An algebraic number is any real number that is a solution of some single-variable polynomial equation whose coefficient s are all integer s. While this is an abstract notion, theoretical mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.
The general form of a single-variable polynomial equation is:
a + a 1 x + a 2 x 2 + a 3 x 3 + ... + a n x n = 0
where a , a 1 , a 2 , ..., a n are the coefficients, and x is the unknown for which the equation is to be solved. A number x is algebraic if and only if there exists some equation of the above form such that a , a 1 , a 2 , ..., a n are all integers.
All rational number s are algebraic. Examples include 25, 7/9, and -0.245245245. Some irrational number s are also algebraic. Examples are 2 1/2 (the square root of 2) and 3 1/3 (the cube root of 3). There are irrational numbers x for which no single-variable, integer-coefficient polynomial equation exists with x as a solution. Examples are pi (the ratio of a circle's circumference to its diameter in a plane) and e (the natural logarithm base). Numbers of this type are known as transcendental number s.