Algebra Know-It-ALL

128 Irrational and Real Numbers

wires. The rational-number line is gray. When you pick it up, you find that it’s weightless. The real-number

line is black. When you lift it, you discover that it’s heavy. The real-number line contains more “stuff.”

Here’s a challenge!

Let’s try a little exercise that involves some plain-language logic, some set theory, and some Venn dia-

gram reading skill. Which of the following statements are true, based on our knowledge of the number

hierarchy?

• All rational numbers are real.


• All integers are real.


• Some integers are irrational.


• Some irrational numbers are real.


• All irrational numbers are real.

Solution

We can figure all of these out by looking at Fig. 9-2. The answers, bullet-by-bullet, are as follows.

• Set Q is entirely contained in set R. Therefore, all rationals are real.


• Set Z is entirely contained in set R. Therefore, all integers are real.


• Set Z is completely separate from set S. Therefore, no integers are irrational.


• Set S has elements in common with set R. Therefore, some irrationals are real.


• Set S is entirely contained in set R. Therefore, all irrationals are real.

0

1

2

Positive-

number

line

This point

corresponds to

the square root of 2

1 unit

1 unit

Figure 9-3 If you take a unit square and place it diagonally along

the positive-number line with one corner at the point

for 0, the opposite corner is at the point for 21/2,

which is irrational.