71 lines
1.7 KiB
Plaintext
71 lines
1.7 KiB
Plaintext
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"# Square root convergents (Euler Problem 57)"
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]
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},
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"[https://projecteuler.net/problem=57](https://projecteuler.net/problem=57)\n",
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"\n",
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"It is possible to show that the square root of two can be expressed as an infinite continued fraction.\n",
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"\n",
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"√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...\n",
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"\n",
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"By expanding this for the first four iterations, we get:\n",
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"\n",
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"1 + 1/2 = 3/2 = 1.5\n",
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"\n",
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"1 + 1/(2 + 1/2) = 7/5 = 1.4\n",
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"\n",
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"1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...\n",
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"\n",
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"1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...\n",
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"\n",
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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.\n",
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"\n",
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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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]
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