Home Systems of Equations Adding and Subtracting Rational Expressions with Different Denominators Graphing Linear Equations Raising an Exponential Expression to a Power Horizontal Line Test Quadratic Equations Mixed Numbers and Improper Fractions Solving Quadratic Equations by Completing the Square Solving Exponential Equations Adding and Subtracting Polynomials Factorizing simple expressions Identifying Prime and Composite Numbers Solving Linear Systems of Equations by Graphing Complex Conjugates Graphing Compound Inequalities Simplified Form of a Square Root Solving Quadratic Equations Using the Square Root Property Multiplication Property of Radicals Determining if a Function has an Inverse Scientific Notation Degree of a Polynomial Factoring Polynomials by Grouping Solving Linear Systems of Equations Exponential Functions Factoring Trinomials by Grouping The Slope of a Line Simplifying Complex Fractions That Contain Addition or Subtraction Solving Absolute Value Equations Solving Right Triangles Solving Rational Inequalities with a Sign Graph Domain and Range of a Function Multiplying Polynomials Slope of a Line Inequalities Multiplying Rational Expressions Percent of Change Equations Involving Fractions or Decimals Simplifying Expressions Containing only Monomials Solving Inequalities Quadratic Equations with Imaginary Solutions Reducing Fractions to Lowest Terms Prime and Composite Numbers Dividing with Exponents Dividing Rational Expressions Equivalent Fractions Graphing Quadratic Functions Polynomials Linear Equations and Inequalities in One Variable Notes on the Difference of 2 Squares Solving Absolute Value Inequalities Solving Quadratic Equations Factoring Polynomials Completely Using Slopes to Graph Lines Fractions, Decimals and Percents Solving Systems of Equations by Substitution Quotient Rule for Radicals Prime Polynomials Solving Nonlinear Equations by Substitution Simplifying Radical Expressions Containing One Term Factoring a Sum or Difference of Two Cubes Finding the Least Common Denominator of Rational Expressions Conjugates Multiplying Rational Expressions Expansion of a Product of Binomials Solving Equations Exponential Growth Factoring by Grouping Solving One-Step Equations Using Models Solving Quadratic Equations by Factoring Adding and Subtracting Polynomials Rationalizing the Denominator Rounding Off The Distributive Property What is a Quadratic Equation Laws of Exponents and Multiplying Monomials The Slope of a Line Factoring Trinomials by Grouping Multiplying and Dividing Rational Expressions Solving Linear Inequalities Multiplication Property of Exponents Multiplying and Dividing Fractions 3 Formulas Dividing Monomials Multiplying Polynomials Adding and Subtracting Functions Dividing Polynomials Absolute Value and Distance Multiplication and Division with Mixed Numbers Factoring a Polynomial by Finding the GCF Roots Adding and Subtracting Polynomials The Rectangular Coordinate System Polar Form of a Complex Number Exponents and Order of Operations Graphing Horizontal and Vertical Lines Invariants Under Rotation The Addition Method Solving Linear Inequalities in One Variable The Pythagorean Theorem

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Complex Conjugates

Two complex numbers of the form a + bi and a - bi are called complex conjugates. Notice that complex conjugates have the same real part. Their imaginary parts are the same, except they have opposite signs.

Here are several pairs of complex conjugates:

 7 + 2i-3 + 6i 12 - 8i andand and 7 - 2i-3 - 6i 12 + 8i

Note:

Here are several pairs of complex numbers that are NOT complex conjugates:

 4 - 7i3 + 5i 7 + 2i andand and 7 - 4i-3 - 5i -7 + 2i

Example 1

Find: (6 + 4i)(6 - 4i)

 Solution Multiply using FOIL. Multiply the factors in each term. Replace i2 with -1. Combine like terms. Note that the i terms cancel. So, (6 + 4i)(6 - 4i) = 52. (6 + 4i)(6 - 4i) = 6 Â· 6 - 6 Â· 4i + 4i Â· 6 - 4i Â· 4i = 36 - 24i + 24i - 16i2 = 36 - 24i + 24i - 16(-1) = 52
Notice that (6 + 4i) and (6 - 4i) are complex conjugates.

Their product is 52, a real number.

The next example demonstrates that the product of two complex conjugates is always a real number.

Example 2

Find: (a + bi)(a - bi)

 Solution Multiply using the FOIL method. Multiply the factors in each term. Replace i2 with -1.  Combine like terms. Note that the i terms add to zero.Thus, (a + bi)(a - bi) = a2 + b2 (a + bi)(a - bi) = a Â· a - a Â· bi + bi Â· a - bi Â· bi = a2 - abi + abi - b2i2 = a2 - abi + abi - b2(-1) = a2 - abi + abi + b2 = a2 + b2

 All Right Reserved. Copyright 2005-2018