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Rational or not? This basic math question took decades to answer
It’s surprisingly difficult to prove one of the most basic properties of a number: whether it can be written as a fraction. A broad new method can help settle this ancient question.
When asked where his formulas came from, he claimed, “They grow in my garden.” Mathematicians greeted his assertions with hoots of laughter, called out to friends across the room, and threw paper airplanes. His construction used mathematics that dated back centuries — one article called it “a proof that Euler missed.” But even after mathematicians came to understand his method, they were unable to extend his success to other numbers of interest. In this case, mathematicians construct a “power series” — a mathematical expression with infinitely many terms, such as 3 + 2 x+ 7 x 2+ 4 x 3+ … — where you determine each coefficient by combining the number you’re studying with one fraction in the sequence, according to a particular formula.
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