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 and k, nnk=1k, as both are equal to k. The following generalizes this fact.
Theorem. Given two fractions nm and lk, suppose there is a nonzero whole number c so that k=cm and l=cn. Then nm=lk
Proof. Let k=cm and l=cn for whole numbers c, k, l, m, and n. We will prove that nm=lk. In other words, we will prove: nm=cncm.
The fraction nm is the m-th point in the sequence of n-ths. Now divide each of the segments between consecutive points in the sequence of n-ths into c equal parts. Thus each of [0,1],[1,2],[2,3], ... is now divided into cn equal parts. This the sequence of n-ths together with the new division points become the sequence of cn-ths. A simple reasoning shows that the m-th point in the sequence of n-ths must be the cm-th point in the sequence of cn-ths. This is another way of saying nm=cncm. Thus the proof is complete.
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 and k, nnk=1k, as both are equal to k. The following generalizes this fact.
Theorem. Given two fractions nm and lk, suppose there is a nonzero whole number c so that k=cm and l=cn. Then nm=lk
Proof. Let k=cm and l=cn for whole numbers c, k, l, m, and n. We will prove that nm=lk. In other words, we will prove: nm=cncm.
The fraction nm is the m-th point in the sequence of n-ths. Now divide each of the segments between consecutive points in the sequence of n-ths into c equal parts. Thus each of [0,1],[1,2],[2,3], ... is now divided into cn equal parts. This the sequence of n-ths together with the new division points become the sequence of cn-ths. A simple reasoning shows that the m-th point in the sequence of n-ths must be the cm-th point in the sequence of cn-ths. This is another way of saying nm=cncm. Thus the proof is complete.
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