New Proof Solves 80-Year-Old Irrational Number Problem | Math - Scientific American

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Leila Sloman, PhD student in Mathematics at Stanford University explains, Mathematicians have finally proved a conjecture on approximating numbers with fractions.

Golden ratio is one of the most famous irrational numbers, which run on forever and cannot be expressed accurately without infinite space. Now scientists have proved a conjecture about how to use fractions to approximate them.
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Most people rarely deal with irrational numbers—it would be, well, irrational, as they run on forever, and representing them accurately requires an infinite amount of space. But irrational constants such as  π  and √2—numbers that cannot be reduced to a simple fraction—frequently crop up in science and engineering. These unwieldy numbers have plagued mathematicians since the ancient Greeks; indeed, legend has it that Hippasus was drowned for suggesting irrationals existed. Now, though, a nearly 80-year-old quandary about how well they can be approximated has been solved.

Many people conceptualize irrational numbers by rounding them to fractions or decimals: estimating  π as 3.14, which is equivalent to 157/50, leads to widespread celebration of Pi Day on March 14th. Yet a different approximation, 22/7, is easier to wrangle and closer to  π. This prompts the question: Is there a limit to how simple and accurate these approximations can ever get? And can we choose a fraction in any form we want?

In 1941 physicist Richard Duffin and mathematician Albert Schaeffer proposed a simple rule to answer these questions. Consider a quest to approximate various irrational numbers...

The upshot is that either you can approximate almost every number arbitrarily well, or almost none of them. “There’s a striking dichotomy,” says Dimitris Koukoulopoulos, a mathematician at the University of Montreal. 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.

Although it seems logical that small errors make it harder to approximate numbers, Duffin and Schaeffer were unable to prove their conjecture—and neither was anybody else. The proof remained “a landmark open problem” in number theory, says Christoph Aistleitner, a mathematician at Graz University of Technology in Austria who has studied the problem. That is, until this summer, when Koukoulopoulos and his co-author James Maynard announced their solution in a paper posted to the preprint server arXiv.org.
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Source: Scientific American