User blog comment:Planterobloon/How do you compare huge numbers to each other?/@comment-11227630-20170812012304/@comment-30754445-20170812195457

Usually there's no need to work with "inequalities".

Just expand the numbers (in their respective notations) precisely until the argument is too large to write in full. By that point, all common notations are pretty much equivalent to one another for all practical purposes:

For example, for such numbers, we can safely write:

fω+1(n) ≈ {10,n,1,2}  = Kn ≈ E10##10#n ≈ n![2] ≈ ...

And then, by comparing the arguments, we can tell immediately which number is larger. No need for all the complex math at all (unless you have a very accurate symbolic representation of n for both numbers, and actually care which one of them is a teeny-weeny-bit larger than the other)