The deep recesses of the number line are not as forbidding as they might seem. That’s one consequence of a major new proof about how complicated numbers yield to simple approximations.

The proof resolves a nearly 80-year-old problem known as the Duffin-Schaeffer conjecture. In doing so, it provides a final answer to a question that has preoccupied mathematicians since ancient times: Under what circumstances is it possible to represent irrational numbers that go on forever—like pi—with simple fractions, like 22/7? The proof establishes that the answer to this very general question turns on the outcome of a single calculation.

“There’s a simple criterion for whether you can approximate virtually every number or virtually no numbers,” said James Maynard of the University of Oxford, co-author of the proof with Dimitris Koukoulopoulos of the University of Montreal.

Mathematicians had suspected for decades that this simple criterion was the key to understanding when good approximations are available, but they were never able to prove it. Koukoulopoulos and Maynard were able to do so only after they reimagined this problem about numbers in terms of connections between points and lines in a graph—a dramatic shift in perspective.

“They had what I’d say was a great deal of self-confidence, which was obviously justified, to go down the path they went down,” said Jeffrey Vaaler of the University of Texas, Austin, who contributed important earlier results on the Duffin-Schaeffer conjecture. “It’s a beautiful piece of work.”

The Ether of Arithmetic

Rational numbers are the easy numbers. They include the counting numbers and all other numbers that can be written as fractions.

This amenability to being written down makes rational numbers the ones we know best. But rational numbers are actually rare among all numbers. The vast majority are irrational numbers, never-ending decimals that cannot be written as fractions. A select few are important enough to have earned symbolic representations, such as pi, e and the square root of 2. The rest can’t even be named. They are everywhere but untouchable, the ether of arithmetic.

So maybe it’s natural to wonder—if we can’t express irrational numbers exactly, how close can we get? This is the business of rational approximation. Ancient mathematicians, for instance, recognized that the elusive ratio of a circle’s circumference to its diameter can be well approximated by the fraction 22/7. Later mathematicians discovered an even better and nearly as concise approximation for pi: 355/113.

“It’s hard to write down what pi is,” said Ben Green of Oxford. “What people have tried to do is to find explicit approximations to pi, and one common way of doing that is with rationals.”

Lucy Reading-Ikkanda/Quanta Magazine

In 1837 the mathematician Gustav Lejeune Dirichlet found a rule for how well irrational numbers can be approximated by rational ones. It’s easy to find approximations so long as you’re not too particular about the error. But Dirichlet proved a straightforward relationship between fractions, irrational numbers and the errors separating the two.

He proved that for every irrational number, there exist infinitely many fractions that approximate the number evermore closely. Specifically, the error of each fraction is no more than 1 divided by the square of the denominator. So the fraction 22/7, for example, approximates pi to within 1/72, or 1/49. The fraction 355/113 gets within 1/1132, or 1/12,769. Dirichlet proved that there is an infinite number of fractions that draw closer and closer to pi as the denominator of the fraction increases.

“It’s a rather beautiful and remarkable thing that you can always approximate a real number by a fraction and the error is no more than 1 over [the denominator squared],” said Andrew Granville of the University of Montreal.

In a 1913 manuscript, the mathematician Srinivasa Ramanujan used the fraction 355/113 as a rational approximation for pi. Wikicommons

Dirichlet’s discovery was, in a sense, a narrow statement about rational approximation. It said that you can find infinitely many approximating fractions for each irrational number if your denominators can be any whole number, and if you’re willing to accept an error that’s 1 over the denominator squared. But what if you want your denominators to be drawn from some (still infinite) subset of the whole numbers, like all prime numbers, or all perfect squares? And what if you want your approximation error to be 0.00001, or any other values you might choose? Will you succeed at producing infinitely many approximating fractions under such specific conditions?