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Long-Standing Problem of ‘Golden Ratio’ and Other Irrational Numbers Solved With ‘Magical Simplicity’

Irrational numbers are rarely handled commonly as they run on forever making their accurate representation an infinite system. But irrational constants like π and √2 are numbers that cannot be reduced to a simple fraction, yet they frequently feature in science and engineering. These cumbersome numbers have plagued mathematicians since the ancient Greeks; as legend has it that Hippasus was drowned for suggesting that irrationals existed. Now, finally, a nearly 80-year-old dilemma about how well they can be approximated has been solved.

Most people conceptualize irrational numbers by rounding them off to fractions or decimals. Thus, approximating π as 3.14, equivalent to 157/50, led to the widespread celebration of Pi Day on March 14th. But, a different estimate to 22/7 is easier to accept and considered closer to π. This prods the question: Is there a limit to how simple and accurate such approximations can be? And can we pick a fraction in any form we want?

Back in 1941 physicist Richard Duffin and mathematician Albert Schaeffer proposed a simple rule to answer these questions. Consider a mission to approximate various irrational numbers. First, decide how close your resulting approximation should be for fractions of a particular denominator. You might choose that simplified fractions of the n/2 form can approximate any irrational number whose true value falls within 1/10 of them, thus giving the approximation an “error” of 1/10. Fractions that appear like n/10 are placed closer together on the number line than those with a denominator of 2, so you can limit the error in such cases to only 1/100 i.e. those fractions can approximate anything within 1/100th of them.

Typically, larger denominators are associated with smaller errors. Also, there are infinitely many denominators that can be used to approximate a number to within the matching error, then by increasing the denominator the approximation can be improved further. Duffin and Schaeffer’s rule measures when this can be achieved based on the size of the errors.

If the chosen errors are small enough in aggregate, a randomly picked irrational number x will have only a limited number of good approximations: it might fall into the gaps between approximations with particular denominators. But if the errors are large enough, there will be infinitely many denominators that create a good approximating fraction. In this case, if the errors also diminish as the denominators get bigger, then you can choose an approximation that is as accurate as you want.

The upshot is that either you can approximate almost every number arbitrarily well, or almost none of them.

Dimitris Koukoulopoulos, a mathematician at the University of Montreal, says, “There’s a striking dichotomy. Moreover, you can choose errors however you want, and as long as they are large enough in aggregate most numbers can be approximated infinitely many ways. This means that, by choosing some errors as zero, you can limit the approximations to specific types of fractions—for example, those with denominators that are powers of 10 only”.

Even though it seems logical that small errors make it harder to estimate numbers, Duffin and Schaeffer were unable to verify their conjecture and anybody else could either. The evidence persisted as “a landmark open problem” in number theory, as per Christoph Aistleitner, a mathematician at the Graz University of Technology in Austria who has studied the problem. Until this summer, when Koukoulopoulos and his co-author James Maynard publicized their solution in a paper posted to the preprint server arXiv.org.

Maynard, a professor at the University of Oxford, says, “The Duffin-Schaeffer conjecture has this magical simplicity in an area of maths that’s normally exceptionally difficult and complicated”.

He stumbled upon the problem by accident as he is a number theorist studying prime numbers which is not in the same area as most Duffin-Schaeffer experts. A University of York professor advised Maynard to tackle the Duffin-Schaeffer conjecture after he gave a talk there.

Maynard recalls, “I think he had an intuition that it might be beneficial to get someone slightly outside of that immediate field”.

That intuition turned out to be correct and bore fruit albeit several years later. Later, Maynard asked Koukoulopoulos to collaborate on a feeling that his colleague had relevant expertise.

Maynard and Koukoulopoulos realized that previous work in the field had condensed the problem to one about the prime factors of the denominators, those prime numbers that, when multiplied together, yield the denominator.

Maynard proposed thinking about the problem as shading in numbers, “Imagine, on the number line, coloring in all the numbers close to fractions with denominator 100.”

 

The Duffin-Schaeffer conjecture states that if the errors are large enough and one does this for every possible denominator, nearly every number will be colored in infinitely many times.

For any particular denominator, only part of the number line will be colored in. If mathematicians could show that for each denominator, sufficiently different areas were colored, they would ensure almost every number was colored. If they could also demonstrate those sections were overlapping, they could conclude that happened many times. One way of executing this idea of different-but-overlapping areas is to prove the regions colored by different denominators had nothing to do with one another—they were independent.

But this is not actually true, especially in case two denominators share many prime factors. For instance, the possible denominators 10 and 100 share factors 2 and 5 and the numbers that can be approximated by fractions of the form n/10 reveal annoying overlaps with those that can be approximated by fractions n/100.

Graphing The Problem

Maynard and Koukoulopoulos resolved this conundrum by reframing the problem into mathematical networks called graphs which are basically a bunch of dots with edges viz. some lines connecting the dots. The dots in their graphs signified possible denominators that intended for use by the researchers for approximating the fraction, and two dots were linked with an edge if they had many prime factors in common. The graphs of cases where the precisely allowed denominators had unwanted dependencies had a lot of edges.

The two mathematicians were able to visualize the problem in a new way using graphs.

Maynard says, “One of the biggest insights you need is to forget all the unimportant parts of the problem and to just home in on the one or two factors that make [it] very special. Using graphs not only lets you prove the result, but it’s really telling you something structural about what’s going on in the problem.”

Maynard and Koukoulopoulos inferred that graphs with many edges belong to a particular, highly structured mathematical situation that could be analyzed separately.

The duo’s solution surprised many in the field.

Aistleitner says, “The general feeling was that this was not close to being solved”.

Jeffrey Vaaler, a retired professor at the University of Texas, Austin, who proved a special case of the conjecture in 1978, says, “The technique of using [graphs] is something that maybe in the future will be regarded as just as important [as]—maybe more important than—the actual Duffin-Schaeffer conjecture”.

Other experts may take several months to understand the full details.

Aistleitner says, “The proof now is a long and complicated proof. It’s not sufficient just to have one striking, brilliant idea. There are many, many parts that have to be controlled.”

The paper comprises of 44 pages of such dense, technical mathematics that even leading mathematical minds need time to wrap their heads it.

However, the community seems optimistic as Vaaler says, “It’s a beautiful paper. I think it’s correct.”

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