Transcendental proofs vs. Irrational proofs

Question

Why are proofs of the transcendence of certain numbers usually harder than irrationality proofs of those same numbers (for example, Lindemann's proof of the transcendence of pi vs. Niven's proof of the irrationality of pi?)

Answer 1

Proving a number irrational only involves showing that the assumption that the number is rational leads to a contradiction.

In other words, the number cannot be the root of a linear equation with integer coefficients.

To prove that a number is transcendental, you must show that the assumption that the number is a root of $any$ polynomial of $any$ degree with integer coefficients leads to a contradiction. This is far more difficult.

This is why the Greeks were able to show that $\sqrt{2}$ is irrational (this can be done using only geometry, not algebra), while it took until 1844 for the existence of transcendental numbers to be proved by Liouville. Hermite proved $e$ was transcendental in 1873, Cantor proved that almost all numbers are transcendental (by showing that the algebraic numbers formed a countable set), Lindeman proved that $\pi$ was transcendental in 1882, and so on.

Other names worthy of study are Weierstrass, Hilbert, Gelfond, Schneider, and Baker.