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Quick, Does 23/67 Equal 33/97? A Mathematician’s Secret from Euclid to Today

David Pengelley
The American Mathematical Monthly
Vol. 120, No. 10 (December 2013), pp. 867-876
DOI: 10.4169/amer.math.monthly.120.10.867
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.10.867
Page Count: 10
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Quick, Does 23/67 Equal 33/97? A Mathematician’s Secret
from Euclid to Today
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Abstract

Abstract How might we determine in practice whether 23/67 equals 33/97? Is there a quick alternative to cross-multiplying? How about reducing? Cross-multiplying checks equality of products, whereas reducing is about the opposite, factoring and cancelling. Do these very different approaches to equality of fractions always reach the same conclusion? In fact, they wouldn’t, but for a critical prime-free property of the natural numbers more basic than, but essentially equivalent to, uniqueness of prime factorization. This property has ancient, though very recently upturned, origins, and was key to number theory even through Euler’s work. We contrast three prime-free arguments for the property, which remedy a method of Euclid, use similarities of circles, or follow a clever proof in the style of Euclid, as in Barry Mazur’s essay [22].

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