How to determine if a number is a combination of common values $(\sqrt 2, \varphi, e, \pi, \text{etc})$

Question

Say I come across a number that's approximately $2.236$. I might wonder if this is a well known value, or the result of a combination of well known values. I might start by looking at square roots of natural numbers:

$$ 1.41421… \\ 1.73205… \\ 2 \\ 2.23606… \\ $$

Ah, it looks like this might be $\sqrt 5$. But what if it had been $1.6449$ i.e. $\frac{{\pi}^2}{6}$? Or $4.5842$ i.e. $(e-\gamma)^2$?

There seems to be an endless combination of constants and powers to try, which is rather tedious by hand.

What software is commonly used to find some approximate representation of a number using such combinations of well-known constants?

Answer 1

Try the following:

1. Inverse Symbolic Calculator (Original)

2. Inverse Symbolic Calculator (CARMA)

3. WolframAlpha

4. RIES (RILYBOT Inverse Equation Solver).

5. OEIS (Click and see)

6. Recognize[] function of Mathematica (for algebraic numbers)

Answer 2

The usual method employs an integer relation algorithm, such as the Lenstra–Lenstra–Lovász lattice basis reduction algorithm.

Answer 3

If you tipe a number in wolfram|alpha it provides a list of possible closed forms.

Answer 4

The PSLQ algorithm works well. Given a set of numbers $\{ x_1, x_2, \ldots, x_n\}$, it looks for a non-trivial integer linear combination that makes this close to zero. It has been used successfully to find many suspected relationships (one must then use other techniques to prove that the relationship is indeed true).