1.1Real numbers and the real line

Real Numbers

Much of calculus is based on properties of the real number system. Real numbers are numbers that can be expressed as decimals, such as

The dots in each case indicate that the sequence of decimal digits goes on forever. Every conceivable decimal expansion represents a real number, although some numbers have two representations. For instance, the infinite decimals and represent the same real number 1. A similar statement holds for any number with an infinite tail of 9’s.

The real numbers can be represented geometrically as points on a number line called the real line.

The symbol R denotes either the real number system or, equivalently, the real line. 

The properties of the real number system fall into three categories: algebraic properties, order properties, and completeness. The algebraic properties say that the real numbers can be added, subtracted, multiplied, and divided (except by 0) to produce more real numbers under the usual rules of arithmetic. You can never divide by 0.