Abstract
This chapter discusses systems of rational numbers and real numbers. It discusses addition, multiplication, ordering, and division of rational numbers. There are an infinite number of rational numbers between any two rational numbers. The set of all real numbers that are not rational is called the set of irrational numbers. It can be shown that between any two irrational numbers, there are an infinite number of irrational numbers. Because there are an infinite number of rational numbers between any two rational numbers, it appeared as though the rational numbers are rather tightly packed on the number line. However, there is a fundamental difference between the rational numbers and the real numbers. The real number line is continuous, whereas the rational number line is not continuous. The development of calculus is highly dependent on this special property of the real numbers. Any two rational numbers have a product and their product is a rational number as the denominators' being nonzero guarantees that the product has nonzero denominator.
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