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# Almost integers

Almost integers (or near integers[1][2]) are numbers which, although not being integers on account of having a fractional part, are very close to integers and can be mistaken for such as the result of a loss of machine precision during a calculation. For example, 99.99999999999 is an almost integer that is almost the integer 100.

## Ramanujan's constant

Ramanujan's constant is amazingly close to an integer, the first 12 digits after the decimal point being 9. The decimal expansion of Ramanujan's constant is

${\displaystyle e^{\pi {\sqrt {163}}}=262537412640768743.9999999999992500725971981856888793538563373369908627075374103782\ldots \,}$

A060295 Decimal expansion of e^(Pi*sqrt(163)).

{2, 6, 2, 5, 3, 7, 4, 1, 2, 6, 4, 0, 7, 6, 8, 7, 4, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 5, 0, 0, 7, 2, 5, 9, 7, 1, 9, 8, 1, 8, 5, 6, 8, 8, 8, 7, 9, 3, 5, 3, 8, 5, 6, 3, 3, 7, 3, 3, 6, 9, 9, 0, 8, 6, 2, 7, 0, 7, 5, 3, 7, 4, 1, 0, ...}

## Pi^3

Pi^3 is somewhat close to an integer, the first 2 digits after the decimal point being 0. The decimal expansion of ${\displaystyle \scriptstyle \pi ^{3}\,}$ is

${\displaystyle \pi ^{3}=31.00627668029982017547631506710139520222528856588510769414453810380639\ldots \,}$

A091925 Decimal expansion of pi^3.

{3, 1, 0, 0, 6, 2, 7, 6, 6, 8, 0, 2, 9, 9, 8, 2, 0, 1, 7, 5, 4, 7, 6, 3, 1, 5, 0, 6, 7, 1, 0, 1, 3, 9, 5, 2, 0, 2, 2, 2, 5, 2, 8, 8, 5, 6, 5, 8, 8, 5, 1, 0, 7, 6, 9, 4, 1, 4, 4, 5, 3, 8, 1, 0, 3, 8, 0, 6, 3, 9, 4, 9, 1, 7, 4, 6, 5, 7, ...}

Also, observe that the decimal expansion of ${\displaystyle \scriptstyle \pi ^{3}-31\,}$ is nearly

${\displaystyle {\frac {2\pi }{10^{3}}}=0.0062831853071\ldots \,}$

## Notes

1. One might have wanted to distinguish between the two cases of near integers: slightly less than an integer (almost integers) and slightly more than an integer (quasi integers), but according to common usage, almost integers means near integers. Also, quasi-integers (hyphenated) (Ahlswede and Khachatrian 1996) refers to a different concept.
2. Rudolf Ahlswede and Levon H. Khachatrian (Bielefeld), Sets of integers and quasi-integers with pairwise common divisor, ACTA ARITHMETICA, LXXIV.2 (1996).