# MandysNotes

Tuesday, 18 March 2014 23:11

## What Does a Fraction Really Mean?

By  Gideon
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We can view the expression:

$\frac{X}{Y}$

How many $$Y$$s are there in each $$X$$?

So for example if $$X = 1$$ and $$Y = 2,$$

then the fraction:

$\frac{X}{Y} = \frac{1}{2} ,$

How many twos are there in one?

If, on the other hand, $$X = 2,$$ and $$Y = 1,$$ then:

$\frac{X}{Y} = \frac{2}{1} ,$

How many ones are there in two?

In this case the answer is two:

$\frac{2}{1} = 2 .$

We refer to the expression:

$\frac{X}{Y}$

as a ratio or as a fraction.

We refer to $$X,$$ the number upstairs, as the numerator.

We refer to $$Y,$$ the number downstairs, as the denominator.

Note that for any number $$X$$, including zero, the ratio:

$\frac{X}{1} ,$

How many ones are in $$X$$?

And the answer is always $$X$$.

For instance if $$X= 0,$$ then there are no ones in $$X,$$,so

$\frac{0}{1} = 0 .$

If $$X = \pi,$$ then there are $$\pi$$ ones in $$\pi$$ and so:

$\frac{\pi}{1} = \pi .$

in general, when $$X$$ is not equal to zero, there are $$X$$ ones in $$X$$, so:

$\frac{X}{1} = X .$

Now note that for any nonzero number $$X$$, if we ask the question:

How many $$Xs$$ are in $$X?$$

The answer is always: one; there is one $$X$$ in $$X.$$

For example, if $$X = 10,$$ then

$\frac{10}{10} = 1 ;$

if $$X = \frac{1}{2},$$ then:

$\frac{ \dfrac{1}{2} }{ \tfrac{1}{2} } = 1 .$

And in general, for any nonzero number $$X \neq 0.$$

$\frac{X}{X} = 1 .$