Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks)
Page 7
Which Is Larger: ab or ba?
In the preceding section, it was established that the total length of our “polygonal spiral” was eπ = 23.1407. Now, we are no longer interested in the previous topics such as complex numbers and vector diagrams. We are interested only in the quantity eπ = 23.1407. Suppose we switch the positions of π and e and then compute πe. Recall that π = 3.14159 and e = 2.71828. With our hand calculator we determine that πe = 22.4591, which is a bit less than eπ = 23.1407. Interesting, but so what?
Well, let's generalize. Suppose that a and b are positive numbers, not necessarily integers, and that a is larger than b. We ask the question: which is the larger quantity, ab or ba, that is, a to the bth power or b to the ath power? Incidentally, this problem was suggested by two somewhat similar problems that appear in Dörrie (1965) and Dunn (1980).
To start with, here are three numerical examples:
1. If a = 3 and b = 2, then ab = 32 = 9 and ba = 23 = 8. In this case, ab > ba.
2. If a = 4 and b = 2, then ab = 42 = 16 and ba = 24 = 16. In this case, ab = ba.
3. If a = 5 and b = 2, then ab = 52 = 25 and ba = 25 = 32. In this case, ab < ba.
We note that modest changes in the magnitude of a, with the value of b held constant in these three examples, entirely alter the relative magnitudes of ab and ba. This seems strange.
It will be very helpful in our analysis if we know when the two quantities, ab and ba, are equal. That is,
Unfortunately, it is impossible to obtain a solution to this equation in a form that expresses a in terms of b or vice versa. Consequently, we must settle for a parametric solution. Taking the logarithms of both sides of equation (8.6) gives
We let a = kb, where k is a number larger than one (since a > b); k is called the parameter. Substituting this relationship into equation (8.7) and carrying out some algebra, we obtain the expressions
Assigning various numerical values to the parameter k in these expressions gives the magnitudes of a and b that satisfy equation (8.6).
An interesting question is, what are the values of a and b when k approaches one? We note that the relationships of equation (8.8) are indeterminate for k = 1. Never mind. In these two expressions, we make the substitution n = 1 /(k - 1). This yields
It is clear that if k = 1 then n = ∞. Utilizing equation (7.5), it is easily established that in this limiting (k = 1) case we have the values a = e and b = e.
The results of our analysis are displayed in figure 8.3. For your information, table 8.1 lists some coordinates for the curve ab = ba shown in the figure. Two interesting observations: (1) The values a = 4, b = 2 are the only integers for which ab = ba. (2) The values a = e, b = e, as the end point (k = 1) coordinates on the ab = ba curve, represent an unexpected appearance of our very remarkable number e.
FIG. 8.3
Graphical display of the regions in which ab is greater than, equal to, and less than ba.
TABLE 8.1
The main feature of the display of figure 8.3 is the identification of region I in which ab > ba and region II where ab < ba. You might want to confirm these results by computation with some selected values for a and b, including a = π and b = e. Also, you will undoubtedly want to show that the slope of the ab = ba curve at the point (e, e) is (da/db) = –1, i.e., it is perpendicular to the line a = b.
9
Great Number Sequences: Prime, Fibonacci, and Hailstone
What Are Number Sequences?
The answer to this question is that number sequences are simply lists of numbers appearing in a particular order. For example, the sequence 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is the list of the natural numbers or integers. The sequence 1, 3, 5, 7, 9, 11 is the list of the odd integers and the sequence 1, 4, 9, 16, 25, 36 is the list of the squares of the integers.
Another example: The sequence 1, 2, 4, 8, 16, 32, 64 is the list of numbers generated by doubling successive numbers. Sometimes, as in this case, a mathematical formula can produce the sequence of numbers. In this example, the formula is N = 2n, where N is the magnitude of the nth number of the sequence. This particular sequence is the one that describes geometric growth; it is closely related to so-called exponential growth. This sequence is quite famous. In a quantitative way, it reflects the gloomy prediction of Thomas Malthus, the eighteenth-century English clergyman-economist, regarding the explosive growth of human population.
Here is an interesting sequence: 6, 28, 496, 8,128, and so on. This is the sequence of so-called perfect numbers. A perfect number is a number composed of the sum of all its divisors except the number itself. For example, it is clear that 6 and 28 are perfect numbers because 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.
These numbers were known and studied by the Greeks more than two thousand years ago. An interesting fact is that the next perfect number in the sequence, following the 8,128, is 33,550,336. After that come 8,589,869,056 and then 137,438,691,328. At present (1999), we know the values of thirty-six perfect numbers. In recent years, the rapid development of computers has accelerated our search for these numbers.
There are a great many sequences in mathematics; some are important or interesting, others are trivial or silly. Over the years, a very large number of sequences have been discovered or devised. A collection of over 2,300 from all branches of science and mathematics has been nicely assembled by Sloane (1973). On the following pages, we shall investigate a few.
Mostly about Prime Numbers
Suppose we have the following sequence of positive numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and so on. Does this sequence represent anything in particular?
It certainly does. It is the sequence of so-called prime numbers. That is, it is a list of the numbers that are exactly divisible only by 1 and the number itself. For example, 13 is a prime number because it can be exactly divided only by 1 and 13. On the other hand, 12 is not a prime number; in addition to 1 and 12, it can also be divided by 2, 3, 4, and 6.
The natural numbers or integers are 1, 2, 3, 4, 5, 6, 7, 8, and so on. If a natural number is neither 1 nor a prime it is called a composite number. Following the 37, the above sequence of prime numbers continues: 41, 43, 47, 53, 59, 61, 67, and so on. Altogether, there are 168 prime numbers less than 1,000 and, with the exception of 2, they are all, of course, odd numbers. By the way, mathematicians usually do not consider 1 to be a prime number.
It turns out that the sequence of prime numbers goes on forever. The famous Greek geometer Euclid was the first to prove this; the year, around 300 B.C. Ever since then, mathematicians have been trying to discover or develop a method or an equation to determine whether or not a particular number is a prime number. So far, they have not been successful.
Long ago, numerous formulas were devised that generated prime numbers. A good example is the expression N = n2 – 79n + 1601. This formula produces prime numbers for all values of n up to and including n = 79. You might want to check a few. However, when n = 80, the formula fails because N = 1681 = 41 × 41. Thus, 1681 is a composite number. Other early expressions for generating prime numbers were N = n2 – n + 41 and N = n2 + n + 17. With the benefit of hindsight, it now seems obvious that these polynomial-type expressions are entirely inadequate in producing a lengthy list of only prime numbers.
The noted French mathematician Pierre de Fermat (1601-1665) also searched for a formula that would yield only prime numbers. He invented the following equation, which produces what we now call the Fermat numbers:
This expression generates the numbers F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65,537. All of these numbers are prime. However, F5—which is a ten-digit number—is not prime and neither are the Fermat numbers corresponding to n = 6, 7, 8, 9, 11, and many more numbers calculated with larger values of n. So Fermat was mistaken about equation (9.1) producing only prime numbers. Indeed, a present-day question in number theory is, are any Fermat numbers, larger than n = 4, prime numbers? So far, the question has not been answered.
Mathematicians have l
ong since abandoned the search for formulas that generate prime numbers; there are a great many more interesting subjects in number theory for them to investigate. The following is an important example.
How Are the Prime Numbers Distributed?
It is well known that the sequence of prime numbers goes on forever. However, as the numbers become larger and larger, their frequency decreases, that is, the gaps between successive primes increase. This feature provided the historical basis for the statement of the fundamental theorem of prime numbers. This theorem, for which there are now numerous proofs, postulates that as n increases to very large values, the number of primes not exceeding n is given by the expression
The problem attracted the attention of many of the leading mathematicians of the past. The German mathematician Carl Friedrich Gauss (1777-1855), one of the greatest of all time, examined the problem of prime number distribution in 1792 when he was only fifteen years of age.
Table 9.1 may help illustrate our point concerning the distribution of primes. In the table, column 2 indicates that there are 168 prime numbers less than 1,000. Column 3 says that the number of primes, computed from equation (9.2), is 145, which is somewhat less than the actual number. As shown in column 4, the ratio of column 3 to column 2 is 0.863.
TABLE 9.1
Source: Lines (1986).
Now observe that as the number n increases, the ratio of the two quantities of prime numbers, shown in column 4 of the table, gradually increases and appears to approach the limiting value 1.0. Indeed, if n = 1 billion the ratio is 0.949 and if n = 10 billion the ratio is 0.954. This is precisely the result postulated by the prime number theorem.
The entire matter can be carried a significant step further. Studies have shown that an even more accurate answer to the prime number theorem is given by the expression
where Li(n) is the so-called logarithmic integral. The indicated integral is tabulated in numerous mathematical handbooks.
For the sake of completeness, the number of primes determined by equation (9.3) is listed in column 5 of table 9.1. If these numbers are compared with those shown in column 2 of the table, we note very close agreement. Thus, the accuracy of equation (9.3) in predicting the distribution of prime numbers is quite remarkable.
What Is the Largest Prime Number?
Of course, one of the greatest challenges to mathematicians is to discover ever-larger prime numbers. Over the years, the magnitude of the largest known prime number has steadily increased; clearly, the very rapid advances in computer technology have greatly assisted these efforts. Currently (1999), the largest known prime number is 23,021,377 – 1. This number is so enormous that it requires 909,526 digits to express it.
Not surprisingly, some of the greatest mathematicians of all time have worked on the theory of prime numbers. If you are interested in the historical aspects of the subject, Bell (1956) and Boyer (1991) are good places to start. Elementary presentations of topics on prime numbers are given by Beiler (1964), Lines (1986), and Ogilvy and Anderson (1988). A more advanced coverage is presented in the very readable book by Ribenboim (1995), which includes a lengthy list of references.
Concerning Fibonacci Numbers
One of the most remarkable sequences of numbers in mathematics is the Fibonacci sequence or simply the Fibonacci numbers.
The story all began about eight hundred years ago when an Italian mathematician named Leonardo of Pisa (1170–1250)—better known as Fibonacci—wrote a book entitled Liber Abaci (first published in 1202; revised in 1228). He was born in Pisa, Italy, at about the time construction was begun on what is now called the Leaning Tower of Pisa (which was completed around 1300). Incidentally, he lived long before Galileo Galilei (1564–1642) carried out his famous weight-dropping experiments from the top of the tower.
During much of his early life, Fibonacci lived in North Africa and studied algebra and geometry under Arabic mathematics teachers. As a result, he learned a great deal about the Arabic number system and decimal notation concepts; he included these topics in his Liber Abaci. Thank goodness! Without question, he has our eternal gratitude for hastening the demise of the ghastly Roman numeral system. (Try dividing CCCXXI by XLIX.)
In any event, Fibonacci was undoubtedly the greatest European mathematician of the Middle Ages. Among a great many other things, he presented a mathematical problem in his Liber Abaci which is still used as a popular way to introduce the subject of Fibonacci numbers. This is his famous rabbit problem, as presented by Vajda (1989):
A pair of newly born rabbits is placed in a confined enclosure. This pair, and every later pair, produces one new pair every month, starting in their second month of age. How many pairs will there be after one, two, three,…, months?
TABLE 9.2
Fibonacci's rabbits
End of month number Number of pairs of rabbits
1 1
2 2
3 3
4 5
5 8
6 13
7 21
8 34
9 55
10 89
Source: Vajda (1989).
The answer to the problem is displayed in table 9.2. From the table, we note that the sequence representing the number of pairs of rabbits is 1, 2, 3, 5, 8, 13, 21, 34, and so on. These are the famous Fibonacci numbers. Without doubt, you have already discovered how the sequence increases: each number is the sum of the two previous numbers. So the recurrence relationship is
The next few numbers are 144, 233, 377, 610, and so on. Now here is an important point. If you divide each number in the sequence by the preceding number, you get closer and closer to the quantity 1.618034. This is a very interesting result; we shall come back to it shortly.
The diversity of places where the Fibonacci numbers make appearances is absolutely incredible. In one form or another they show up not only in numerous topics of mathematics but also in biology, botany, music, art, and architecture. At the elementary level, suggested references are Huntley (1970), Lines (1990), and Dunlap (1997); more advanced coverage is given by Vajda (1989). It is noteworthy that in 1963, a Fibonacci Association was created, which began publication of the Fibonacci Quarterly. Over the years, hundreds of articles have been published in the Quarterly dealing with a great many subjects involving Fibonacci numbers.
FIG. 9.1
Rectangles of various ratios of height to width.
Closely related to the Fibonacci sequence is one called the Lucas sequence. As before, each term of this sequence is the sum of the previous two terms. However, instead of starting with the numbers (1, 2) as in the Fibonacci sequence, the Lucas sequence begins with (1, 3). This yields 1, 3, 4, 7, 11, and so on.
Fibonacci Numbers and the Golden Section
In figure 9.1, four rectangles are shown with height-to-width ratios ranging from the tall slender rectangle on the left (1) to the square on the right (4). Which one of the four shapes do you like the best? That is, which, if any, do you think is the most esthetically appealing?
Well, different people like different things, but psychologists have found that most people “like” rectangle 2 best, because it is neither too slender nor too square; it “seems about right.” In any event, rectangle 2 of figure 9.1 is rotated 90° and dimension symbols added in figure 9.2.
The ancient Greeks, especially their architects, were probably the first to believe that there was a ratio between the length and the height of a rectangle that gave the most pleasing artistic proportion. Historians suspect that the famous Parthenon in Athens was designed and built with this ratio in mind.
FIG. 9.2
Definition sketch for the golden rectangle or golden section.
With reference to figure 9.2, our problem commences by simply constructing the pleasing ratio
From here on, the problem is one of algebra. The above equation yields the expression L2 – HL – H2 = 0. If you remember how to solve quadratic equations like this one, you will easily obtain the answer,
So we have deter
mined the length-height ratio of a rectangle that gives the most pleasing appearance, at least in the opinion of the early architects and artists. This ratio, L/H, has been given a special name: the golden ratio or golden section. Furthermore, since mathematicians consider it to be a very important constant, it has also been given a special symbol, ø. That is,
Now, are you ready for an interesting surprise? The numerical value of the golden section is ø = 1.618034.…Does this quantity look familiar? It is the ratio of successive Fibonacci numbers we looked at a few moments ago. Fantastic, right? Why should the breeding habits of a bunch of prolific rabbits have anything in common with ancient Greek architecture? Mathematics is not only beautiful, it can also be very intriguing!
A few more items about ø. The early Egyptians may also have been aware of the golden section. In chapter 20, “How to Make Mountains Out of Molehills,” we discuss the fact that the main dimensions of the Great Pyramids of Egypt (built during the period 2650 to 2500 B.C.) seem to conform to the geometrical proportions directly related to ø.