Download Lecture Notes for Section 1.4 (Complex Numbers)

College Algebra Lecture Notes
Section 1.4
Page 1 of 4
Section 1.4: Complex Numbers
Big Idea: .Calling the square root of negative one “i” allows us to state the solution of many
algebra equations that would otherwise be unsolvable with real numbers.
Big Skill: You should be able to perform arithmetic (add, subtract, multiply, and divide) with
complex numbers.
A. Identifying and Simplifying Imaginary and Complex Numbers
Imaginary Numbers and the Imaginary Unit
 Imaginary Numbers are those of the form  k , where k is a positive real number.
 The imaginary unit i represents the number whose square root is -1:
i2 = -1 and i  1
Rewriting imaginary numbers

For any positive real number k,
Practice:
1. Evaluate
k  1  k  i k .
25 .
2. Evaluate
2 .
3. Evaluate
48 .
4. Evaluate 12 36 .
Complex Numbers
 Complex numbers are numbers that can be written in the form a + bi, where a and b are
real numbers and i  1 is the imaginary unit.
 The real number a is called the real part of the complex number.
 The real number b is called the imaginary part of the complex number.
 The standard form of a complex number is a + bi.
Practice:
5. Write in standard form:
18  2 25
.
10
College Algebra Lecture Notes
Section 1.4
Page 2 of 4
The Subset Hierarchy of Numbers:
B. Add and Subtract Complex Numbers
 Adding, subtracting, multiplying, or dividing complex numbers results in an answer that
is also a complex number.
 To add complex numbers, add “like terms”:
 a  bi    c  di    a  c   b  d  i

To subtract complex numbers, subtract “like terms”:
 a  bi    c  di    a  c   b  d  i
Practice:
6.  2  3i    6  7i  
7.
5 
 
8.
 4  2i    2  7i  

36  2  49 
College Algebra Lecture Notes
Section 1.4
Page 3 of 4
C. Multiply Complex Numbers and Find powers of i


To multiply complex numbers, use the distributive property, then combine “like terms”
(like using FOIL):
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 
 25 4 
  5i  2i 
 100
 10i 2
 10
 10

Product of Complex Conjuagtes
 For a complex number a + bi and its conjugate a – bi, theor product (a + bi)( a – bi), is
the real number a2 + b2.
 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.
Practice:
100  81 
9.
10. 2i  5  3i  
11.  5  2i  1  3i  
 2
2  2
2 

i 

i  
12. 
 2
2
2
2



College Algebra Lecture Notes
Section 1.4
Page 4 of 4
13. i 27 
14. i 38 
D. Divide Complex Numbers
 To divide complex numbers, multiply the numerator and denominator by the conjugate
of the denominator, then multiply and simplify. Note: the denominator will always
multiply out to a difference of squares, which will be a real number.
Example:
4  3i 4  3i 1  2i


1  2i 1  2i 1  2i
 4  3i 1  2i 

1  2i 1  2i 

4  8i  3i  6i 2
12   2i 
4  11i  6
1  4i 2
2  11i

1 4
2 11i


5
5

Practice:
6  5i

15.
7i
16.
2i

4  3i
2