Tuesday, 18 March 2014 23:11

What Does a Fraction Really Mean?

By  Gideon
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We can view the expression:

\[ \frac{X}{Y} \]

as asking, and answering, the question:

How many \(Y\)s are there in each \(X\)?

So for example if \(X = 1 \) and \(Y = 2,\) 

then the fraction:

\[ \frac{X}{Y} = \frac{1}{2} ,\]

is asking:

How many twos are there in one?

And the answer is one-half.

If, on the other hand, \(X = 2,\) and \(Y = 1,\) then:

\[ \frac{X}{Y} = \frac{2}{1} ,\]

is asking:

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:

\[ \frac{X}{1} ,\]

asks the question:

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 \(X \neq 0. \)

\[ \frac{X}{X} = 1 .\]


Read 1700 times Last modified on Wednesday, 19 March 2014 15:26
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