Chapter 1 – Section 3 – Complex Numbers
1
Section 1.3 – Complex Numbers
The set of real numbers does not include all the numbers needed in algebra. For example, there is no real number solution
of the equation
x
2 =−1
because no real number, when squared, gives -1. To extend the real number system to include solutions of equations of
this type, the number i is defined as:
Complex numbers are formed by adding real numbers and multiples of i.
Two complex numbers 𝑎 + 𝑏𝑖 and 𝑐 + 𝑑𝑖 are equal provided that their real parts are equal and their imaginary parts are
also equal — that is, they are equal if and only if 𝑎 = 𝑐 and 𝑏 = 𝑑.
The form 𝑎 + 𝑏𝑖 (or 𝑎 + 𝑖𝑏) is standard form. (The form 𝑎 + 𝑖𝑏 is used to write expressions such as 𝑖√5, because
√5𝑖 could be mistaken for √5𝑖 .)
Chapter 1 – Section 3 – Complex Numbers
2
The relationships among the subsets of the complex numbers are shown in the figure below:
Using the following definition of the imaginary unit, we can define the square root of the expression √−𝑎
Example 1.3.1: Write each number as the product of a real number and i
a.
−16
b.
−70
c.
−48
Chapter 1 – Section 3 – Complex Numbers
Example 1.3.2: Find each product or quotient. Simplify the answer:
a.
−7 −7
b.
−6 −10
c.
d.
−20
−2
−48
24
3
Chapter 1 – Section 3 – Complex Numbers
4
Example 1.3.3: Write the expression in standard form (a+bi):
−8 + −128
4
With the definitions 𝑖 2 = −1 and √−𝑎 = 𝑖 √𝑎 1a for all a > 0, all properties of real numbers are extended to complex
numbers. As a result, complex numbers are added, subtracted, multiplied, and divided using real number properties and
the following definitions.
Example 1.3.4: Find each sum or difference. Write answers in standard form.
a.
( 3 − 4i ) + ( −2 + 6i )
b.
( −4 + 3i ) − ( 6 − 7i )
Chapter 1 – Section 3 – Complex Numbers
5
The product of two complex numbers is found by multiplying as though the numbers were binomials and using the fact
that 𝑖 2 = −1, as follows.
Example 1.3.5: Find each product, write your answer in standard form:
a.
( 2 − 3i )( 3 + 4i )
b.
( 4 + 3i )
c.
( 6 + 5i )( 6 − 5i )
2
Chapter 1 – Section 3 – Complex Numbers
Example 1.3.6: Find each quotient. Write each answer in standard form:
a.
3 + 2i
5−i
b.
3
i
6
Chapter 1 – Section 3 – Complex Numbers
Example 1.3.7: Simplify each power of i
a.
i15
b.
i −3
7
0
