The PROOF: e and pi are transcendental

What people said

Recommends Mason · 2019-11 · Mathematics Stack Exchange · on “Is $e$ raised to an irrational number rational?”

Here's [this video] a sketch of some of the important arguments which I think you might enjoy. Mathologer does a good job of showing the important elements without letting the details (or a commitment to rigor) obsfucate the big picture.

Recommends clathratus · 2019-12 · Mathematics Stack Exchange · on “Why is $e$ transcendental, even though it can be written as a sum of algebraic numbers?”

What, essentially, you are wondering is why limits do not preserve algebraicity. Specifically, you have noted that even though the partial sums $$S_n=\sum_{k=0}^{n}\frac1{k!}$$ are all rational, their limit $\lim_{n\to\infty}S_n=e$ is not. And frankly, this is the case because of the way we have defined real numbers. That is, any $x\in\Bbb R$ can be written as a decimal $$x=p.q_1q_2q_3...$$ where $p\in\Bbb Z$ and $q_k\in\{0,1,...,9\}$ . Hence every number $$a_M=p.q_1...q_M=10^{-M}\lfloor 10^M x\rfloor$$ is rational and the sequence $(a_M)_{M\ge0}$ converges to $x$ . In other words, every real number, regardless of whether or not it's rational, has a sequence of rationals which converges to i

Recommends Gevorg Hmayakyan · 2021-09 · Mathematics Stack Exchange · on “Irrationlity of $e$”

Improving of the "proof" based on this answer brings as to regular proof:) There is a nice source here [this video]

Recommends D.R. · 2024-01 · Mathematics Stack Exchange · on “How hard is the proof of $\pi$ or $e$ being transcendental?”

Here's a beautifully presented video by Mathologer [this video] He presents the proof for e in a pretty detailed manner (takes ~20 minutes, not counting the initial motivation starting from the much easier irrationality proof of e), and sketches the proof for $\pi$

Cites Micael · 2025-02 · Mathematics Stack Exchange · on “How to prove that any number which can be written w/ the four operations and radicals on rationals are roots of polynomials with integer coefficients?”

Few days ago I watched this ( [this video] ) amazing video by Mathologer on $e$ and $π$ transcendence proof. Then I remembered a question I have about $e$ and $π$ : can these numbers be expressed as a finite combination of the 4 basic operations and radicals evaluated at rationals? I mean, are $e$ or $π$ equal to some "algebraic" expression like ${\sqrt[5]{3+{\sqrt7}}}$ or ${\sqrt[3]{31 - \sqrt5}}$ $+$ ${\sqrt[4]9}$ or any other possibly even harder and bigger with a finite expression with roots like these? My attempt was to show that these expressions are algebraic numbers (roots of polynomial equations with integer coefficients), but I failed. If we can prove so, $e$ and $π$ can't be written li