Vignette 7
The Square Root of 2 and Other Irrational Numbers

You have probably heard the claim that is an irrational number. That means that cannot be expressed as a quotient of two integers.  How many other irrational numbers can you think of?  While you may be able to come up with a few other examples, it is much easier for us to think of examples of rational numbers!  Which kind of number is more plentiful?  In this vignette, we will see not only why is irrational, but we will also see evidence that the irrational numbers are far more plentiful than rationals!

is Irrational

To see that must be irrational, we will use the method of proof by contradiction introduced in the vignette on prime numbers.  Suppose that were a rational number.  Recalling that  refers only to the positive square root of 2, this means that  could be written as the quotient of two positive integers, say , where the fraction is in lowest terms -- that is, a and b have no common factors.  Then simple algebra gives us the following:

From this we can conclude that a2 is an even number -- that is, a2 has 2 as a factor.  But since , it follows that 2 must actually be a factor of a.  That is, a itself is an even number.  But then a2 actually has 4 as a factor, and thus 2b2 has 4 as a factor.  Thus b2 has 2 as a factor, so b2 is an even number.  Repeating the argument just used for the number a, we see that b must also be an even number.  But then the fraction would not be in lowest terms after all, since both a and b have 2 as a factor.  This contradiction shows that cannot be a rational number, and instead is irrational.

How Many Irrational Numbers are There?

The argument above is not unique to the number 2, and actually applies to any prime number.  Thus, if p is a prime number, then  is an irrational number.  Since there are infinitely many primes (as seen in an earlier vignette), it follows that there are infinitely many irrational numbers.

Now in the vignette titled How Big is Infinity?, we saw that the set of rational numbers is countably infinite.  Could it be that the set of irrational numbers is also countably infinite?  (Pay attention now, because this is tricky -- but really cool!)  We also saw that the set of numbers in [0, 1] is uncountable, and of course it follows that the (larger) set of all real numbers must be uncountable as well.  But the set of all real numbers is simply the union of the set of rational numbers with the set of irrational numbers.  So if the set of irrational numbers were countable, then we would have a union of two countable sets producing an uncountable set.  We will now see that this is impossible.

Suppose we have two sets A and B, both of which are countable.  To see that the set is also countable, it suffices to exhibit a one-to-one correspondence between the set of positive integers and the set .  Since  and are countable, we have such one-to-one correspondences for these sets, say

and
respectively.  To build the needed one-to-one correspondence between the positive integers and , we need only do the following:
This argument shows that a union of two countable sets is countable.

Consequently, if the set of irrational numbers were countable, we would have an argument that the set of real numbers is countable -- since the set of real numbers would then be a union of two countable sets.  But we know that the set of real numbers is uncountable.  Thus, the set of irrational numbers is uncountable.

Where ARE All of Those Irrationals?

OK, we now know that there are uncountably many irrational numbers.  That means that there are far more irrationals than rationals.  But we are in the uncomfortable position of not knowing so many of them!  Well, we are pretty comfortable with our understanding of rational numbers as fractions -- that is, as quotients of integers.  Using decimal representation of numbers, the rational numbers are precisely those whose decimal representation includes a repeating block of digits at the end.  (We are counting the terminating decimals as having a repeating 0 at the end.)  The irrational numbers, then, are those whose decimal representations do not terminate or repeat.  Some examples include the famous numbers

and

The rational numbers and the irrationals are interwoven on the number line, so that between any two rationals (no matter how close they are) there is an irrational, and between any two irrationals (no matter how close they are) there is a rational.  Thus, there are never any blocks of "consecutive" rationals or irrationals.  As an example, between the rational numbers  and , we can find the irrational number .  And between the irrational numbers 7.4367110893... and 7.4367110894... we can find the rational number .

Further Exploration