The number system consists of several different sets of numbers:

1. Natural Numbers (N) - the set of counting numbers, {1, 2, 3, ...}
2. Whole Numbers (W) - the set of counting numbers and zero {0, 1, 2, 3, ...}
3. Negative numbers and zero first appeared in the seventh century A.D.
   Integers (I) - the set of positive and negative numbers and zero {..., -3, -2, -1, 0, 1, 2, 3, ...}. Interesting Fact
4. Rational Numbers - numbers that can be written in the form where a and b are integers and b0. Rational Numbers include all integers, fractions, perfect roots and all terminating and repeating decimals. (All terminating and repeating decimals can be expressed as fractions.)
   Examples:
   
5. Irrational Numbers - numbers that cannot be expressed as fractions. The decimal expansion of an irrational number neither terminates or repeats. Therefore, irrational numbers include all non-perfect roots, all non-terminating, non-repeating decimals and =3.1415....
   Examples:
6. Real Numbers () - all numbers that can be expressed as decimals.
   

   Real numbers correspond to every point on the number line and include all rational and irrational numbers.

   
   
7. Imaginary Numbers - the square roots of negative numbers.
   Go to the site http://www.csun.edu/~hcmth014/comics/cb23.html and check out the Calvin and Hobbs comic strip for Hobb's hilarious explanation of imaginary numbers.
   
   Previously, we learned that an equation such as has no solution in the set of real numbers since in calculating the no integer multiplied by itself equals -16. However, by extending the number system, we can give meaning to the solution of this equation. We do this by defining the number, i, with the property that .
   "Since there is no real number () with the property that its square is negative, the number i is not a real number, . It cannot be expressed as a decimal, and it cannot be represented by a point on the number line. For these reasons, the square roots of negative numbers were called imaginary numbers. This is an unfortunate name because it suggests that these numbers are somehow less valid than the real or decimal numbers to which we are accustomed. However, all numbers are imaginary in the sense that the are abstractions. Once mathematicians had learned to understand and work with this new kind of number, they found that the numbers had many applications in science, egineering, and electronics." (Kelly, B., Alexander, B., Atkinson, P. and Ditto, G. Mathematics 12. (1991). Don Mills, Ont.: Addison and Wesley Publishers Ltd.) For further explanations regarding the existence and applications of imaginary numbers, go to the following site: http://www.math.toronto.edu/mathnet/answers/imaginary.html.