Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Adding, Subtracting and Multiplying Complex Numbers

Addition and subtraction of complex numbers are performed as if the complex numbers were algebraic expressions with i being a variable.


Example 1

Addition and subtraction of complex numbers

Find the sums and differences.

a) (2 + 3i) + (6 + i)

b) (-2 + 3i) + (-2 - 5i )

c) (3 + 5i) - (1 + 2i)

d) (-2 - 3i) - (1 - i )


a) (2 + 3i) + (6 + i) = 8 + 4i

b) (-2 + 3i) + (-2 - 5i ) = -4 - 2i

c) (3 + 5i) - (1 + 2i) = 2 + 3i

d) (-2 - 3i) - (1 - i ) = -3 -2i

We can give a symbolic definition of addition and subtraction as follows.


Addition and Subtraction of Complex Numbers

The sum and difference of a + bi and c + di are defined as follows:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i


Complex numbers are multiplied as if they were algebraic expressions. Whenever i2 appears, we replace it by -1.


Example 2

Products of complex numbers

Find each product.

a) 2i(1 + i)

b) (2 + 3i)(4 + 5i)

c) (3 + i)(3 - i)


a) 2i(1 + i) = 2i + 2i2 Distributive property
  = 2i + 2(-1) i2 = -1
  = -2 + 2i  

b) Use the FOIL method to find the product:

(2 + 3i)(4 + 5i) = 8 +10i + 12i + 15i2  
  = 8 + 22i + 15(-1) Replace i2 by -1.
  = 8 + 22i - 15  
  = -7 + 22i  

c) This product is the product of a sum and a difference.

(3 + i)(3 - i) = 9 - 3i + 3i - i2  
  = 9 - (-1) i2 = -1
  = 10  

We can find powers of i using the fact that i2 = -1. For example, i3 = i2 · i = -1 · i = -i.

The value of i4 is found from the value of i3:

i4 = i3 · i = -i · i = -i2 = 1.

In the next example we find more powers of imaginary numbers.


Example 3

Powers of imaginary numbers

Write each expression in the form a + bi.

a) (2i)2

b) (-2i)2

c) i6


a) (2i)2 = 22 · i2 = 4(-1) = -4

b) (2i)2 (2)2  i2 4i2 4(1) 4

c) i6 = i2 · i4 = -1 · 1 = -1

For completeness we give the following symbolic definition of multiplication of complex numbers. However, it is simpler to find products as we did in Examples 2 and 3 than to use this definition.


Multiplication of Complex Numbers

The complex numbers a + bi and c + di are multiplied as follows:

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

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