Some Almost-Integer Numbers

[lpetrich]

Almost Integer -- from Wolfram MathWorld discusses several real numbers that are surprisingly close to integers, and it mentions some particularly odd ones, including one attributed to mathematician Srinivasa Ramanujan, who lived a century ago (Ramanujan Constant -- from Wolfram MathWorld). I calculated them with Mathematica, and I verified that that is indeed the case.

exp(pi*sqrt(22)) = 2 508 951.998 257...
exp(pi*sqrt(37)) = 199 148 647.999 978 047...
exp(pi*sqrt(58)) = 24 591 257 751.999 999 822 213...
exp(pi*sqrt(163)) = 262 537 412 640 768 743.999 999 999 999 250 073...

These results can be derived with the help of "modular functions", in particular, the "j-function" (j-Function -- from Wolfram MathWorld). That last one has the help of the algebraic field Q(sqrt(-163)), the rational numbers extended with sqrt(-163), and the others are similar.

 

[rjh01]

If you consider any irrational number that is an integer if you only go to two decimal places then one in hundred irrational numbers will be integers. If you go to three decimal places then the answer is one in a thousand. There are an infinitely large number of irrational numbers.

 

[lpetrich]

So it's a matter of looking at enough numbers? That will work, to a limited degree, but it does not explain the likes of Ramanujan's constant, exp(pi*sqrt(163)). From  Heegner number,

This number was discovered in 1859 by the mathematician Charles Hermite.[7] In a 1975 April Fool article in Scientific American magazine,[8] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.

This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.

Which the article then explains.

That number is
(262 537 412 640 768 744) - (7.499 274 ... * 10^(-13))
where the integer part is (640 320)^3 + 744

I thought of proving the falsity of Martin Gardner's statement with the  Lindemann–Weierstrass theorem, but it fails there. A consequence of that theorem is that there is only one algebraic number that has an algebraic natural logarithm: 1. But if exp(pi*sqrt(163)) is an integer, then pi*sqrt(163) must be transcendental, and therefore pi must be transcendental, something already established with that theorem.

 

[fromderinside]

Ever think of statistical ordinal ordering approaching integer (interval data) status? Work by Tversky: https://deepblue.lib.umich.edu/bits...d=62AB55DF37F5B8D9477779CE3B6E0AFE?sequence=1 I'm thinking about in relation to this matter in conjoint measurement.