More About Irrationals and Reals 129
More About Irrationals and Reals
Let’s explore “infinity” for a few minutes. Then we’ll prove that the square root of 2 is irratio-
nal. Get into the mood for some serious abstract thinking!
Number lists
Mathematicians use the symbol א 0 (called aleph-null) to describe the number of elements in
the set N of natural numbers. This is the same as the number of elements in the set Z of inte-
gers, as we saw in the solution to the final practice exercise in Chap. 3. We can create “implied
lists” of both sets, and be confident that if we go far enough out, we’ll always hit any natural
number or integer we care to choose.
Cardinality of a set
The number of elements in a set is called the cardinality of the set. The cardinality of N is א 0 ,
and the cardinality of Z is also א 0. Even the elements of Q,the set of rationals, can be defined
in terms of an “implied list.” Figure 9-4 is an example. So, as counterintuitive as it may seem,
the cardinality of Q is the same as the cardinality of N or Z, that is, א 0. If we can make an
“implied list” of the elements in an infinite set, that set is said to be denumerably infinite (or
simplydenumerable), and by definition it has cardinality א 0.
The irrationals and reals can’t be “listed”
The elements of S or R cannot be denoted in any type of list. We can’t even make an “implied
list” of all the nonnegative irrational numbers smaller than 1 in decimal-expansion form.
Table 9-1, and the following explanation, should give you some idea of why this is so.
Suppose we try to list the irrational numbers between 0 and 1 (including 0, but not 1) by
writing down the numerals in their expanded-decimal forms. To the left of the decimal point,
every numeral will have a single 0 and nothing else. The first numeral will have an endless
string of digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to the right of the decimal point. Let’s
call those digits a 11 ,a 12 ,a 13 , and so on. The second number will have an endless string of digits
that we can call a 21 ,a 22 ,a 23 , and so on, different from the first string. We can keep on listing
irrational numbers like this forever. The nth number in our list (corresponding to the nth row
in Table 9-1) will be of the form
0.an 1 an 2 an 3 ...
Now imagine that we have listed one irrational number for every possible value of n. (This
could not actually be done by any mortal human, because it would take forever. But in the
world of mathematics, our imaginations let us do infinitely many tasks in a finite amount
of time!) Suppose that no two irrational numbers in this list are the same. It is tempting to
believe that this list of irrational numbers, taking the form of a matrix that extends forever to
the right and downward from what we see in Table 9-1, must contain all possible irrationals.
After all, there are infinitely many of them, and we haven’t listed any of them twice.
But no! Even this infinite list is not complete. There are still more irrationals. Here is one
of them. Imagine building an irrational number of this form:
0 .b 1 b 2 b 3 ...
