Algebra Know-It-ALL

130 Irrational and Real Numbers

where the digit b 1 is different from a 11 in the first list entry, the digit b 2 is different from a 22 in

the second list entry, the digit b 3 is different from a 33 in the third list entry, and so on forever.

This new irrational number cannot be in the list denoted by Table 9-1. No matter which of

the entries in that list (with the a’s and subscripts) we choose, our new number (with the b’s

and subscripts) has at least one digit that doesn’t match.

When we can’t make a list—or even concoct an infinitely long “implied list” in the math-

ematical cosmos—of all the elements of a set, the set is said to be non-denumerable.

Are you confused?

The notion of non-denumerability is difficult to grasp. Don’t feel bad if you don’t fully understand it. It’s

a little like trying to envision space with more than three dimensions. Some things that can be defined in

Figure 9-4 All the elements in the set of rational numbers can be

arranged in an “implied list.” Follow the dashed “square

spiral” in this two-dimensional list, starting at the center

and going outward as shown. Eventually, you’ll hit the

box for any ratio of integers that can exist.