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

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].

Additional information

Notes on contributors

David Pengelley

DAVID PENGELLEY is professor emeritus at New Mexico State University. His research is in algebraic topology and history of mathematics. He develops the pedagogies of teaching with student projects and with primary historical sources, and created a graduate course on the role of history in teaching mathematics. He relies on student reading, writing, and mathematical preparation before class to enable active student work to replace lecture. He has received the MAA's Haimo teaching award, loves backpacking and wilderness, is active on environmental issues, and has become a fanatical badminton player.