In modern mathematics, more and more stress is placed on the context in which statements are true. In elementary mathematics this generally means an emphasis on a clear understanding of which number systems possess certain properties. We begin, then, by describing briefly the number systems with which we will be concerned. These number systems have developed through successive enlargements of previous systems.

At one time a "number" meant one of the natural numbers: 1,2,3,4,5,6,... . The next numbers to be introduced were the fractions:1/2, 1/3, 2/3, 1/4, 3/4, 1/5,..., and later the set of numbers was expanded to include zero and the negative integers and fractions. The number system consisting of zero and the positive and negative integers and fractions is called the system of rational numbers, the word "rational" being used to indicate that the numbers are ratios of integers. The integers themselves can be thought of as ratios of integers since 1=1/1, -1= -1/1, 2=2/1, -2=-2/1, 3=3/1, etc.

The need to enlarge the rational number system became evident when mathematicians proved that certain constructible lengths, such as the lengthof a diagonal of a unit square, cannot be expressed as rational numbers. The system of real numbers then came into use. The real numbers include all the natural numbers; all the fractions; numbers such asand which represent constructible lengths; numbers such asand  which do not represent lengths constructible from a given unit with compass and straightedge; and the negatives of all these numbers. Modern technology and science make great use of a still larger number system, called the complex numbers, consisting of the numbers of the form a + bi with a and b real numbers and i² = -1.