Leila Sloman in Scientific American:
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?