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自然常数 e 的分数近似值e approximation
数学常数 e 存在一个精度极高的有理数近似值：2721/1001。与直接截断小数位的简单做法（如 2718/1000 仅能提供四位有效数字精度）相比，2721/1001 在分母大小相近的情况下，能够将精度大幅提升至七位，甚至接近八位有效数字。这种相对于其分母规模而言异常出色的逼近能力，展示了有理数在数值计算中的奇妙特性。该发现为需要快速计算且对精度有要求的算法场景提供了一种极其高效的替代方案。
John
I ran across the approximation
e ≈ 2721/1001
recently. What makes this remarkable is its accuracy relative to the size of the denominator.
You can create a trivial approximation just by truncating a decimal expansion
e ≈ 2718/1000
but this is only good to four significant figures, but 2721/1001 is good to seven, almost eight, significant figures.
e = 2.71828182…
2721/1001 = 2.71828171…The comparison is more impressive in binary.
$ bc -l
>>> obase=2
>>> 2721/1001
10.10110111111000010100…
>>> e(1)
10.10110111111000010101…The denominator is a 10-bit number but the approximation is accurate to 21 bits.
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