Algebra Know-It-ALL

The Number Hierarchy 127

S Q

R= Everything inside the large rectangle

ZN

S

N Q

Z

Figure 9-2 The relationship among the sets of real numbers (R),

irrationals (S), rationals (Q), integers (Z), and natural

numbers (N).

You can define a point between 1 and 2 that doesn’t correspond to any rational number. Take a square
measuring exactly 1 unit on each edge, known as a unit square, and place it on the number line so one
corner is at the point for 0 and the other corner is on the line between the points for 1 and 2, as shown in
Fig. 9-3. The opposite corner of the square falls exactly on the point for the square root of 2, because the
diagonal of a unit square is precisely 21/2 units long. (That fact comes from basic geometry.) The number
2 1/2 is irrational.
If you build a line using only the points corresponding to rational numbers, that line will be full of “holes.”

Are you still confused?

The rational numbers, when depicted as points along a line, are “dense,” but not as dense as the points on
a line can get. No matter how close together two rational-number points on a line might be, there is always
another rational-number point between them. But there are points on a continuous geometric number line that
don’t correspond to any rational quantity. The set of reals is more “dense” than the set of rational numbers.
“All right,” you say. “This game is strange, but I’ll play along. How many times more dense is the real-
number line than the rational line? Twice? A dozen times? A hundred times?” The answer might astonish
you. The set of real numbers, when assigned to points on a line, is infinitely more “dense” than the set of
rational numbers.
Here’s a crude analogy. Suppose you’re exploring the “planet Maths” and you stumble across a rational-
number line and a real-number line lying in a field. Both lines resemble straight, infinitely long, thin, rigid