You probably know how Cantor proved the countability of the rational numbers. Give it a little thought and you'll likely be able to reconstruct the zigzag path the nineteenth-century German mathematician took through the upper half-plane to derive the result.
What, though, is the 25th rational number in Cantor's bijection? What natural number does the fraction 5/17 correspond to?
Aimeric Malter, Dierk Schleicher, and Don Zagier ask these questions in the first of the three sections that comprise their paper "New Looks at Old Number Theory," which appears in the March 2013 special issue of The American Mathematical Monthly.
Their goal in Part I is to "breathe new life into the . . . hoary theorem" that the rational numbers are countable, to provide a bijection that's different from—and "nicer" than—the standard one.
And that's just the beginning. As Malter, Schleicher, and Zagier write in the paper's abstract:
We present three results of number theory that all have classical roots but also modern aspects. We show how to (1) systematically count the rational numbers by iterating a simple function, (2) find a representation of any prime congruent to 1 modulo 4 as a sum of two squares by using simple properties of involutions and pairs of involutions, and (3) find counterexamples to Euler's conjecture that a fourth power can never be the sum of three fourth powers by using properties of quadratic polynomials with rational coefficients.
The authors wrote "New Looks at Old Number Theory" with the intent to "entertain and edify." Read the paper and you'll get a lot more than just a new way to visualize enumeration of the rationals.
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About the March Monthly
Comprised entirely of contributions from distinguished presenters at the 2011 International Mathematics Summer School for Students in Bremen, Germany, the March 2013 special issue—the first in more than 20 years—of The American Mathematical Monthly captures the spirit of the summer school in journal form. A full-color and generously illustrated 100 pages, the issue allows readers—be they high schoolers, college students, mathematics educators, or lifelong learners—to engage with and learn from leading international mathematicians. Papers range over such topics as unprovable arithmetic statements, bike tire tracks, the uncountability of the rationals, and how objects roll.