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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Operating with Complex Numbers

Some equations have no real solutions. For instance, the quadratic equation

x2 + 1 = 0 Equation with no real solution

has no real solution because there is no real number x that can be squared to produce -1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as

Imaginary unit

where i2 = -1. By adding real numbers to real multiples of this imaginary unit, we obtain the set of complex numbers. Each complex number can be written in the standard form, a + bi.


Definition of a Complex Number

For real numbers a and b, the number a + bi is a complex number. If b ≠ 0, a + bi is called an imaginary number, and bi is called a pure imaginary number.

To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately.


Addition and Subtraction of Complex Numbers

If a + bi and c + di are two complex numbers written in standard form, their sum and difference are defined as follows.

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

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


The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a + bi is

-(a + bi) = -a - bi Additive inverse

Thus, you have (a + bi) + (-a - bi) = 0 + 0i = 0.


Example 1

Adding and Subtracting Complex Numbers

a. (3 - i) + (2 + 3i) = 3 - i + 2 + 3i Remove parentheses.
  = 3 + 2 - i + 3i Group like terms.
  = (3 + 2) + (- 1 + 3)i  
  = 5 + 2i Standard form
b. 2i + (-4 - 2i) = 2i - 4 -2i Remove parentheses.
  = -4 + 2i - 2i Group like terms.
  = -4 Standard form
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