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As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal? of the [square root of two]? to three [decimal place]?s is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator? 192099600 is 384199200, which differs from the numerator? by only one. p = 19601 and q = 13860 satisfies the [Diophantine equation]? 2q2+1=p2. Any fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two. |
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As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal? of the [square root of two]? to three [decimal place]?s is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator? 192099600 is 384199200, which differs from the numerator? by only one. p = 19601 and q = 13860 satisfies the [Diophantine equation]? 2q2 + 1 = p2. Any fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two. |
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:nq2+1=p2 |
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:nq2 + 1 = p2 |
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