The fundamental theorem of fractions

Aion

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Hey, I'm currently trying to understand a proof. And I have trouble understanding a specific part of it which I will highlight. I find this part hard to visualize. Maybe someone can draw me a diagram of what is happening.

Recall that two fractions are equal if they are the same point on the number line. We obvserved that for all nonzero whole numbers n\displaystyle n and k\displaystyle k, nkn=k1\displaystyle \frac{nk}{n} =\frac{k}{1}, as both are equal to k\displaystyle k. The following generalizes this fact.

Theorem. Given two fractions mn\displaystyle \frac{m}{n} and kl\displaystyle \frac{k}{l}, suppose there is a nonzero whole number c\displaystyle c so that k=cm\displaystyle k=cm and l=cn\displaystyle l = cn. Then mn=kl\displaystyle \frac{m}{n} = \frac{k}{l}

Proof. Let k=cm\displaystyle k = cm and l=cn\displaystyle l = cn for whole numbers c\displaystyle c, k\displaystyle k, l\displaystyle l, m\displaystyle m, and n\displaystyle n. We will prove that mn=kl.\displaystyle \frac{m}{n} = \frac{k}{l}. In other words, we will prove: mn=cmcn\displaystyle \frac{m}{n} = \frac{cm}{cn}.

The fraction mn\displaystyle \frac{m}{n} is the m\displaystyle m-th point in the sequence of n\displaystyle n-ths. Now divide each of the segments between consecutive points in the sequence of n\displaystyle n-ths into c\displaystyle c equal parts. Thus each of [0,1],[1,2],[2,3]\displaystyle [0,1], [1,2], [2,3], ... is now divided into cn\displaystyle cn equal parts. This the sequence of n\displaystyle n-ths together with the new division points become the sequence of cn\displaystyle cn-ths. A simple reasoning shows that the m\displaystyle m-th point in the sequence of n\displaystyle n-ths must be the cm\displaystyle cm-th point in the sequence of cn\displaystyle cn-ths. This is another way of saying mn=cmcn\displaystyle \frac{m}{n} = \frac{cm}{cn}. Thus the proof is complete.
 
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This seems to be part of some particular development of fractions, and depends on previous definitions. (In a normal context, it would be trivial to prove.) I could draw pictures of what it seems to mean, but I can't fully understand why they are doing what they are without the context. Where does this come from, and what definitions are we supposed to "recall"?

Okay, I searched for a phrase and found this source: https://books.google.com/books?id=c1OiDAAAQBAJ&pg=PA28 .

Do you have that source? It includes examples of what it is talking about, so maybe that's all you need. If not, please explain what is missing in your mind.
 
Thanks I didn't know it was a book. I've already bought it now and will look into it! :)
 
I just had in mind looking at the context as far as it can be seen online. I don't know whether your goal is to learn to deeper facts lying behind arithmetic (which the book is about), or just to learn to work with fractions (for which a different book might be more suitable).

I'm curious, of course, as to where you did get the proof, if not from the book (which was the only source I found with a search). But that doesn't matter.
 
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