A question about how to express a fraction as <mrow class="MJX-TeXAtom-ORD"> 1

Akira Huang

Akira Huang

Answered question

2022-05-21

A question about how to express a fraction as 1 q 1 + 1 q 2 + + 1 q N
Let x be a positive rational number, strictly between 0 and 1. Prove that there is a finite strictly increasing list of positive integers 2 q 1 < q 2 < < q N such that
x = 1 q 1 + 1 q 2 + + 1 q N .
I have tried many methods, such as mathematical induction. I know it is obvious that 1 m is right for assumption, but when I begin to prove 2 m I feel it is hard to find some patterns for 2 m . Thus, maybe my thoughts are wrong.
Also, I still tried to change the fraction so that the numerator will be smaller and smaller. But I still cannot find a way to lower the numerator. Can someone help me solve the question? Or, can someone give me some hints. I will appreciate you very much!

Answer & Explanation

mseralge

mseralge

Beginner2022-05-22Added 10 answers

Let q = m / n rational, with q ( 0 , 1 ), and hence
1 k m n < 1 k 1
for some integer k > 1 . If 1 k = m n , we have nothing to prove. Otherwise, we have
m n 1 k = k m n k n = m n > 0.
But 1 k 1 > m n implies that n > m ( k 1 ) , or m > k m n = m > 0
Hence, after we subtracted the the nearest and smallest fraction of the form 1 / k , the remainder m / n has smaller numerator.
Hence, in at most m steps this procedure ends, and thus we obtain the sought for representation.

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