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

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