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Good Math by Mark C. Chu-Carroll

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Continued Fractions

A continued fraction is a very neat thing. Here’s the idea: take a number where you don’t know its fractional form. Pick the nearest simple fraction 1/n that’s just a little bit too large. If you were looking at, say, 0.4, you’d take 1/2, because it’s a bit bigger. That gives you a first approximation of the number as a fraction. But it’s a little bit too big. If the value of a fraction is a little bit larger than you want it to be, that means that the denominator of that fraction is a little bit too small, and to fix it you need to add a correction to the denominator to make it a little bit bigger. A continued fraction works on that basic principle. Just keep adjusting the denominator; you approximate the correction to the ...

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