The word rationalize does not mean strictly to make rational in mathematics-speak.
It is considered poor form to have a radical in the denominator.
Therefore instead of we rationalize that number writing it as by multiplying by .
In other words we make the denominator rational.
Before good calculators and/or CAS it was very difficult to divide by an irrational number (actually it still is in practice).
Consider over against which is easier to do by ‘hand’?
They are both equal. But the second is easier to work with.
Edit: Plato beat me to it . . . *sigh*
. . . .But I refuse to delete all this typing . . . so there!
Why is it better to have an irrational number in the numerator
instead of the denominator?
Probably the same reason we prefer positive denominators.
We know that is read "two-thirds".
And is read "negative-two thirds".
And is read "negative two-thirds".
But how do we read ? . . . Maybe "two negative-thirds" ?
Interesting . . . This problem never came up before. .Why?
Because "they" always use positive denominators at the start.
(And didn't bother to tell us that it's a Rule.)
A bit of trivia . . .
A fraction has three signs: .
So that is actually:
We can change any two signs without changing the value of the fraction.
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Back to the question: Why do we "rationalize" denominators?
The main reason arose back in 1000 B.C. (before calculators).
If we need the value of , we have a lot of work to do!
First, look up on a square-root table and get, say,
Then we must divide: . . . . a very unplesant task!
It starts like this:
We "move the decimal points" and divide:
So, to four decimal places, we have: .
If rationalize first, we have: .
And the division looks like this:
Which way do you prefer?