Multiplying two complex numbers is accomplished in a manner similar to
multiplying two binomials. You can use the FOIL process of
multiplication, distributive multiplication, or your personal favorite means
of multiplication.
|
Distributive Multiplication: |
(2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 +
5i)
= 8 + 10i + 12i + 15i2
= 8 + 22i + 15(-1)
= 8 + 22i -15
= -7 + 22i Answer
Be sure to replace
i2 with (-1) and proceed with
the
simplification. Answer should be in a + bi
form. |
|
The product of two complex
numbers is a complex number. |
 |
(a+bi)(c+di) = a(c+di) + bi(c+di)
= ac + adi + bci + bdi2
= ac + adi + bci + bd(-1)
= ac + adi + bci - bd
= (ac - bd) + (adi + bci)
= (ac -bd) + (ad + bc)i
answer
in
a+bi
form |
|
The conjugate
of a complex number a + bi is the complex
number a - bi.
For example, the conjugate of 4
+ 2i is 4 - 2i.
(Notice that only the sign of the bi term is changed.)
|
The product of a complex number and its
conjugate
is a real number, and is always positive. |
(a + bi)(a - bi) = a2 + abi - abi
- b2i2
= a2 - b2 (-1)
(the middle terms drop out)
= a2 + b2 Answer
This is a real number (
no i's ) and since both
values are squared, the answer
is positive. |
 |
|
When dividing two complex
numbers, |
| 1. |
write the problem in
fractional form, |
| 2. |
rationalize the denominator by multiplying the numerator and
the denominator by the conjugate of the denominator. |
| |
(Remember that a complex number times its conjugate will give a real
number.
This process will remove the i from the
denominator.) |
Example:
| Dividing using
the conjugate: |
Answer |
|
 |
Find out how to use your
TI-83+/84+ graphing calculator for multiplying and dividing complex
numbers.
It will be very helpful!
Click
here. |
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