Big Data/Analytics Zone is brought to you in partnership with:

John Cook is an applied mathematician working in Houston, Texas. His career has been a blend of research, software development, consulting, and management. John is a DZone MVB and is not an employee of DZone and has posted 173 posts at DZone. You can read more from them at their website. View Full User Profile

Recognizing Special Numbers with nsimplify

05.01.2013
| 2315 views |
  • submit to reddit

I was playing around with SymPy, a symbolic math package for Python, and ran across nsimplify. It takes a floating point number and tries to simplify it: as a fraction with a small denominator, square root of a small integer, an expression involving famous constants, etc.

For example, suppose some calculation returned 4.242640687119286 and you suspect there’s something special about that number. Here’s how you might test where it came from.

>>> from sympy import *
>>> nsimplify(4.242640687119286)
3*sqrt(2)

Maybe you do a calculation numerically, find a simple expression for the result, and that suggests an analytical solution.

I think a more common application of nsimplify might be to help you remember half-forgotten formulas. For example, maybe you’re rusty on your trig identities, but you remember that cos(π/6) is something special.

>>> nsimplify(cos(pi/6))
sqrt(3)/2

Or to take a more advanced example, suppose that you vaguely remember that the gamma function takes on recognizable values at half integer values, but you don’t quite remember how. Maybe something involving π or e. You can suggest that nsimplify include expressions with π and e in its search.

>>> nsimplify(gamma(3.5), constants=[pi, E])
15*sqrt(pi)/8

You can also give nsimplify a tolerance, asking it to find a simple representation within a neighborhood of the number. For example, here’s a way to find approximations to π.

>>> nsimplify(pi, tolerance=1e-5)
355/113

With a wider tolerance, it will return a simpler approximation.

>>> nsimplify(pi, tolerance=1e-2)
22/7

Finally, here’s higher precision approximation to π that isn’t exactly simple:

>>> nsimplify(pi, tolerance=1e-7)
exp(141/895 + sqrt(780631)/895)

Published at DZone with permission of John Cook, author and DZone MVB. (source)

(Note: Opinions expressed in this article and its replies are the opinions of their respective authors and not those of DZone, Inc.)