Tuesday, October 31, 2017

[xwnzahyz] Some nice irrational numbers

Some ways of getting some nice irrational numbers from integers or rational numbers, where "nice" is subjective.

sqrt(d) where d is a positive squarefree integer.

Fractional powers of integers.

atan2(y,x).  If x and y are both normally distributed, then this yields a number uniformly distributed between -pi and pi.  Avoid the temptation to scale the result to between -1 and 1 because that will result in rational numbers sometimes.

log(x) where x is an integer greater than 1.  log(x)/log(y) where gcd(x,y)=1.  x and y could also be positive rational numbers, equivalently quotients of the form (log(a) - log(b)) / (log(c) - log(d)).  Previously.

sin(r*pi) where r is a rational number.  By Niven's Theorem, one only needs to avoid r={0,1,1/6}.

gamma(r) where r is a rational number.  Staying between 0.0953252 and 4.39008 keeps values below 10.  Inspired by Watson's triple integrals and elliptic integral singular values.

zeta(r) within some nice range, avoiding negative integers.

exp(r) where r is a rational number between -1 and 1.  This one seems less aesthetically satisfying because terms like that don't seem to naturally occur in math and science.  exp(z*pi) does occur, with z often complex, but the famous examples, z=i and z=sqrt(163) result in an integer and an almost-integer, which is precisely what we are trying to avoid: we want irrational.


Multiply any of the above by a rational number.

Sums of any of the above.  We enter realms in which we're not sure whether values are irrational, but they probably are.  d=1 in sqrt(d) and r=0 in exp(r) above are obviously not irrational, but keeping them around allows us to form (a+b*sqrt(d))/c, the quadratic surd.  We limited d to squarefree above to avoid situations like sqrt(2)-sqrt(8)/2=0.

Need to be careful with logarithms, e.g., 4/3*log(8)-log(16)=0.

If positive, square root of any of the above.  Probably any non-integral rational power.  Maybe other functions.  Need to be careful to avoid things inverting and becoming rational.

Some care needed to avoid numbers from becoming too big or too small in absolute value.  Central limit theorem might be useful.


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