transcendentals

In mathematics, transcendentals refer to numbers that are not algebraic. An algebraic number is a root of a non-zero polynomial equation with integer coefficients. In simpler terms, an algebraic number can be expressed as the solution to a polynomial equation like ax^n + bx^(n-1) + ... + z = 0, where a, b, ..., z are integers.

Transcendental numbers, on the other hand, cannot be expressed as the solution to any such polynomial equation. This makes them fundamentally different from algebraic numbers. The set of transcendental numbers is much larger than the set of algebraic numbers, although proving a specific number is transcendental can be very difficult.

The first number to be proven transcendental was e, the base of the natural logarithm, by Charles Hermite in 1873. Later, Ferdinand von Lindemann proved that pi was transcendental in 1882. Other notable transcendental numbers include e^a and pi^a for any non-zero algebraic number a. The existence of transcendental numbers implies that there are infinitely many numbers that cannot be constructed using a ruler and compass, which has significant implications in geometry, such as the impossibility of squaring the circle.