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 + 15i^{2
}
= 8 + 22i + 15(1)
= 8 + 22i 15
= 7 + 22i Answer
Be sure to replace
i^{2} 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 + bdi^{2
} = 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) = a^{2} + abi  abi
 b^{2}i^{2
}= a^{2}  b^{2} (1)
(the middle terms drop out)
= a^{2 }+ b^{2} 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
TI83+/84+ graphing calculator for multiplying and dividing complex
numbers.
It will be very helpful!
Click
here. 


