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:
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 
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