Section 2.2

The Complex Numbers

Imaginary Numbers

• Invented in order to find the square root of a negative number.

• Imaginary numbers are numbers that can be written using i.

Imaginary NumbersBy definition the following new number was created:

Imaginary Unit

Which means. . .

1i

1i2

Square Root of Negative Number

If a 0, then

- a i a

Caution: This rule should be used before applying any other rules for radicals.

Complex Number System

• Mathematicians invented the complex number system in order to make it possible to solve all quadratic equations.

• What is a complex number?A real number plus an imaginary number

Complex Number- is an expression of the form a + bi where a and b are real numbers and

a + bi

2i 1

Real Part Imaginary Part

2 parts of the definition

1. 2. By definition . . .

2i 1

Standard Form

Adding/Subtracting Complex Numbers

** Procedure is the same as adding and subtracting like terms **

ex. 3x + 5x ex. 3 + 8w – 12 – 4w

8x -9 + 4w

To Add/Subtract Complex Numbers1. Add/Subtract the real parts and

add/subtract imaginary parts

Multiplying and Dividing Square Roots

• Rule when a and b are positive . . .

a b a b a a

orbb

Multiplying and Dividing Square Roots

• This rule DOES NOT apply for negative numbers.

NO. . . NO. . . NO. . . NO!

- a - b - a b

a aor

bb

Multiplication of Complex Numbers

1. Use FOIL (like multiplying binomials)

2. Remember . . .

2i 1

Dividing Complex Numbers

Complex Conjugate- Conjugate of

the divisor is used to find the quotient of two complex

numbers in standard form.

Conjugate of a+bi is a-biConjugate of i is -i