# Check algebraic transformation with trial numbers?

I did some algebraic transformations on equations with multiple real variables $x_i$, and I'd like to check whether the transformed equation is still valid. The equations are basically only rational functions with rational coefficients.

I found https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem and made a guess that using $x_i=\exp{\sqrt{p_i}}$ ($p_i$ is $i$-th prime number) for trial numbers would be a good idea, since plain arithmetic operations are unlikely to introduce spurious equalitities?! So, I plug in the trial numbers and see if the equations still hold (given enough calculator precision).

Is my interpretation correct? Am I guaranteed to detect transformation errors this way, if my equations consist of basic arithmetics $+,-,*,/$ only? Or under what circumstances would the test be reliable?

• The question is still whether a single test is enough. Can you give an example equation which includes $\exp\sqrt{2}$ and $\exp\sqrt{3}$, has only basic arithmetics and has conflicting truth values after transformations? Aug 27 '15 at 5:23