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The set of algebraic numbers is countably [[infinite]].  This follows from the fact that the set of finite degree polynomials with integer coefficients is countably [[infinite]]. Conversely, the set of transcendental numbers is [[uncountable]] because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

The set of algebraic numbers is countably [[infinite]].  This follows from the fact that the set of finite degree polynomials with integer coefficients is countably [[infinite]]. Conversely, the set of transcendental numbers is [[uncountable]] because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

The algebraic numbers are closed under [[addition]], [[subtraction]], [[multiplication]] and [[division]]. These properties show that the algebraic numbers are a [[field]]. In addition every root of a polynomial of finite degree with algebraic coefficients is an algebraic number, so it is an [[algebraically closed]] field. The field of algebraic numbers is denoted by <math>\mathbb{A}</math> or <math>\bar{\mathbb{Q}}</math>.

The algebraic numbers are closed under [[addition]], [[subtraction]], [[multiplication]] and [[division]]. These properties show that the algebraic numbers are a [[Field_(mathematics)|field]]. In addition every root of a polynomial of finite degree with algebraic coefficients is an algebraic number, so it is an [[algebraically closed]] field. The field of algebraic numbers is denoted by <math>\mathbb{A}</math> or <math>\bar{\mathbb{Q}}</math>.

If a is an algebraic number and t is a transcendental number, then a+t, a-t, at, a/t, a^t, and t^a are all transcendental.  

If a is an algebraic number and t is a transcendental number, then a+t, a-t, at, a/t, a^t, and t^a are all transcendental.