Introduction
Recall that the imaginary unit i is equal to -1. A fraction
with i in the denominator does not have a rational
denominator, since -1 is not a rational number.
Similar to rationalizing a fraction with an irrational square
root in the denominator, fractions with i in the
denominator can also have the denominator
rationalized.
1
4.3.4: Dividing Complex Numbers
Key Concepts
• Any powers of i should be simplified before dividing
complex numbers.
• After simplifying any powers of i, rewrite the division of
two complex numbers in the form a + bi as a fraction.
• To divide two complex numbers of the form a + bi and
c + di, where a, b, c and d are real numbers, rewrite
the quotient as a fraction.
c + di
( c + di ) ¸ ( a + bi ) = a + bi
2
4.3.4: Dividing Complex Numbers
Key Concepts, continued
• Rationalize the denominator of a complex fraction by
using multiplication to remove the imaginary unit i
from the denominator.
• The product of a complex number and its conjugate is
a real number, which does not contain i.
• Multiply both the numerator and denominator of the
fraction by the complex number in the denominator.
• Simplify the rationalized fraction to find the result of
the division.
3
4.3.4: Dividing Complex Numbers
Key Concepts, continued
• In the following equation, let a, b, c, and d be real
numbers.
c + di ) ( a - bi ) ac + bd + adi - bci
(
=
=
a + bi ( a + bi ) ( a - bi )
a +b
c + di
2
2
4
4.3.4: Dividing Complex Numbers
Common Errors/Misconceptions
• multiplying only the denominator by the complex
conjugate
• incorrectly determining the complex conjugate of the
denominator
5
4.3.4: Dividing Complex Numbers
Guided Practice
Example 2
Find the result of (10 + 6i ) ÷ (2 – i ).
6
4.3.4: Dividing Complex Numbers
Guided Practice: Example 2, continued
1. Rewrite the expression as a fraction.
(10 + 6i ) ¸ ( 2 - i ) =
10 + 6i
2- i
7
4.3.4: Dividing Complex Numbers
Guided Practice: Example 2, continued
2. Find the complex conjugate of the
denominator.
The complex conjugate of a – bi is a + bi, so the
complex conjugate of 2 – i is 2 + i.
8
4.3.4: Dividing Complex Numbers
Guided Practice: Example 2, continued
3. Rationalize the fraction by multiplying
both the numerator and denominator by
the complex conjugate of the denominator.
10 + 6i
2-i
=
=
10 + 6i ) ( 2 + i )
(
=
(2 - i )(2 + i )
20 + 12i + 10i + 6i 2
4 + 2i - 2i - i 2
20 + 22i - 6
4 +1
=
14 + 22i
4.3.4: Dividing Complex Numbers
5
9
Guided Practice: Example 2, continued
4. If possible, simplify the fraction. The
answer can be left as a fraction, or
simplified by dividing both terms in the
numerator by the quantity in the
denominator.
14 + 22i
5
=
14
5
+
22
5
i
✔
10
4.3.4: Dividing Complex Numbers
Guided Practice: Example 2, continued
11
4.3.4: Dividing Complex Numbers
Guided Practice
Example 3
Find the result of (4 – 4i) ÷ (3 – 4i 3).
12
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
1. Simplify any powers of i.
i 3 = –i
13
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
2. Simplify any expressions containing a
power of i.
3 – 4i 3 = 3 – 4(–i) = 3 + 4i
14
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
3. Rewrite the expression as a fraction,
using the simplified expression. Both
numbers should be in the form a + bi.
4 - 4i
( 4 - 4i ) ¸ ( 3 + 4i ) = 3 + 4i
15
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
4. Find the complex conjugate of the
denominator.
The complex conjugate of a + bi is a – bi, so the
complex conjugate of 3 + 4i is 3 – 4i.
16
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
5. Rationalize the fraction by multiplying
both the numerator and denominator by
the complex conjugate of the denominator.
4 - 4i ) ( 3 - 4i )
(
=
3 + 4i ( 3 + 4i ) ( 3 - 4i )
4 - 4i
=
=
12 - 12i - 16i + 16i 2
9 + 12i - 12i - 16i 2
12 - 28i - 16
9 - ( -16 )
=
-4 - 28i
4.3.4: Dividing Complex Numbers
25
17
Guided Practice: Example 3, continued
6. If possible, simplify the fraction. The
answer can be left as a fraction, or
simplified by dividing both terms in the
numerator by the quantity in the
denominator.
-4 - 28i
25
=-
4
25
-
28
25
i
✔
18
4.3.4: Dividing Complex Numbers
Guided Practice: Example 3, continued
19
4.3.4: Dividing Complex Numbers