3.1 F15 O’Brien
CA 6th ed HLR
3.1: Complex Numbers
I.
The Imaginary Unit, i
i
II.
2
1 and i  1
The Principal Square Root of a Negative Number
 a  a i or
 a  i a Note: The imaginary unit, i, is never written under a radical.
Whenever you see a square root of a negative number, “pull the i out” before you do anything else.
Examples 1 and 2
III.
Complex Numbers
A complex number is a number written in the form a + bi, where a is the real part and bi is the
imaginary part. b is the real-number coefficient of the imaginary unit, i.
Every real number can be written in complex form as a + 0i, thus every real number is a complex
number. However, not every complex number is a real number. If a = 0 and b ≠ 0, the resulting
number 0 + bi can also be written as bi. In this form it is called a pure imaginary number.
IV.
Operations on Complex Numbers
A.
Addition and Subtraction of Complex Numbers
Add or subtract like parts - real to real, imaginary to imaginary.
Example 3
B.
Multiplication of Complex Numbers
F.O.I.L.
Remember: i2 = –1, so b·i2 = –b and –c·i2 = c
Example 4
The conjugate of a + bi is a – bi and the conjugate of –c – di is –c + di.
The product of a complex number and its conjugate is a real number.
Example 5
C.
Simplifying Powers of i
1.
Powers of i
i1 i
2.
i 2  1
i 3  i
i4 1
To reduce a positive power of i, i

“i clock”
n
Divide the exponent (n) by 4. The remainder will always be 0, 1, 2, or 3.
n
0
i n i1 i
i n i 2  -1
i n i 3 -i
Therefore, the possible answers are: i i 1
Hint: Never leave a power of i in an answer - always simplify to i, –1, –i or 1.
Example 6
1
3.1 F15 O’Brien
CA 6th ed HLR
3.
To reduce a negative power of i, i
n
First rewrite so the exponent is positive.
Then divide the exponent (n) by 4.
Example 7
D.
Division of Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator.
Example 8
2
CA 6th ed HLR
3.1 F15 O’Brien
3
CA 6th ed HLR
3.1 F15 O’Brien
4