### ALGEBRAIC AND TRANSCENDENTAL NUMBERS

It was known in ancient Greece (Pythagoras, 6th century B.C.) that the square root of 2 is irrational. However it does satisfy a polynomial equation with integer coefficients:

x - 2 = 0

*Definition*: A real number which satisfies a polynomial equation with integer coefficients is called **algebraic**. The set of algebraic numbers is often denoted by a capital A with a double bar down the left, but I don't have one of those available here, so I'll denote it **A**.

A real number which is not algebraic is called **transcendental**.

e.g. **A**: it satisfies x - 5 = 0

** A** : it satisfies x - x - 1 = 0

**Q** ** A** : satisfies nx - m = 0.

Are there *any* transcendental numbers ?

Hermite proved e to be transcendental in 1873.

Lindemann proved to be transcendental in 1882.

Meanwhile (1874 onwards) Cantor answered the question in a different way, by introducing the theory of sets and cardinal numbers.

Cantor's Theory

Foundations of Mathematics

History of Mathematics Module

Links to other History of Mathematics sites

[ Comments ] [ Module Leader ]

These pages are maintained by M.I.Woodcock.