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Division of Complex Numbers



Dividing complex numbers is not what you might think it to be.
The goal of the division is to obtain one complex number of the form a + bi where a and b are real numbers. Thus, the solution (the quotient) will not have an "i" in a denominator.


The process of division for complex numbers is more a process of
"
rationalizing the denominator" so it does not contain any values of "i " (since "i " is actually a square root, iii). This "rationalization" is accomplished by using the conjugate of the denominator of the division fraction.
This process is similar to the process for dividing radicals.
conjugatesee

When dividing two complex numbers,
1. write the problem in fractional form,
2. remove the i from the denominator by multiplying the numerator and denominator by the complex conjugate of the denominator.

We have seen that a complex number times its conjugate will yield a real number. This process removes the i from the denominator.
cpd2
Read more about complex conjugates under the Multiplication section.

d
Always check to see if you are expected to express your final answer in a + bi form. For the problem above,
cpd3

are all correct and acceptable answers. But if asked to express the answer in simplest a + bi form, only the last answer will be accepted.


Division Rule:    cdp4

dividerdash

ex1    Simplify: (3 + i) รท (1 + 2i)

cdp6

ex2   Simplify: cpd7

cpd8


ex3   Simplify: cpd9

cpd10
Multiplying by "i " creates a (-1) denominator, eliminating the i denominator.


ex4   Simplify: cpd11

cpd12

 

ti84
How to use your
TI-83+/84+
calculator with
complex numbers.
Click here.

divider

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