e approximation
by John from John D. Cook on (#76SSK)
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>>> 2721/100110.10110111111000010100...>>> e(1)10.10110111111000010101...
The denominator is a 10-bit number but the approximation is accurate to 21 bits.
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