The theorems just presented were probably not surprising to you and their proofs were pretty straightforward. The fun begins when we prove theorems that are less intuitive. So far, we have mostly been dealing with integers, but now it’s time to graduate to fractions. The rational numbers are those numbers that can be expressed as a fraction. To be more precise, we say r is rational if r = a/b, where a and b are integers (and b = 0). Rational numbers include, for example, 23/58, −22/7, and 42 (which equals 42/1). Numbers that are not rational are called irrational. (You may have heard that the numberπ =3.14159 . . . is irra-tional, and we will have more to say about that in Chapter 8.)
For the next theorem, it may be helpful to recall how to add frac-tions. It is easiest to add fractions when they have a common denomi-nator. For example,
Otherwise, to add the fractions, we rewrite them to have the same de-nominators. For example,
In general, we can add any two fractions a/b and c/d by giving them a common denominator, as follows:
We are now ready to prove a simple fact about rational numbers.
Theorem: The average of two rational numbers is also rational.
Proof: Let x and y be rational numbers. Then there exist integers a, b, c, d where x =a/b and y= c/d. Notice x and y have average
x+y
2 = a/b+c/d
2 = ad+bc 2bd
and that average is a fraction where the numerator and denominator are integers. Consequently, the average of x and y is rational.
Let’s think about what this theorem is saying. It says that for any two rational numbers, even if they are really, really close together, we can always find another rational number in between them. You might be tempted to conclude that all numbers are rational (as the ancient Greeks believed for a while). But, surprisingly, that’s not the case. Let’s consider the number√
2, which has decimal expansion 1.4142 . . . . Now, there are many ways to approximate√
2 with a fraction. For example,√ 2 is approximately 10/7 or 1414/1000, but neither of these fractions has a square that is exactly 2. But maybe we just haven’t looked hard enough?
The following theorem says that such a search would be futile. The proof, as is usually the case for theorems about irrational numbers, is by contradiction. In the proof below, we will exploit the fact that every fraction can be reduced until it is in lowest terms, where the numerator and denominator share no common divisors bigger than 1.
Theorem: √
2 is irrational.
Proof:Suppose, to the contrary, that√
2 is rational. Then there must exist positive integers a and b for which
√2=a/b
where a/b is a fraction in lowest terms. If we square both sides of this equation, we have
2=a2/b2 or equivalently,
a2 =2b2
But this implies that a2 must be an even integer. And if a2is even, then a must also be even (since we previously showed that if a were odd, then a times itself would be odd). Thus a = 2k, for some integer k.
Substituting that information into the equation above tells us that (2k)2 =2b2
So
4k2=2b2 which means that
b2 =2k2
and therefore b2 is an even number. And since b2 is even, then b must be even too. But wait a second! We’ve just shown that a and b are both even, and this contradicts the assumption that a/b is in lowest terms.
Hence the assumption that √
2 is rational leads to trouble, so we are forced to conclude that√
2 is irrational.
I really love this proof (as evidenced by the smiley) since it proves a very surprising result through the power of pure logic. As we will see in Chapter 12, irrational numbers are hardly rare. In fact, in a very real sense, virtually all real numbers are irrational, even though we mostly work with rational numbers in our daily lives.
Here is a fun corollary to the previous theorem. (A corollary is a the-orem that comes as a consequence of an earlier thethe-orem.) It takes ad-vantage of the law of exponentiation, which says that for any positive
numbers a, b, c,
ab
c
=abc For example, this says that
532
=56, which makes sense, since 532
= (5×5×5)×(5×5×5) =56
Corollary: There exist irrational numbers a and b such that ab is rational.
It’s pretty cool that we can prove this theorem right now, even though we currently know only one irrational number, namely√
2. We call the following proof an existence proof, since it will show you that such values of a and b exist, without telling you what a and b actually are.
Proof:We know that√
2 is irrational, so consider the number√ 2
√2
. Is this number rational? If the answer is yes, then the proof is done, by letting a=√
2 and b=√
2. If the answer is no, then we now have a new irrational number to play with, namely√
2
2, then, using the law of exponentiation, we get
ab=√2
√2√2
=√2
√2×√
2=√22=2
which is rational. Thus, regardless of whether√ 2
√2
is rational or irra-tional, we can find a and b such that abis rational. Existence proofs like the one above are often clever, but sometimes a little unsatisfying, since they might not tell you all the information that you are seeking. (By the way, if you are curious, the number √
2
√2
is irrational, but that’s beyond the scope of this chapter.)
More satisfying are constructive proofs, which show you exactly how to find the information you want. For example, one can show that ev-ery rational number a/b must either terminate or repeat (since in the long division process, eventually b must divide a number that it has previously divided). But is the reverse statement true? Certainly a ter-minating decimal must be a rational number. For example, 0.12358 = 12,358/100,000. But what about repeating decimals? For example, must the number 0.123123123 . . . necessarily be a rational number? The an-swer is yes, and here is a clever way to find exactly what rational num-ber it is. Let’s give our mystery numnum-ber a name, like w (as in waltz), so that
w=0.123123123. . . Multiplying both sides by 1000 gives us
1000w=123.123123123. . . Subtracting the first equation from the second, we get
999w=123 and therefore
w= 123999 = 33341
Let’s try this with another repeating decimal, where we don’t start repeating from the very first digit. What fraction is represented by the decimal expansion 0.83333 . . .? Here we let
x=0.83333. . . Therefore
100x=83.3333. . . and
10x=8.3333. . .
When we subtract 10x from 100x everything after the decimal point cancels, leaving us with
90x= (83.3333. . .)−(8.3333. . .) =75 Therefore
x= 75 90 = 5
6
Using this procedure, we can constructively prove that a number is rational if and only if its decimal expansion is either terminating or repeating. If the number has an infinite decimal expansion that doesn’t repeat, like
v=.123456789101112131415. . . then that number is irrational.