By Sheldon Axler
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0 1 Two key points on the real line. 1 The Real Line 3 Once the points 0 and 1 have been chosen on the line, everything else is determined by thinking of the distance between 0 and 1 as one unit of length. For example, 2 is one unit to the right of 1. Then 3 is one unit to the right of 2, and so on. The negative integers correspond to moving to the left of 0. Thus −1 is one unit to the left of 0. Then −2 is one unit to the left of −1, and so on. 3 2 1 0 1 2 3 Integers on the real line. 1 If n is a positive integer, then n is to the right of 0 by the length obtained by dividing the segment from 0 to 1 into n segments of equal length.
If you want to see how this result was computed, click Show steps on the right side of the result box and you will see (in more detail that you probably want) how the distributive property was used multiple times to produce the result. The next example is presented mainly to show that parentheses must sometimes be used with symbolic calculators even when parentheses do not appear in the usual mathematical notation. example 11 Use Wolfram|Alpha to simplify the expression 1 y−b − b As shown here, Wolfram|Alpha does not mind if you insert extra spaces to make the input more easily readable by humans.
The undefined intuitive notions (such as “no gaps”) can be made precise using more advanced mathematics. In this book, we let our intuitive notions of the real line serve to define the system of real numbers. Is Every Real Number Rational? We know that every rational number corresponds to some point on the real line. Does every point on the real line correspond to some rational number? In other words, is every real number rational? If more and more labels of rational numbers were placed on the figure above, the real line would look increasingly cluttered.