Transcript MTH 112 Section 2.2

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 Numbers
By definition the following new number
was created:
Imaginary Unit
i  1
Which means. . .
i  1
2
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
2
real numbers and i  1
2 parts of the definition
1.
a + bi
Real Part
2. By definition . . .
i  1
2
Imaginary Part
Standard Form
Adding/Subtracting Complex Numbers
** Procedure is the same as adding
and subtracting like terms **
ex.
3x + 5x
8x
ex. 3 + 8w – 12 – 4w
-9 + 4w
To Add/Subtract Complex Numbers
1. 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 or
a
a

b
b
Multiplying and Dividing Square Roots
• This rule DOES NOT apply for negative
numbers.
-a  -b  -a  b or
a
a

b
b
NO. . . NO. . . NO. . . NO!
Multiplication of Complex Numbers
1. Use FOIL (like multiplying
binomials)
2. Remember . . .
i  1
2
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-bi
Conjugate of i is -i