Algebra Tutorials!    
         
  Tuesday 19th of March      
 
   
Home
Scientific Notation
Notation and Symbols
Linear Equations and Inequalities in One Variable
Graphing Equations in Three Variables
What the Standard Form of a Quadratic can tell you about the graph
Simplifying Radical Expressions Containing One Term
Adding and Subtracting Fractions
Multiplying Radical Expressions
Adding and Subtracting Fractions
Multiplying and Dividing With Square Roots
Graphing Linear Inequalities
Absolute Value Function
Real Numbers and the Real Line
Monomial Factors
Raising an Exponential Expression to a Power
Rational Exponents
Multiplying Two Fractions Whose Numerators Are Both 1
Multiplying Rational Expressions
Building Up the Denominator
Adding and Subtracting Decimals
Solving Quadratic Equations
Scientific Notation
Like Radical Terms
Graphing Parabolas
Subtracting Reverses
Solving Linear Equations
Dividing Rational Expressions
Complex Numbers
Solving Linear Inequalities
Working with Fractions
Graphing Linear Equations
Simplifying Expressions That Contain Negative Exponents
Rationalizing the Denominator
Decimals
Estimating Sums and Differences of Mixed Numbers
Algebraic Fractions
Simplifying Rational Expressions
Linear Equations
Dividing Complex Numbers
Simplifying Square Roots That Contain Variables
Simplifying Radicals Involving Variables
Compound Inequalities
Factoring Special Quadratic Polynomials
Simplifying Complex Fractions
Rules for Exponents
Finding Logarithms
Multiplying Polynomials
Using Coordinates to Find Slope
Variables and Expressions
Dividing Radicals
Using Proportions and Cross
Solving Equations with Radicals and Exponents
Natural Logs
The Addition Method
Equations
   
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.


Complex Numbers

A complex number is an expression of the form x + iy where x and y are real numbers.

Either x or y may be equal to zero: real numbers and imaginary numbers are special cases of complex numbers.

Given a complex number z = x + iy the real number x is known as the real part of z and the real number y is known as the imaginary part of z.

The notation Re is used for real part and Im is used for imaginary part.

Note that the imaginary part of z is a real number y (NOT iy).

 

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, we add or subtract their real and imaginary parts separately:

 

 

Examples:

1) (2 + 3i) + (4 + i) = 6 + 4i

2) (3 - 5i) - 7i = 3 - 12i

3) (5 - 4i) + (3 + 2i) - (8 + i) = -3i

4) 12 - 4i + (3 + 4i) = 15

 

Multiplying complex numbers

To multiply complex numbers, we use the usual rules and the identity i2 = -1:

 

Examples:

1) (2 + 3i)(4 + 5i) = (2 × 4) + (3i × 4) + (2 × 5i) + (3i × 5i)
  = 8 12i + 10i - 15
  = -7 + 22i
2) (2 + 3i)3 = (2 + 3i)(2 + 3i)(2 + 3i)
  = (-5 + 12i)(2 + 3i)
  = -46 + 9i

 

The complex conjugate

Given a complex number z = x + iy, the complex conjugate is given by z* = x - iy (pronounced “z star”):

To find the complex conjugate of any expression, replace i by –i.

 

Examples:

1) (3 + 4i)* = 3 - 4i

2) (5 - 2i)* = 5 + 2i

3) (6i)* = -6i

4) [(2 + 4i)*]* = [2 - 4i]* = 2 + 4i

 

The last expression demonstrates that

A complex number and its complex conjugate have the property that:

 

Examples:

1) (3 + 4i) + (3 + 4i)* = 3 + 4i + 3 - 4i = 6

2) (3 + 4i) - (3 + 4i)* = 3 + 4i - 3 + 4i = 8i

3) 5i + (5i)* = 5i - 5i = 0

4) 5i - (5i)* = 10i

 

Complex conjugate of a product

 

The modulus of a complex number

A complex number z = x + iy multiplied by its complex conjugate is a real number:

The modulus of a complex number z = x + iy is written | z | and is equal to the positive square root of the sum of the squares of its real and imaginary parts:

 

Examples:

The modulus is a real number

 

The quotient of complex numbers

In order to simplify an expression with a complex number in the denominator, we multiply both the numerator and denominator by the complex conjugate:

In this way, we get a real number in the denominator (the modulus squared of a + bi).

 

Examples:

In general:

Copyrights © 2005-2024