Transcendental constant

A number is transcendental if it is not the root of any non-zero polynomial with integer coefficients.

Common examples

• π ≈ 3.14159…, proved transcendental by Ferdinand von Lindemann in 1882.
• e ≈ 2.71828…, proved transcendental by Charles Hermite in 1873.
• e^π, transcendental (Gelfond's theorem).

Not transcendental

• √2, algebraic; root of x² − 2.
• φ (golden ratio), algebraic; root of x² − x − 1.
• Any rational number, trivially algebraic.

Why it shows up in EML

e drops out of the EML operator itself: eml(1, 1) = exp(1) − ln(1) = e. So the constant e is depth 1 in EML.

The paper's search replaces the traditional choice of "which transcendental to keep" (usually e or π) with a structural choice: keep 1, and let the operator produce transcendentals on demand.

Producing π from EML requires going through complex intermediates, see Euler's formula and why EML needs complex arithmetic.