9514 1404 393
Answer:
1921/496
Explanation:
There are many rational approximations to √15, some better than others.
A linear approximation is often used:
√15 ≈ 3 +(15-3^2)/(4^2-3^2) = 3 6/7 = 27/7
That can be refined by one iteration of the Babylonian method of determining the root:
√15 ≈ (27/7 +15/(27/7))/2 = (3 6/7 +3 8/9)/2 = 3 55/63 = 244/63
This value is equivalent to the root rounded to 4 decimal places.
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Another iteration of the Babylonian method gives the approximation ...
√15 ≈ 119071/30744, equivalent to the root rounded to 9 decimal places.
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The approximation 1921/496 is the best approximation that has a denominator under 1000.
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Additional comment
Successive rational convergents of the continued fraction approximation of √15 can be found as ...
a'/b' = (3a +15b)/(a +3b)
This method adds approximately one more good decimal place per iteration.
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Successive rational approximations can be found using the Babylonian method (Newton's method iteration) as ...
a'/b' = (a² +15b²)/(2ab)
This method has "quadratic" convergence. It approximately doubles the number of good decimal places with each iteration.
You can use 3/1, 4/1, or 27/7 to begin either of these iterations.