Transcendental Numbers

Transcendental numbers, named after the Latin expression meaning to climb beyond, are complex numbers that exist beyond the realm of algebraic numbers. Mathematicians have defined algebraic numbers as those that can function as a solution to polynomial equations consisting of x and powers of x. Two well-known examples of transcendental numbers are the mathematical constants π (pi, which approximately equals 3.14) and e (sometimes called Euler’s number or Napier’s constant, which approximately equals 2.718). In 1844, French mathematician Joseph Liouville (1809–82) was the first person to prove the existence of transcendental numbers.

Much earlier, German mathematician Gottfried Wilhelm Leibniz (1646–1716) was probably the first mathematician to consider the existence of numbers that do not satisfy the solutions to polynomial equations. Then, in 1744, the Swiss mathematician Leonhard Euler (1707-83) established that a large variety of numbers (for example, whole numbers, fractions, imaginary numbers, irrational numbers, negative numbers, etc.) can function as a solution to a polynomial equation, thereby earning the attribute algebraic. However, Euler pointed to the existence of certain irrational numbers that cannot be defined as algebraic. Thus, , π, and e are all irrationals, but they are nevertheless divided into two fundamentally different classes. The first number is algebraic, which means that it can be a solution to a polynomial equation. For example, is the solution of x² - 2 = 0.

However, π and e cannot solve a polynomial equation, and are therefore defined as transcendental. While π, which represents the ratio of the circumference of a circle to its diameter, had been known since antiquity, its transcendence took many centuries to prove: In 1882 German mathematician Carl Louis Ferdinand von Lindemann (1852–1939) finally solved the problem of squaring the circle by establishing that there was no solution. There are infinitely many transcendental numbers, as there are infinitely many algebraic numbers. However, in 1874, German mathematician Georg Ferdinand Ludwig Philipp Cantor (1845–1918) showed that the former are more numerous than the latter, suggesting that there is more than one kind of infinity.

See also e (number); Irrational number; Pi; Polynomials.