Number

Definition and Properties

• The complex conjugate of $a + bi$ is $a - bi$.

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

$(a + bi)(a - bi) = a^2 + b^2$

$(a - bi)(a + bi) = a^2 + b^2$

Complex Number Division

Method

The goal of dividing complex numbers is to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator.

• Given $\frac{a + bi}{c + di}$, multiply numerator and denominator by $c - di$.

• This results in a denominator that is a real number.

Example:

• Divide $\frac{5 + 4i}{4 + i}$:

• Multiply numerator and denominator by $4 - i$:

• $\frac{(5 + 4i)(4 - i)}{(4 + i)(4 - i)}$

• Numerator: $5 \times 4 + 5 \times (-i) + 4i \times 4 + 4i \times (-i) = 20 - 5i + 16i - 4i^2 = 20 + 11i - 4(-1) = 20 + 11i + 4 = 24 + 11i$

• Denominator: $4^2 - (i)^2 = 16 - (-1) = 17$

• Result: $\frac{24 + 11i}{17}$

Principal Square Root of a Negative Number

Definition

• For any positive real number $b$, the principal square root of $-b$ is defined as:

$\sqrt{-b} = i\sqrt{b}$

Quadratic Equations with Complex Solutions

Solving Using the Quadratic Formula

• Quadratic equations may have complex solutions when the discriminant ($b^2 - 4ac$) is negative.

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example:

• Solve $x^2 - 2x + 2 = 0$

• $a = 1$, $b = -2$, $c = 2$

• Discriminant: $(-2)^2 - 4(1)(2) = 4 - 8 = -4$

• Solutions: $x = \frac{2 \pm \sqrt{-4}}{2} = \frac{2 \pm 2i}{2} = 1 \pm i$

Summary Table: Complex Number Operations

Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(5 - 2i) + (3 + 3i) = 8 + i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(2 + 6i) - (12 - i) = -10 + 7i$
Multiplication	$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$	$(2 - 9i) = 63 + 14i$
Conjugate	Conjugate of $a + bi$ is $a - bi$	Conjugate of $3 + 4i$ is $3 - 4i$
Division	$\frac{a + bi}{c + di} \times \frac{c - di}{c - di}$	$\frac{5 + 4i}{4 + i} = \frac{24 + 11i}{17}$
Square Root	$\sqrt{-b} = i\sqrt{b}$	$\sqrt{-16} = 4i$

Additional info: These notes are based on textbook slides and include expanded academic context for clarity and completeness.