e approximation

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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…<br>2721/1001 = 2.71828171…

The comparison is more impressive in binary.

$ bc -l<br>>>> obase=2<br>>>> 2721/1001<br>10.10110111111000010100…<br>>>> e(1)<br>10.10110111111000010101…

The denominator is a 10-bit number but the approximation is accurate to 21 bits.

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John D. Cook, PhD

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