Multiplying Complex Numbers
Multiplying complex numbers is very much like multiplying polynomials.
When we simplify the result, we replace each occurrence of i2 with -1.
We often write the final result in the form a + bi.
Example 1
Find: 5i · 7i
| Solution
Multiply 5 · 7 and multiply i · i.
Replace i2 with -1.
Multiply.
So, 5i · 7i = -35. |
5i · 7i
= 35 · i2
= 35 · (-1)
= -35 |
Note:
We write -35 in the form a + bi like this:
0 + (-35)i
Example 2
Find: 4i(9 - 6i)
| Solution
Distribute 4i.
Multiply the factors in each term.
Replace i2 with -1.
Simplify.
Write the result in the form a + bi.
So, 4i(9 - 6i) = 24 + 36i. |
4i(9 - 6i)
= 4i · 9 - 4i
· 6i
= 36i - 24i2
= 36i - 24(-1)
= 36i + 24
= 24 + 36i |
Example 3
Find: (7 - 4i)(10 + 5i)
| Solution
Multiply using the FOIL method.
Multiply the factors in each term.
Replace i2 with -1.
Combine like terms.
So, (7 - 4i)(10 + 5i) = 90 - 5i. |
(7 - 4i)(10 + 5i)
= 7 · 10 + 7
· 5i - 4i
· 10 - 4i · 5i
= 70 + 35i - 40i - 20i2
= 70 + 35i - 40i - 20(-1)
= 70 + 35i - 40i + 20
= 90 - 5i |
|
