Hi,

A rational number is described as being a fraction where both the numerator and denominator are integers. If this is so, why does my book say that is a rational number?

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- August 26th 2010, 02:20 PMwebguyRational Number
Hi,

A rational number is described as being a fraction where both the numerator and denominator are integers. If this is so, why does my book say that is a rational number? - August 26th 2010, 02:23 PMskeeter
- August 27th 2010, 01:18 PMwebguy
Book says: To rationalise the fraction , multiply both the numerator and denominator by giving: .

- August 27th 2010, 01:45 PMPlato
There is a failure to communicate there. At least a failure to translate correctly.

**The word***rationalize*does not mean strictlyin mathematics-speak.__to make rational__

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. - August 27th 2010, 04:13 PMwebguy
Understood, thanks :). However, why is it better to have an irrational number as the numerator instead of the denominator?

- August 27th 2010, 05:20 PMskeeter
- August 27th 2010, 05:23 PMPlato
Yours is actually a very good question.

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. - August 27th 2010, 06:30 PMSoroban
Hello, webguy!

Edit: Plato beat me to it . . . *sigh*

. . . .But I refuse to delete all this typing . . . so there!

Quote:

Why is it better to have an irrational number in the numerator

instead of the denominator?

Probably the same reason we preferdenominators.*positive*

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 hassigns: .*three*

So that is actually:

We can change*any*__two____signs__

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 aof work to do!*lot*

First, look up on a square-root table and get, say,

Then we must divide: . . . . aunplesant task!*very*

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?

- August 28th 2010, 04:33 AMwebguy
Thank you for all of your answers! I think I can declare this thread as solved :)