hyperop is a small library for representing really, really, ridiculously large numbers in pure python. It does so using hyperoperations.
- Hyperoperation 0,
H0is the successor function,
H0(None, 4) = 5
H1(2,4) = 2 + (1+1+1+1) = 6
H2is multiplication (repeated addition),
H2(2,4) = 2+2+2+2 = 8
H3is exponentiation (repeated multiplication),
H3(2,4) = 2*2*2*2 = 16
H4is tetration (repeated exponentiation)
H4(2,4) = 2^(2^(2^(2))) = 65536
- Hyperoperation n is repeated Hyperoperation (n-1)
Fundamentally, hyperop works recursively by applying a fold-right operation:
H[n](x,y) = reduce(lambda x,y: H[n-1](y,x), [a,]*b)
pip install hyperop
To install the latest version use:
pip install git+https://github.com/thoppe/python-hyperoperators
from hyperop import hyperop H1 = hyperop(1) print H1(2,3), H1(3,2), H1(5,4) # >> 5, 5, 9 H3 = hyperop(3) print H3(2,3), H3(3,2), H3(5,4) # >> 8, 9, 625 from math import log H = hyperop(4) print H(2,5) >>> 200352993040684646497....45587895905719156736 print log(log(log(log(H(2,5),2.0),2.0),2.0),2.0) == 2 >>> True
Approximate infinite tetration. Show that sqrt(2)^sqrt(2)^... where the tower continues an infinite amount of times is 2.
H4 = hyperop(4) print H4(2**0.5, 200) # >> 2.0
Calculate the incomprehensibly large, but finite Graham's number:
def GrahamsNumber(): # This may take awhile... g = 4 for n in range(1,64+1): g = hyperop(g+2)(3,3) return g
Plot the phase angle on the complex plane over tetrating four times
from hyperop import hyperop import mpmath H = hyperop(4) f = lambda z: H(z,4) mpmath.cplot(f, verbose=True, points=100000)
Sometimes, especially in the case of small complex numbers, you only care about numbers that stay bounded during the calculation.
That is, you'd only like to keep the result for some bound z such that
H[n](a,b) <= z.
bounded_hyperop does just that:
from hyperop import bounded_hyperop Hb = bounded_hyperop(4, bound=1000) print Hb(2,3), Hb(2,4) # >> 16 inf
Higher order hyperoperations (from tetration and above) are not associative, thus tetration
H4(2,4) = 2^(2^(2^(2))) = 65536 is not
H4(2,4) != 2^(2*2*2) = 256.
Since tetration is not defined for non-integral heights, the second argument of tetration and both arguments of pentation and above are restricted to non-negative integer values.
Talks & Press
Hyperop was featured in issue #231 of Python Weekly!
This library was first presented at DC's Hack && Tell (Feb. 2016). Talk link.
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