19 lines
765 B
Markdown
19 lines
765 B
Markdown
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
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~~~
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sqrt(2) = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...
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~~~
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By expanding this for the first four iterations, we get:
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~~~
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1 + 1/2 = 3 /2 = 1.5
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1 + 1/(2 + 1/2) = 7 /5 = 1.4
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1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
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1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...
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~~~
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The next three expansions are `99/70`, `239/169`, and `577/408`, but the eighth expansion, `1393/985`, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
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In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?
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