Worked solution: Lindemann's transcendence proof of $\pi$ settles squaring the circle (1882)

Frequently asked questions

Why is proving

e

transcendental easier (or earlier) than proving

\pi

transcendental?

Because $e$ is directly defined by its own power series and exponential growth rate, which meshes naturally with the integral estimates Hermite used. Extending the method to $\pi$ requires handling $e^\alpha$ for an arbitrary algebraic $\alpha$ (via Euler's identity, as the next step shows) — a genuinely harder technical generalization, which took Lindemann nine more years.

Source

Ferdinand von Lindemann, 1882

Über die Zahl π

Step 2 of 6: The template: Hermite's 1873 proof that

e

is transcendental

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In plain words

Proving a specific number is transcendental is hard precisely because you have to rule out every possible polynomial with rational coefficients, not just check a handful of guesses. Charles Hermite found a way to do this for $e$ in 1873: build a special auxiliary expression, made from integrals, that — if $e$ satisfied some integer-coefficient polynomial equation — would have to equal a nonzero whole number.

The trick is that clever estimates also show this same expression must be smaller than $1$ in size once the construction is tuned large enough. A nonzero whole number smaller than $1$ cannot exist, so the assumption (that $e$ satisfies such an equation) must be false. Lindemann's entire strategy for $\pi$ , in the steps ahead, borrows this template directly.

a_0 + a_1 e + a_2 e^2 + \cdots + a_n e^n = 0,\ a_i\in\mathbb{Z} \implies a_0=a_1=\cdots=a_n=0

Detailed analysis

Hermite's 1873 paper 'Sur la fonction exponentielle' proves $e$ is transcendental: no nonzero integers $a_0,\ldots,a_n$ satisfy $a_0+a_1e+a_2e^2+\cdots+a_ne^n=0$ . The method constructs an auxiliary polynomial $f(x)$ of high degree (built using a large prime $p$ as a parameter) and studies the quantities $J_k=\int_0^k e^{k-x}f(x)\,dx$ for $k=0,1,\ldots,n$ ; integrating by parts repeatedly turns each $J_k$ into a sum involving $e^k$ and derivatives of $f$ evaluated at $0$ and $k$ .

The combination $a_0J_0+a_1J_1+\cdots+a_nJ_n$ can be shown, by construction, to equal a nonzero integer not divisible by the chosen large prime $p$ (so certainly nonzero) if the assumed relation among powers of $e$ held. But separate estimates of the integrals show the same combination has absolute value tending to $0$ as $p\to\infty$ . A nonzero integer cannot have arbitrarily small absolute value, so the assumed relation cannot exist: $e$ is transcendental.

This proof supplies the template — auxiliary integral, integer-vs-small-estimate contradiction — that Ferdinand von Lindemann adapted nine years later to handle not just $e$ itself but $e^\alpha$ for any nonzero algebraic $\alpha$ , the key extension needed for $\pi$ .

Terms in this step

Auxiliary function method: A proof technique that constructs a specially designed function or integral whose value can be bounded two contradictory ways — forced to be a nonzero integer by one argument, and forced arbitrarily small by another — to derive a contradiction.

Knowledge used in this step

Exponential and logarithmic functions