I'm looking for a method of checking whether a number is close to a transcendental number, like pi or e, in simple terms, without using things like calculus or infinite series.

For algebraic numbers it's easy.
A positive number $x$ is close to $\sqrt{2}$ if $x^2 - 2$ is close to 0.
A positive number $x$ is close to $\varphi$ if $x^2 - x - 1$ is close to 0.

What's a similar method I could use for pi, for example? Or e?