4.6 Perform Operations with Complex Numbers

 

Imaginary unit i

The imaginary unit i is defined as

 

Complex number

A number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part.

 

Imaginary number

A complex number a + bi where b ¹ 0

 

Complex conjugates

Two complex numbers of the form a + bi and a - bi

 

Complex plane

A coordinate plane where each point (a, b) represents a complex number a + bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

 

Absolute value of a complex number

The absolute value of a complex number z = a + bi, denoted êz ê, is a nonnegative real number defined as

êz ê

 

THE SQUARE ROOT OF A NEGATIVE NUMBER

Property

Example

1.   If r is a positive real number, then

2.     By Property (1), it follows that

 

 

 

Example 1

Solve a quadratic equation

 

2x2 + 15 = -35

 

Original equation

_2x2 = - 50_

Subtract _15_ from each side.

_x2 = - 25_

Divide each side by _2_.

_x = ±_______

Take square roots of each side.

_x = ± i_____

Write in terms of i.

__x = ± 5i__

Simplify radical.

The solutions are __5i__ and __-5i__.

 

SUMS AND DIFFERENCES OF COMPLEX NUMBERS

To add (or subtract) two complex numbers, add (or subtract) their __real__ parts and their __imaginary__ parts __separately__.

Sum of complex numbers:

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

Difference of complex numbers:

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

 

Example 2

Add and subtract complex numbers

 

Write as a complex number in standard form.

a.     (6 + 3i) - (4 - i)

= ( 6 - 4 ) + ( 3 + 1 )i                       Complex subtraction

= _2 + 4i_                                         Standard form

b.   (2 + 5i) + (7 - 2i)

= (_2 + 7_) + (_5 - 2_)i                    Complex addition

= __9 + 3i__                                      Standard form

 

Example 3

Multiply complex numbers

Write the expression (2 + i)(-5 + 2i) as a complex number in standard form.

(2 + i)( -5 + 2i)

= -10 + 4i - 5i+ 2i2                    Multiply using FOIL.

= _-10 - i+ 2(-1)_                    Simplify and use i2 = -1.

= _-10 – i - 2_                          Simplify.

= _-12 -i_                                Write in standard form.

 

Example 4

Divide complex numbers

Write the quotient             in standard form.

                                              Multiply numerator and denominator by _2 - i_, the complex conjugate of 2 + i.

 


=                                                 Multiply using FOIL.

=                                                 Simplify.

=                                                 Write in standard form.

 

Write the expression as a complex number in standard form.

 

1.      (12 - 2i) -(16 + 3i)

-4 - 5i

2.      -4i(9 + 5i)

20-36i

3.       

2 - 2i

4.      (4 + 4i) + (-6 + 3i)

-2 + 7i

 

Example 5

Plot complex numbers

 

Plot the complex numbers in the same complex plane.

a. 4 + 3i           b. -5 - 4i

 

Solution

a.      To plot 4 + 3i, start at the origin, move _4 units to the right_, and then move _3 units up_.

b.      To plot -5 - 4i, start at the origin, move _5 units to the left_, and then move _4 units down__.

 

 


ABSOLUTE VALUE OF A COMPLEX NUMBER

The absolute value of a complex number z = a + bi, denoted | z |, is a _nonnegative_ real number defined as | z | =         This is the distance of z from the _origin__ in the complex plane.

 

Example 6

Find absolute values of complex numbers

Find the absolute value of (a) 6 - 8i and (b) -6i.

a.    ï6 - 8iï =                      =            = __10__

b.    ï-6iï =                     =           = __6__