Math CentralQuandaries & Queries


Question from Anandmay, a student:

I saw one of your answered questions:Why are equivalent fractions equal?
The same question I had for why i searched and ended up entering this wonderful site.
The answer was so much experimental based. I mean,for example,it was explained how
3/4 was equal to 6/8 by dividing a unit into 4 equal parts and taking four of the parts,and then dividing the same unit into 8 equal parts and then taking 6 of them.The result was that both the taken parts were equal. (That is,3 out of 4 equal parts were equal to 6 of the 8 equal parts.)
But,this was very much non-generalized and experimental-based result(proof). And we all know that in mathematics,we confirm some particular thing for all the numbers iff we generalize it.

So,can you explain more clearly 'HOW' the 3 parts of the 4 equal parts and the 6 of the 8 equal parts of the unit equal(that is the same)?I don't want experimental proof. I want clear proof,and thus please give a generalized proof for all fractions and their equivalents.

Hi Anandmay,

This is a very insightful question.

You are correct. My response to Why are equivalent fractions equal? was really an argument to say that whatever fractions are and however we define two fractions to be equal, if you want these definitions to agree with our everyday understanding of fractions, three quarters and six eights should be equal. Hence the question is, what is a fraction as a mathematical object and what does it mean for two fractions to be equal?

I am going to assume we both understand the integers and the standard operations on them. I am going to look at ordered pairs of integers. If $a$ and $b$ are integers then I will write $(a,b)$ as an ordered pair where ordered means that $(a,b)$ and $(b,a)$ are different pairs unless $a = b.$ My only stipulation is that I am only going to consider ordered pairs $(a,b)$ where $b \neq 0.$ Said in a more formal way let

\[P = \{(a,b)| \mbox{ where }a \mbox{ and } b \mbox{ are integers and } b \neq 0 \}.\]

Next define equivalence as follows.

\[\mbox{ If } (a,b) \mbox{ and }(c,d) \mbox{ are in }P \mbox{ then } (a,b) \mbox{ is equivalent to } (c,d) \mbox{ iff } ad = bc.\]

Finally if $(a,b)$ is in $P$ then let

\[ [(a,b)] \mbox{ be the set of all } (c,d) \mbox{ in } P \mbox{ such that } (a,b) \mbox{ and } \mbox{ and } (c,d) \mbox{ are equivalent.}\]

This mathematical object $[(a,b)]$ is called the equivalence class of $(a,b).$

We normally don't write this equivalence class as $[(a,b)]$ but rather we write is as $\frac{a}{b}.$ Thus $\frac{a}{b}$ can be seen as the collection of all ordered pairs in $P$ that are equivalent to $(a,b).$

There are many definitions to follow and details to prove to show that the fractions (rational numbers) defined this way behave the way we want fractions to behave. In particular since $[(a,b)]$ is a set we can define $\frac{a}{b} = \frac{c}{d}$ if $[(a,b)] = [(c,d)]$ as sets. With this development and this definition of the equality of fractions you can see that since $(3,4)$ is equivalent to $(6,8)$ it follows that $\frac34 = \frac{6}{8}.$

I hope this helps,

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