Algebra Tutorials!
   
 
 
Tuesday 21st of November
 
   
Home
Solving Linear Equations
Systems of Linear Equations
Solving Linear Equations Graphically
Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Percents
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Algebra
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
fractions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Quadratics
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Decimals
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
   
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Polar Form of a Complex Number

Example 1

Writing a Complex Number in Polar Form

Write the complex number in polar form.

Solution

The absolute value of z is

and the angle θ is given by

Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. Thus, the polar form is

(See the figure below)

The polar form adapts nicely to multiplication and division of complex numbers. Suppose you are given two complex numbers

z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 )

The product of z1 and z2 is

z1 z2 = r1 r2 cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 )

= r1 r2 [(cos θ1cos θ2 - i sin θ1sin θ2) + i(sin θ1cos θ2 + cos θ1sin θ2)]

Using the sum and difference formulas for cosine and sine, you can rewrite this equation as

z1 z2 = r1 r2 [(cos1+ θ2) + i sin (θ1+ θ2)]

This establishes the first part of the following rule. Try to establish the second part on your own.

 

Product and Quotient of Two Complex Numbers

Let z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 ) be complex numbers.

z1 z2 = r1 r2 [(cos1+ θ2) + i sin (θ1+ θ2)] Product
Quotient

Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments.

 

Copyrights © 2005-2017