 #### What are rational, irrational, real, imaginary, complex, algebraic, and transcendental numbers?

The set of counting, or natural, numbers, N Î {1, 2, 3, 4, 5, 6, ...}.

The integers are the set of natural numbers, their reflections (negatives), and zero; Z Î {..., -3, -2, -1, 0, 1, 2, 3, ...}. Consequently, the complete set of integers contains the set of natural numbers.

A rational number is defined as the ratio (division) of two integers, as long as the denominator is not zero; for example, 1/2, -15/4, and so on. It should also be noted that all integers are rational numbers; for example, the rational number, -20/4=-5, is integer.

A number that cannot be be written as the ratio of two integers is called irrational.

The complete set of real numbers, R, is made up exclusively of rational and irrational numbers .

An algebraic number is the root (solution) of the polynomial equation

anxn + an-1xn-1 + ... + a2x2 + a1x + a0 = 0, where ak Î Z, and an ¹ 0.

By definition all rational numbers, a/b, are algebraic, as they are the root of the linear equation, bxa=0. Although algebraic numbers can be irrational, not all irrational numbers are algebraic. The set of irrational numbers that are not algebraic (the root of a polynomial with integral coefficients) are called the set of transcendental numbers; examples would include p and e.

In order to solve equations like, x² + 1 = 0, we require the set of imaginary numbers, I.

By combining real and imaginary numbers we produce the set of complex numbers, C, which allows us to find all three solutions of the equation x3=1 and solve equations like, sin(x)=2.

Johann Carl Friedrich Gauss (1777-1855), perhaps one of the greatest mathematicians to have lived, proved that the solution of any polynomial equation, with complex coefficients, is closed with the set of complex numbers. In other words, there will never be any need of other sets of numbers outside of complex numbers. This result is so important that it is called the Fundamental Theorem of Algebra.