How a 19th-century theorem proved that pi is not a simple fraction
In short: Most numbers are transcendental, meaning they cannot be roots of any non-zero polynomial with rational coefficients. Yet, proving this for specific constants like pi or e is notoriously difficult. The Lindemann–Weierstrass theorem provides the rigorous machinery to bridge this gap, turning complex analysis into a definitive statement about the nature of numbers.
The Lindemann–Weierstrass theorem is a cornerstone of transcendental number theory. It establishes that if $\alpha_1, \dots, \alpha_n$ are algebraic numbers that are linearly independent over the rationals, then $e^{\alpha_1}, \dots, e^{\alpha_n}$ are algebraically independent. In simpler terms, this means you cannot create a polynomial equation using these exponential values that would equal zero. This result is a massive generalization of earlier work; for instance, Charles Hermite first proved a simpler version where the exponents were restricted to rational integers.
The theorem's most famous application is proving the transcendence of $\pi$. The logic follows a proof by contradiction: if $\pi$ were algebraic, then $i\pi$ (an algebraic number) would result in $e^{i\pi}$ being transcendental. However, Euler's identity tells us that $e^{i\pi} = -1$. Since $-1$ is clearly algebraic, $\pi$ cannot be algebraic. Therefore, $\pi$ must be transcendental. This same logic allows mathematicians to prove that trigonometric functions like $\sin(\alpha)$, $\cos(\alpha)$, and $\tan(\alpha)$ are also transcendental for any non-zero algebraic $\alpha$.
The progression of this theorem highlights the evolution of modern mathematics. Ferdinand von Lindemann first proved that $e^\alpha$ is transcendental for non-zero algebraic $\alpha$ in 1882, which was the key to $\pi$. Karl Weierstrass later generalized this in 1885. The theorem has since been extended by the Gelfond–Schneider theorem and Baker's theorem, with the ultimate goal being a full generalization via Schanuel's conjecture. While the theorem is well-established, some analogues—such as those involving the modular function $j$ conjectured by Daniel Bertrand in 1997—remain unsolved mysteries.
The mechanics of the proof often rely on constructing specific polynomials and using the fundamental theorem of symmetric polynomials. By defining polynomials $f_i(x)$ and evaluating integrals $I_i(s)$, mathematicians can create a situation where a sum of terms must be non-zero. By showing this sum is an algebraic integer divisible by a large prime $p$ but not by $p!$, they reach a contradiction that confirms the transcendence of the numbers in question.
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