2.4 Complex Numbers

Students will use the imaginary unit i to write complex numbers.

Students will add, subtract, and multiply complex numbers.

Students use complex conjugates to write the quotient of two complex numbers in standard form.

Students will plot complex numbers in the complex plane.

Complex Numbers: a + bi

The standard form of a complex number is

a + bi. a is the real portion of the complex number while b is the imaginary portion.

b is the imaginary portion because

which is an imaginary number (not a real number on the real number line). Any real or imaginary number can be written in this form.

i 1

Example 1: Adding and Subtracting Complex Numbers

a)

b)

c)

d)

( ) ( )3 2 3 i i

2 4 2i i ( )

3 2 3 5 ( ) ( )i i

( ) ( ) ( )3 2 4 7 i i i

Example 2: Multiplying Complex Numbers

a)

b)

c)

4 16

( )( )2 4 3 i i

( )( )3 2 3 2 i i

Example 2: Multiplying Complex Numbers

d)

e)

4 1 5i i( )

( )3 2 2 i

Complex Conjugates

The conjugate of a complex number a + bi is a complex number a – bi.

There are a couple reasons why these numbers are connected.

One reason being that when you multiply them together, you get a real number as in example 2(c).

Another reason these numbers are linked is because when performing the quadratic equation the symbol

requires that if a + bi is a solution to the quadratic, then so is a – bi.

xb b ac

a

2 4

2

Example 3 Multiplying Conjugates

Multiply by its complex conjugate.3 5 i

Example 4: Writing a quotient of complex numbers in standard formWrite the quotient in standard form. Standard form of a

complex number is a + bi.

2 3

4 2

i

i

Example 5: Plotting Complex Numbers

Plot each complex number in the complex plane.

a) b) c) d)2 3 i 1 2i 4 3iy

x–2

2

Homework

p.133-134 # 3-72x3