mathematics, and in which problems can be neatly worked out, simply defy any attempt at “seeing them in
the mind’s eye.” It’s good enough to remember that there are more real numbers than rational numbers—a
lot more.
It can help if you stop thinking of “infinity” as something you can count toward. Instead, think of
“infinity” as an expression of the size of a set. The cardinality of the set R of real numbers is greater than
א 0 , the cardinality of N, Z, or Q.
You might wonder if there are any “infinities” larger than the cardinality of the set of reals. The mathema-
tician Georg Cantor’s answer to this question was “Yes, infinitely many!” He called these “infinities” transfi-
nite cardinals. During his lifetime, Cantor was scorned by some of his fellow mathematicians for his theory
of transfinite cardinals. Now they are commonly accepted in advanced mathematics.
Here’s a big challenge!
You are now invited to follow along with an “extra-credit” proof. It will take some time and effort to
understand it. But why not try it? It doesn’t involve anything more complicated or sophisticated than facts
of arithmetic you already know. Let’s prove that the value of 21/2 cannot be represented as a ratio of integers
in lowest terms, and therefore that it’s an irrational number. We’ll need two lemmas. Let’s accept them on
faith. Both of them can be proved, but that would be a distraction right now.
- The square of an integer is always an integer. Let’s call this the integer-squared rule.
- The square of an odd integer is always an odd integer. Let’s call this the odd-integer-squared rule.
Solution
Whenever we suspect that a proof might be involved, it’s tempting to try reductio ad absurdum. Let’s use it
now. Table 9-2 does the job. If you can’t follow this proof as a whole, don’t worry. Try to understand each
step, one at a time.
Table 9-1. If we try to create an “implied list” of all the irrational numbers
between 0 and 1 as endless, nonrepeating decimal expansions, we are destined
to fail. Each digit in the bottom row (b with subscript) is chosen so that it’s
different from the boldface digit above it in the same column (a with subscript).
Mathematically, for every positive integer subscript n in the table, bn≠ann,
the endless decimal 0.b 1 b 2 b 3 ... can’t be in the “a” list, even though that list is
infinitely long.
- a 11 a 12 a 13 a 14 a 15 a 16 a 17 →
- a 21 a 22 a 23 a 24 a 25 a 26 a 27 →
- a 31 a 32 a 33 a 34 a 35 a 36 a 37 →
- a 41 a 42 a 43 a 44 a 45 a 46 a 47 →
- a 51 a 52 a 53 a 54 a 55 a 56 a 57 →
- a 61 a 62 a 63 a 64 a 65 a 66 a 67 →
- a 71 a 72 a 73 a 74 a 75 a 76 a 77 →
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ →
0.b 1 b 2 b 3 b 4 b 5 b 6 b 7 →
More About Irrationals and Reals 131
