Algebra Tutorials!  
Tuesday 26th of September
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Dividing Complex Numbers

To divide a complex number by a real number, divide each term by the real number, just as we would divide a binomial by a number. For example,

To understand division by a complex number, we first look at imaginary numbers that have a real product. The product of the two imaginary numbers (3 + i)(3 - i) is a real number, namely (3 + i)(3 - i) = 10

We say that 3 + i and 3 - i are complex conjugates of each other.


Complex Conjugates

The complex numbers a + bi and a - bi are called complex conjugates of one another. Their product is the real number a2 + b2.


Example 1

Products of conjugates

Find the product of the given complex number and its conjugate.

a) 2 + 3i

b) 5 - 4i


a) The conjugate of 2 + 3i is 2 - 3i.

(2 + 3i)(2 - 3i) = 4 - 9i2
  = 4 + 9
  = 13

b) The conjugate of 5 - 4i is 5 + 4i.

(5 - 4i)(5 + 4i) = 25 + 16
  = 41

We use the idea of complex conjugates to divide complex numbers. The process is similar to rationalizing the denominator. Multiply the numerator and denominator of the quotient by the complex conjugate of the denominator.


Example 2

Dividing complex numbers

Find each quotient. Write the answer in the form a + bi.


a) Multiply the numerator and denominator by 3 + 4i, the conjugate of 3 - 4i:

b) Multiply the numerator and denominator by 2 - i, the conjugate of 2 + i:

c) Multiply the numerator and denominator by -i, the conjugate of i:

The symbolic definition of division of complex numbers follows.


Division of Complex Numbers

We divide the complex number a + bi by the complex number c + di as follows:


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