For any positive rational numbers $p/q$ and $m/n$, I can decide which is bigger by comparing $pqn$ and $mqn$.
For irrational numbers, we can't use this test. Furthermore, almost all irrational numbers can't be specified by a finite formula in a given symbolic system (i.e. computed to arbitrary precision with an algorithm whose definition is finite). Some can: $\sqrt[5]{2}$ and $\ln{\pi}$ are two examples. There are algorithms available to compute either of these to arbitrary precision, and at some point, their decimal expansions diverge and we can conclude which one is bigger. This is equivalent to finding a rational number which is bigger than one but less than the other.
However, is it possible to prove a fact like $\sqrt[5]{2} > \ln{\pi}$ without obtaining a rational number between them?
More generally, say we can describe two expressions for real numbers, $r$ and $s$, in terms of $+,-,\times,\div$, some transcendental functions, infinite series, etc., such that we could approximate each of their decimal expansions. Is it possible in general to compute/prove which of the expressions $r$ and $s$ is bigger without finding a decimal expansion or otherwise finding a rational number in between them?
Best Answer
The set of numbers for which there is a formula to approximate their decimal expansions is called the computable numbers. The order relation on the computable numbers is non-computable. If the two numbers are known to be unequal, then it is computable, but the method is successive approximation.