# An Innocent Investigation

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1 An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number is either even or odd, but why. Hardly anyone except the most hardened skeptic would even question that every number is either even or odd. But you might have asked why. Then again, maybe the two questions aren t that different. So, how do we know every number is either even or odd? Experience, for one thing. We know the numbers 2, 4, 6, 8, 10, 12, and so forth, are all even; and the numbers 1, 3, 5, 7, 9, 11, and so forth, are all odd. We see the pattern, namely odd, even, odd, even, odd, even, and so forth, and that s pretty convincing. For another thing, we all learned that numbers are even or odd at a very young age. We learned it so early that it probably didn t even occur to us at the time to question it. That s a sort of knowledge by authority. But even then the authorities probably gave us some reason why it s true, and by now we have forgotten that reason. What are we looking for? Where do we look to find out why all numbers are even or odd? How do we know when we ve found the answer? Are there any guidelines? Let s go blindly forward, and, maybe we ll find some answers. What do we know? Maybe if we look at something simpler we know, and analyze that, we ll get somewhere. Let s start with the statement made above: We know the numbers 2, 4, 6, 8, 10, 12, and so forth, are all even; and the numbers 1, 3, 5, 7, 9, 11, and so forth, are all odd. How do we know 2, 4, 6, 8, 10, 12, and so forth, are all even? What does it even mean that 2, 4, 6, 8, 10, 12, and so forth, are all even? There are a couple of answers that come to mind. One is that the number 2 evenly divides them. Another is that each of them is the sum of some other number taken twice. The second statement seems to depend on the concept of addition, while the first depends on the concept of division, so, maybe the second concept is simpler. Let s analyze it a little more. For instance, we re saying a number, like 10, is even, because it is the sum of some other number, 5, taken twice, that is, 10 is even since 10 = That looks like a pretty good definition. We can say it in words, as we just did, or we can say it more formally, using symbolism, as follows. A number n is said to be even if there exists a number m such that n = m+m. The phrase there exists a number n is an example of an existential quantifier. There is a standard notation for existential quantifiers, and we can use it to abbreviate this definition a little bit. Definition. A number n is even if m : n = m + m. We read that defining clause as there exists an m such that n equals m plus m. Note that the colon for such that is only one abbreviation that has been used for such that, but let s stick to it here for consistency. With this definition, we can show that the numbers 2, 4, 6, 8, 10, and 12, are all even, since 2 = 1+1, 4 = 2+2, 6 = 3+3, 8 = 4+4, 10 = 5+5, and 12 = The and so forth gives us a little 1

3 required that the next number n + 1 be of the form m + m, that is, that n + 1 be even. Now let s consider the claim that an even number is followed by an odd number. Let the even number be n, so that the next number is n + 1. Now, since n is even, we know there is a number m so that n = m + m. How are we going to conclude that the next number n + 1 is odd? In order to show that n + 1 is odd, according to our definition, we need to show that the number after it, n + 2 is even. But n was even, that is, n = m + m, therefore n + 2 = m + m + 2 = (m + 1) + (m + 1), so n + 2 is even, as required. That s a good explanation. Now that we ve found explanations, let s write them up formally for the record. We ll do that by stating the claim as a theorem along with the proof we just found. Theorem. The number after an odd number is even, and the number after an even number is odd. Proof. The first statement is the definition for odd number. Next, let n be an even number. Then n = m+m for some number m. Therefore, n + 2 = (m + 1) + (m + 1) is also an even number. Since the number after n + 1 is even, therefore, by the definition for odd number, n + 1 is odd. q.e.d. We now understand why the pattern goes odd, even, odd, even, odd, even, and so forth. We still have to figure out how to use that knowledge to show every number is either even or odd. General principles. Here s one way that might help. To figure out the parity of n, that is, whether n is even or odd, look at its predecessor n 1. If n 1 is odd, then n is even, but if n 1 is even, then n is odd. Thus if the predecessor n 1 is either even or odd, then so will n be, but with the opposite parity. That helps, but we still have two problems. (1) How do we know n has a predecessor? (2) How do we know the predecessor is either even or odd? Let s look at (1) first. It may be that n doesn t have a predecessor. That happens when n = 1. But, of course, 1 is the smallest number, and doesn t have a predecessor. In this even/odd investigation, that s not a problem since we know 1 is odd. But we have identified two general principles, the first statement so basic that we ll call it an axiom and not try to prove it, and the second we ll prove after we find another axiom. Axiom. 1 is a number that has no predecessor, that is, there is no number m such that 1 = m + 1. Theorem. Every number other than 1 does have a predecessor, that is, if n 1, then m : n = m+1. (Proof to come later.) These two statements take care of problem (1). But we still have problem (2) to address. There are a three of ways to go, all more or less equivalent. Infinite descent. This will be a long trip. We want to know n is either even or odd. If we knew its predecessor n 1 was either even or odd, we d be done. If n 1 were 1, we d be done, but, if not, we d need to know its predecessor n 2 was either even or odd. If that were 1, we d be done, but, if not, we d need to know its predecessor was either even or odd. And so forth. If at any stage we reach 1, we would be done, but if we never reach 1, then we would have an infinite decreasing sequence of numbers n, n 1, n 2,... that never ended. But, of course that can t be. We ve found another general principle of numbers. Axiom. There is no infinite decreasing sequence of numbers. This is a somewhat unsatisfactory axiom because it explicitly mentions infinity. It would be nice to have a general principle that didn t depend on the concept of infinity. Mathematical induction. A standard way to avoid this infinity is to appeal to a different general principle called mathematical induction. Although it s a different principle, it is logically equivalent to infinite descent. For mathematical induction, build from the ground up. In order to show that every number n is either even or odd, we start with the base case when n = 1. We know 1 is odd, so the statement is true for n = 1. Next, we show how to argue that if a statement is true for one value of n, it will 3

5 even, and the number after an even number is odd. Theorem. If a number has a predecessor, then when that number is even its predecessor is odd, but if that number is odd its predecessor is even. Proof. Let the predecessor be n so that the number is n + 1. Suppose that the number n + 1 is even, that is n + 1 = m + m for some m. Then by our definition for odd, that says the predecessor n is odd. Now suppose that the number n+1 is odd. Then n = m + m for some m. First note that m is not 1, for if m = 1, then n = 1 + 1, which implies n+1 = 1, that is, 1 has a predecessor, which it doesn t. Therefore m is not 1, and, thus, has its own predecessor, m = k + 1. Then n = k k + 1, so n = k + k, which says n is even. q.e.d Now we can prove the theorem. Theorem. No number is both even and odd. Proof. We can arrange the proof so that we use induction, but minimization is just a little simpler here. Suppose that some number is both even and odd. Then there is a smallest number n that is both even and odd. This number n is not 1, since we ve already proved that 1 is not even. Therefore n must have a predecessor. But then, since n is both even and odd, by the previous theorem, then its predecessor will be both odd and even, and that contradicts the minimality of n. Thus, there can t be a smallest number that s both even and odd, so there can t be any number that s both even and odd. q.e.d We can summarize our investigation on even and odd numbers their parity in a few sentences. Every number has a parity, that is, it is either even or odd, but not both. The number 1 is odd. The successor of a number has the opposite parity, also, the predecessor of a number, if it has one, has the opposite parity. History. Parity is one of the oldest concepts in formal mathematics. Euclid devotes part of Book IX of his Elements to the concept. propositions such as these three. He includes Proposition IX.21. If as many even numbers as we please are added together, then the sum is even. Proposition IX.26. If an odd number is subtracted from an odd number, then the remainder is even. Proposition IX.29. If an odd number is multiplied by an odd number, then the product is odd. Each of his propositions on even and odd numbers comes with a proof. His definition for even and odd numbers appear earlier in Book VII, the first book in the Elements on number theory. Definition VII.6. An even number is that which is divisible into two equal parts. Definition VII.7 An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number. His definition of even number is the same as ours, but he includes two clauses in his definition of odd number. From his use of this definition in various propositions, it s clear that he means them to be equivalent, that is, he assumes that a number is not even if and only if it differs from an even number by 1. But he never proves that equivalence and he never made that an explicit axiom. The first few books of the Elements are about plane geometry and Euclid includes many axioms for geometry. But he has no axioms for number theory, and it s unclear why. Perhaps he didn t think axioms were necessary for numbers. More likely, he just didn t devote as much time to the foundations of number theory as he did for the foundations of geometry. It appears that he restructured plane geometry into a logical order, but he didn t do the same for number theory. The propositions stated above from Book IX on even and odd numbers are very elementary, but they don t come at the beginning of the discussion of number theory, which begins in Book VII, but are inserted in the last third of Book IX. He might have left them out altogether if they weren t needed to prove his last proposition in Book IX. Proposition IX.36. If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all be- 5

6 comes prime, and if the sum multiplied into the last makes some number, then the product is perfect. (A number is perfect if it is the sum of its proper divisors, Definition VII.22.) It is likely that the propositions about even and odd numbers were taken from some earlier work on number theory, now lost, as a group, and Euclid did not critically analyze them as he did for his propositions on geometry. Indeed, it is likely that these propositions on even and odd numbers are among the earliest propositions that appear in the Elements, and that go back a century or two before Euclid to the Pythagoreans. 6

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