Infinity. The infinite! No other question has ever moved so profoundly the spirit of man. David Hilbert

Transcription

1 ℵ ℵ The infinite! No other question has ever moved so profoundly the spirit of man. David Hilbert

2 ℵℵ The Mathematics of the Birds and the Bee Two birds are racing towards each other in the heat of passion. They are initially km from each other, and they are flying at the speed of.5 km/min. A voyeuristic bee, which can fly at a speed of km/min, starts with one of the birds and flies towards the other. When it reaches the second bird, it turns around and flies back towards the first bird again. The bee keeps this up until the two birds meet. How far does the bee travel before this happens? (The answer is on the bottom of page 88.) I ll bee there beefore the bird. I hate rejection. Nothing stings like double rejection! 84

3 ℵℵ An Early Encounter with the Infinite The paradoxes of Zeno of Elea, a philosopher in ancient Greece, reveal some early encounters with the notion of infinity and its strange properties. About 45 B.C., he made the following assertion known as Zeno s racecourse paradox. Zeno s Assertion: A runner can never reach the end of a racecourse in a finite time. End Statement. The runner must first pass the point located halfway between herself and the finish line before she can finish the race.. It will take a finite time to reach the point. 3. Once reached, there is another halfway point 4 which the runner must reach before she can finish. 4. There are an infinite number of such halfway points which the runner must reach and each will take a finite time. 5. The total time for the race is infinite. Reason is between the runner and the finish line. It is a finite distance from to. The remaining interval is divided in half. Statements,, and 3 repeated an infinite number of times. The sum of an infinite number of finite times is infinite. But we know from experience that the runner can reach the finish line. Where is the flaw in Zeno s argument? 85

4 ℵℵ The Flaw in Zeno s Paradox The following story may help you discover where Zeno s argument breaks down. A famous pirate, Long John Glitter, found a cylindrical bar of gold one meter in length. To divide the bar into more manageable units he cut it in half (perpendicular to its axis of symmetry). One piece he placed in his treasure chest. The remaining piece was cut in half and one of those pieces was then placed in the chest. Again the remaining piece was cut in half and one of the pieces was placed in the chest. This process was repeated n times. Write the length of the n th piece placed in the chest and the length of the remaining piece. Write as a sum of fractions the total length in meters of the first n pieces placed in the chest. Write an expression for the total length of the first n pieces in the chest. Suppose this process could be continued an infinite number of times. Would the total length of the Pieces of the gold bar ever exceed meter? 86 This example shows that the sum of an infinite number of finite quantities may indeed be finite. This calls into question the reason supporting the fifth statement in Zeno s argument above. Paradoxes such as this one prompted mathematicians to put mathematics on firmer foundations by specifying a basic set of self-evident truths or axioms from which all mathematical theorems could be logically and unambiguously derived. This would (hopefully) expose all hidden assumptions and remove paradoxes. Over two millennia later, deeper paradoxes arose which shook the axiomatic foundations of mathematics more profoundly than the challenges made by Zeno. One such paradox appears on page 3.

5 ℵℵ The Harmonic Series An infinite series is a sum of a sequence with an infinite number of terms. When the sum is finite, the sequence is said to converge. Otherwise the series is said to diverge. For example, the series generated by the lengths of the pieces thrown into the basket (see the previous page) is given by: n It converges (to ), but the series clearly does not converge. It is important to recognize that there is a limit implicitly involved here. For example, the value of n is defined to be the limit of the sum of the first n terms as n approaches infinity. That is, the limit of this sequence is the limit of the sequence of partial sums,, +, , The study of convergence is an important one, and often much of an undergraduate course is devoted to understanding how one can tell when a series will converge. The harmonic series is the series: n For many reasons, it comes up repeatedly in many different fields of higher mathematics. There is a beautiful proof that this famous series diverges. If you have never thought about it before, you might want to ponder it yourself (or even sleep on it) before reading this remarkable argument. 87

6 ℵℵ The Classical Proof that the Harmonic Series Diverges This proof is very slick. We ignore the, and instead sum / + /3 + /4 +. By the time we get to /, the partial sum is equal to /. By the time we get to /4, the partial sum is greater than /. By the time we get to /8, the partial sum is greater than 3/.... n By the time we get to/, the partial sum is greater than n/. This means that if we are given some positive number, we can go far enough to make the partial sum bigger than this number. The proof works by considering intervals between adjacent powers of two. Sum between / n- and / n Lower Bound Value / = / = / /3 + /4 > /4 + /4 = / /5 + /6 + /7 + /8 > /8 + /8 + /8 + /8 = / /( n- + ) + +/( n - ) +/ n > / n + + / n + / n = / Adding all of these together, we get: n terms / + /3 + /4 + + / n > / + / + + / = n/ That is, the partial sums increase without limit as n increases. 88

7 ℵℵ There are many variations on this result. Here are a few. Variation Instead of evaluating we can try to evaluate Each term in the second series is only a tiny tiny bit smaller than the corresponding term in the harmonic series. But remarkably, the second sum converges! It converges to something very large, though. If you want to see how large and know a little about integration, you can work it out yourself, using the information that the sum is approximately the same as the following integral: x. dx Variation Instead of evaluating , we can try to sum the reciprocals only 3 4 of those numbers with no 9 s in their decimal representation. So the sum would begin: It doesn t seem as though we ve thrown out many numbers, but strangely, this sum also converges. 89

8 ℵℵ Variation 3 This time we will just sum the reciprocals of the primes, so our series will begin: Now we ve thrown out a lot of numbers. There aren t that many primes there are only 5 less than, 68 less than, and 78,498 less than,, and they keep on getting rarer. But amazingly, especially in light of the previous variation, this sum diverges! Rarer than primes are the so-called twin primes, which are any pair of prime numbers that differ by, such as 5 and 7 or 4 and 43. The series of the reciprocals of all twin primes converges. This tells us that twin primes are very rare indeed. On the other hand, a famous conjecture in number theory, the twin prime conjecture, states that there are an infinite number of pairs of twin primes, and the evidence for this conjecture is overwhelming (but not conclusive). Variation 4 In Prime Numbers in Number Theory (p. 6), we showed using a simple method that there are an infinite number of primes. Leonhard Euler (77-783), the great Swiss mathematician, had another proof based on the fact that the harmonic series diverges. The proof is subtle, but you should be able to figure it out. We use the indirect method we assume the opposite of what we want to prove, and try to find a contradiction. Imagine that there are only a finite number of primes, p, p, p,, p. 3 r Then every positive integer n can be expressed (uniquely) as Consider the product P given by: p p p p p p p p p m m m 3 m r n = p p p 3 p r where the m are integers. 3 Since the number of primes is assumed to be finite, P is finite. (Why? ) i r r r

9 ℵℵ Now imagine that you expanded out all of the brackets (in the same way that when you expand (a + b)(c + d) you get ac + bc + ad + bd you get every possible product of something in the first bracket with something in the second bracket). By analogy, we would expect P to be some huge sum p p p p p p 3 n In this sum should appear every number of the form: p m p m p m 3 p m 3 r r. We said earlier that every positive integer is of the form so P is in fact the harmonic series. Thus P is infinite. m m m 3 m r r p p p 3 p, But we said that P is finite, so we have found a contradiction. There must be an error somewhere, so our original assumption (that there are only a finite number of primes) must be false. Hence there are an infinite number of primes. Variation 5 We can use calculus to get more evidence that the harmonic series diverges. (As if we weren t already convinced!) Define f(x) to be the following infinite series: 3 4 x x x f( x)= x Notice that f() =, and that f() is the harmonic series. Differentiating with respect to x, we get: 3 4 f'( x)= + x + x + x + x + This is an infinite geometric series, so we evaluate this sum to obtain: f'( x)= x Thus f() f() f( = ) = f'( x) dx = x dx = ln ( x) = ( ) (To be fair, substituting x = into ln ( x) is not quite cricket ; rather we should look at the limit of ln ( x) as x approaches from below.) 9

10 Historical Digression 88 The Human Computer John von Neumann was one of the preeminent mathematicians of the th century. At the age of thirty, he was appointed (along with Albert Einstein) as one of the first professors of the Institute for Advanced Study in Princeton. His work in mathematics and physics spanned a wide spectrum of fields and his contributions in any one of these would have won him recognition. Von Neumann s earliest contributions lay in his axiomatization of set theory; that is, he reformulated the axioms of set theory to deal with some of the problems such as Russell s Paradox, discussed in Paradox (p. 3). From this he proceeded to a reformulation of quantum John von Neumann mechanics. He also pioneered the development of game theory, the digital computer, and the study of cellular automata. Near the end of his career he worked on the atomic bomb, performing detailed calculations (often in his head) related to pressures created during implosions. Stories about John von Neumann s remarkable capacity for mental computation abound. On one occasion he was at a cocktail party when a guest challenged him with the problem given on page 8 about the Birds and the Bee. Johnny, as he was fondly called, contemplated the problem for a second or two and then replied, ten kilometers. The surprised interrogator retorted, Oh, you've heard that trick before? In typically innocent fashion, Johnny responded, What trick? I merely summed the infinite series. Answer to The Mathematics of the Birds and the Bee (p. 8) One could take von Neumann s approach and calculate how far the bee travels in each direction. This requires that you find the sum of an infinite geometric series. However, there is a faster solution. The birds each take ten minutes to reach the halfway point; in this time, the bee will travel km because it travels km/min. That s it! (Here is a follow-up question: How many times does the bee turn around during this journey?)

11 Game Theory Revisited Come, Watson, come! The game is afoot. The Return of Sherlock Holmes Sir Arthur Conan Doyle