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

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

Answer 3

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