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*are all integers.

_{n}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.

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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