A real number x is called algebraic if there exists a non-zero polynomial p with

Question

A real number x is called algebraic if there exists a non-zero polynomial p with integer coefficients such that p(x) = 0. For example, all rational numbers are algebraic, since if w = r/g is a quotient of two integers r and q, we have q omega - r = 0. There are also irrational numbers that are algebraic, as squareroot 2 is a solution to the equation x^2 - 2 = 0. Real numbers that are not algebraic are called transcendental. Show that there exist transcendental numbers. You may use that the set of all real numbers is uncountable, and that any non-zero polynomial has only finitely many roots.

Explanation / Answer

If a number tt is algebraic, it is the root of some polynomial with integer coefficients. There are only countably many such polynomials (each having a finite number of roots), so there are only countably many such tt. Since there are uncountably many real (or complex) numbers, and only countably many of them are algebraic, uncountably many of them must be transcendental.

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