Correctly speaking, an algebraic number is a complex number which is a root of a polynomial with integer coefficients. A complex number which is not algebraic is called a transcendental number.

For example, the square root of 2 is a root of the polynomial: x^2 - 2 = 0, so it is an algebraic number.

On the other hand, e and pi are transcendental, as was proved by Hermite and Lindemann. This is the basis for the fact that squaring the circle is impossible.

Algebraic numbers are not to be confused with a rational numbers, which is a number which is a quotient of two integers, like one-half.

More generally, an element x of a commutative ring L that has an identity element is called algebraic over a ring K (where L is an extension of K) with the same properties if there exists a nonzero polynomial f with coefficients in K such that f(x)=0. The most common case is to use the natural or rational numbers for K and the real or complex numbers for L.

A little known (or rather, rarely realized) fact is that if K is countable, then so is the set of all its (finite) polynomials, and thus (due to the fundamental theorem of algebra) the set of all algebraic numbers over K! This means that there are in fact no more polynomial roots of any degree, as well as finite sums, products or powers (or any combination thereof) of polynomial roots than there are natural numbers. While algebraic numbers are the most commonly encountered irrational numbers, they are in fact damn rare and there are far more transcendental ones.

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