Recognizing numbers

Posted on by John

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)

21 thoughts on “Recognizing numbers”

  8. Joe Louderback

    Odd. On my calculator (and matching my trig knowledge) sin(pi/6) = 0.5, not sqrt(3)/2.

  10. Alex

    Doesn’t anybody find this slightly dangerous? I can imagine people finding a result numerically, and giving an analytical form just to make it look like they found it analytically. – Yes, there’s people who would do that.

  13. Alok

    wonderful. Never imagined that one day software will do this!! How fast are such calculations?

  14. John

    Alok: The calculations are quick. Most of the time you get a rational approximation, and algorithms for approximation with the smallest possible denominators are well known and fast. I don’t know how extensive it is in considering square roots etc.

  17. Pseudonym

    @Alex, I was thinking that myself. For some reason, I am reminded of the fine structure constant.

  18. Larry

    Such methods played a part in Euler’s solution of the famous Basel Problem: find the exact value of 1 + 1/2^2 + 1/3^2 + 1/4^2 + …
    He computed the value of the series to six decimal places, then somehow recognized that it seemed to be equal to pi^2 / 6. This was a clue which later led him to a proof that this was indeed the correct value of the series.

  19. nineright

    So cool! Thanks for sharing.

    PS, I find 0.7182818284590453(e-2) which is not well simplified(143656365691809/200000000000000),:)

  20. John

    nineright: If you suggest e is a constant to consider, it’ll work.

    nsimplify(0.7182818284590453, constants=[E]) returns e-2.

  21. Wasicki

    exp((141 + sqrt(780631))/895) is not so bad for an automatic approximation to π. Still, 12 digits and two functions yield only 11 correct digits, which is not quite a good deal.

    The following gives 10 correct digits at the cost of 5 digits and one transcendental function reused twice plus two calls of the same function. Not a good deal either, but at least it is more beautiful. Also, this one was found by hand:

    log(16*log(878)/log(16*log(878)))

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