Rationalization

Rationalization, as the name suggests, is the process of making fractions rational.
The need for rationalization arises when there are irrational numbers, surds or
roots (represented by
) or complex numbers in the denominator of a fraction.

The following are examples of fractions that need to be rationalized:

Although fractions with surds and complex numbers in their denominators are not
wrong, it is a general consensus that they should not be written that way hence
the need to simplify them by rationalization.

Rationalization is all about moving the surd ()
or complex number to the numerator. Rationalization does not change the value of
a number or function but only re-writes it in a more acceptable and most times easier
to understand form.

Rationalization of fractions involves the use of conjugates.

You should observe from above that a conjugate is formed by changing the sign in
front of either the surd or the complex number. This is not a rule but it is a good
practice for the sake of uniformity.

Conjugates are useful because they when a number is multiplied by its conjugate,
the result will not have surds or complex numbers in it.

Rationalization of Surds

As mentioned above, leaving surds in the denominator of a fraction is not good mathematical
practice. So the question becomes how to get rid of them without changing the actual
number or function.

The answer to this is simple: Multiply the surd in the denominator by its conjugate
to get rid of the surd. This works because

But in order not to change the number, you multiply the denominator and the numerator
by the same conjugate which is in effect multiplying by one.

and it should be clear that

So the above becomes

To prove this, let’s use a value of a as 4

therefore if asked to solve for following

This example is relatively simple enough to be a good proof that rationalization
does not change a number

This is the same answer you would get if you solved directly as

A better example of when this would be more useful is solving the following:

This is solved the same as the previous example:

considering only the denominator;

substituting the above into the fraction;

The above algorithm works with any real values of a and b
, as in the example below:

Since there is no obvious way to simplify the above without a calculator, we rationalize
it:

As you can see, the above is much easier to understand than the original expression.
If you want to confirm that they are indeed the same, use a calculator to compare
the values.

One common mistake most students make when rationalizing fractions is to misplace
the signs on the conjugate, for example

is correct, while

and

are both wrong because they change the value of the expression. Therefore, it is
important to remember to be consistent and not misplace any signs as that changes
the entire expression.

One sure way to check if you have made any errors is to observe whether or not the
denominator factors as nicely as in the examples before. If it does not, check your
work for errors.

Rationalization of Complex Numbers

Complex numbers with imaginary numbers in the denominator are rationalized in a
similar manner to the procedure outlined above. The identity that allows for rationalization
of complex numbers is shown below:

because

and from the section above we’ve seen that

therefore

As in surds, in order not to change the number when rationalizing, we multiply both
the numerator and denominator by the same conjugate which is in effect multiplying
by one, i.e.

The conjugate used in rationalizing complex numbers is called a Complex Conjugate
because the imaginary part of the complex number is the one that gets conjugated
(as in the complex conjugate of (a + bi) is (a – bi)).

Rationalization of complex numbers always follows the following algorithm:

Given a complex fraction of the form

To further illustrate the algorithm above, let’s take the following example:

Rationalize the fraction below

As always, begin by multiplying both the numerator and denominator by the complex
conjugate of the denominator.

Quiz on Rationalization

1. Evaluate the following without using a calculator

A.
B.
C.
D.

The correct answer here would be A.

The answer is obtained as follows:

First multiply the numerator and denominator by the conjugate of the denominator
to get rid of the surd in the denominator.

2. Simplify the following expression:

A.
B.
C.
D.

The correct answer here would be D.

The answer is obtained as follows:

3. Rationalize the expression below

A.
B.
C.
D.

The correct answer here would be C.

The answer is obtained as follows:

First step in rationalizing the above is to multiply both the numerator and denominator
by the complex conjugate of the denominator:

4. Simplify the following expression

A.
B.
C.
D.

The correct answer here would be C.

The answer is obtained as follows:

The above is not the final solution, there is a complex number in the denominator
so we have to rationalize further to get rid of it:

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