Download College Algebra Lecture Notes, Section 1.4

Name: ________________________ Date: ___________________ Period: ____________
Algebra 2: Section 4.8 Complex Numbers Notes
In the past we said that you could not take the square root of a negative. Well we can!!
The imaginary unit i represents the number whose square root is -1
i  1 and i2 = -1
Rewriting imaginary numbers

For any positive real number k,
k  1  k  i k .
Practice:
1) Evaluate
25
2) Evaluate
3) Evaluate
48
4) Evaluate 12 36
2
Complex Numbers
 Complex numbers are numbers that can be written in the form a + bi, where a and b are real
numbers
 The real number a is the ________ part of the complex number.
 The real number bi is the ________ part of the complex number.
 The standard form of a complex number is a + bi.
Practice:
5) Write in standard form:
18  2  36
3
6) Write in standard form:
18  2 25
10
Add and Subtract Complex Numbers
 To add and subtract complex numbers, add “like terms”:
 a  bi    c  di    a  c   b  d  i
Practice:
7)
 2  3i    6  7i  
8)
5 
 
9)
 4  2i    2  7i  

36  2  49 
Multiply Complex Numbers


To multiply complex numbers, use the distributive property, then combine “like terms”
One thing to keep in mind: the product property of radicals does not apply when the radicand is
a complex number, so evaluate the square root of the negative number first, then multiply the
radicals.
Correct:
25 4 

25i
 4i 
Incorrect:
25 4 
  5i  2i 

 10i 2
 10
Practice:
100  81 
10)
11)
2i  5  3i  
12)
5  2i  1  3i  
 25 4 
 100

 10
 2
2  2
2 

i 

i 

2 
2 
 2
 2
13)
Divide Complex Numbers

The complex number a + bi has a _____________________ a – bi.

When you multiply a complex number by its conjugate you get a _________ number.
Example:

3  4i 3  4i  
To divide complex numbers, multiply the numerator and denominator by the conjugate of the
denominator.
Practice:
14)
2i

4  3i
15)
6  5i

7i
16)
9  12i

3i
17)
9  12i

3i
Using this new knowledge we can provide more specific solutions:
18) 5m 2  14  4
19)
2 x 2  18  72
To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4.
This works because the powers of i repeat themselves every four factors of i, as can be seen from
making a list of powers of i.
20)
i 27 
21)
i 38 
22)
i 150 
23)
i 111 