by Colin Pask
1, the series also diverges, but for r < 1, it has the finite limit a/(1-r) (since in that case rn approaches zero for large n.) A series with a finite answer is said to converge. An attractive old example is this geometric series (with r = ½):
In general, it is not easy to discover whether an infinite series diverges or converges, and finding the actual sum in the convergent case is usually even harder. But before going further, perhaps I should respond to readers who are asking why we would bother with such questions at all. The answer (apart from natural curiosity and accepting a challenge), is that infinite series are extremely important in the development of mathematics and its applications. An ancient example appears in Archimedes's wonderful calculation of the area enclosed by a parabola and a line cutting it (see the books by Dijksterhuis and Stein in the chapter 2 bibliography). To get his result, Archimedes had to prove that the infinite geometric series with a = 1 and r = ¼ has a sum of 4/3, that is
The development of calculus lead to the use of infinite series (especially by people like Isaac Newton) for representing various functions. For example:
The second expression was very useful for calculating logarithms.
3.3.2 The Problem and the Master
In all of this, one problem became particularly famous and one man gave its solution—along with a great many other results.
The harmonic series in equation (3.3) when extended to an infinite number of terms was proved to diverge by Oresme as long ago as 1350. But what if the terms reduced in size even faster as in
This series does converge; even though there are an infinite number of terms, they reduce in size so quickly that the total sum is finite. But finding the value of the sum caused great difficulties. Because this problem was publicized by Jacob Bernoulli, who lived in the Swiss city of Basel, it became known as the famous Basel problem.
The man to tackle such problems was Leonard Euler (1707–1783). He was actually born in Basel, but in later life he served in academies in Russia and Germany. Euler was surely the most prolific mathematician of all time, and few branches of pure mathematics do not show signs of his pioneering inputs. A poll of readers of the journal Mathematical Intelligencer produced a ranking of the most beautiful results in mathematics;6 the top two and the fifth ranked are results from Euler. He was also an applied mathematician and scientist of great repute with interests spanning many areas. It was Euler who took Newton's mechanics and turned them into the form we use today. He fathered thirteen children and still found time to write textbooks in calculus and algebra. Euler became perhaps the first great popularizer of science. His Lettres á une Princesse d'Allemagne (Letters to a German Princess) became what today we would call a publishing sensation, going through many editions and being translated into German, English, Dutch, Swedish, Italian, Danish, and Spanish. They remain an interesting and informative introduction to science even today. (The book by Dunham gives a good introduction to Euler and his work, and the paper by Kline deals specifically with infinite series.)
When asked about how to tackle mathematical problems, Laplace (the “French Newton”) responded: “Read Euler; He is our master in everything.”7 So it is with series; Euler was truly the master of the topic and his Introduction to Analysis of the Infinite is simply breathtaking in its scope and inventiveness.
Euler solved the Basel problem and gave the answer
For me this is an amazing, neat, and beautiful result (and mathematicians tend to agree as it was fifth in the poll mentioned earlier). Who would have expected that ubiquitous π to turn up here! (Actually, as part of his proof, Euler used expressions like those in equation (3.4) and any link to sine will suggest that π might be involved—see Dunham or Euler's own book for details.)
Euler went further, and he could give results for other even powers such as
These results by the master are my calculation 6, Euler solves the Basel problem.
It should not be thought that these results are mere curiosities (otherwise they would not be in my list of great calculations). Euler developed new approaches to series and the mathematics of functions expressed as power series, which have affected the whole progress of mathematics. One or two examples are given in the next section for those wishing to see some details.
3.3.3 A Technical Comment
Euler showed how to generalize the Basel problem and then how to link it to the theory of prime numbers (more of which in the next section). Euler considered what we now call the zeta function (since it is denoted using the Greek letter ζ):
He was able to study this series as a function of m; for example, he proved that it converges for all m > 1. In one of those steps which only the master might make, Euler linked the zeta function to the prime numbers:
and so the infinite series in equation (3.5) is shown to be equivalent to an infinite product involving just the prime numbers. Euler showed how problems about numbers could be tackled using the methods of analysis. I will return to this in the next section.
3.4 GAUSS, HEROIC CALCULATIONS, AND THE PRIME NUMBER THEOREM
Johann Carl Friedrich Gauss (1777–1855) was six years old when Euler died, and he was already recognized as a child prodigy in mathematics and calculations. Gauss went on to contribute broadly in mathematics and physics and is usually put on the same level as giants like Archimedes and Newton. Gauss gave us the famous hierarchy: “Mathematics is the Queen of the Sciences and Number Theory is the Queen of Mathematics.”8 So it is fitting that Gauss should provide the next great calculation in the field of number theory.
The most special and important of all numbers are the primes (those numbers not divisible by any other number). Already the ancient Greeks knew that any number (and here we are talking about the positive integers) can be written as a unique product of prime numbers. For example, 126 = 2 × 3 × 3 × 7. This result is called the fundamental theorem of arithmetic. Thus the primes are the building blocks for all numbers and a key to understanding arithmetic and number theory results. An obvious question is: How many prime numbers are there? The answer, as Euclid showed in his Elements, is that the list of primes goes on forever; there is no largest prime, or as some people like to say, there are an infinite number of primes. (This is a little disappointing when we compare the physical case of atoms, which are all built up using just three particles: the electron, the proton, and the neutron.)
The next questions come when we look at the primes in more detail. Here are all the primes less than 150:
Looking at the first primes suggests that they come in pairs: 3 and 5, 5 and 7, 11 and 13, 17 and 19. Such pairs of prime numbers that differ by 2 are known as twin primes. However, the next twins do not appear until 41 and 43, then 71 and 73. There are more twins, like 101 and 103, but no obvious pattern emerges. There is probably an infinite number of twin primes—nobody knows for sure. That example is typical; it is hard to see any pattern in the prime numbers.
Another approach is to seek formulas or equations that produce sets of primes. For example, Euler discovered that the polynomial x2 – x + 41 gives prime numbers when x is set equal to 1, 2, 3,…40. (See the books by Wells, and Conway and Guy for more on such things.) Lots of results have been found, but no single overall method for generating the primes is known.
If we step back further, we can ask the seemingly simple question: How many primes are there less than or equal to a given number x? The result is called π(x). (It seems unfortunate that the Greek letter π is used here given that it is so widely used in terms of circle properties. The notation π(x) was introduced by Edmund Landau in a 1909 book about prime numbers.) Counting up the primes listed in the above table gives that π(20) = 8, π(71) = 20 and π(150) = 35. It is not at all obvious how π(x) depends on x. We need to see how π(x) changes as x is increased to larger and larger values. We need to do some calculations!
It is here that we see a perfect counterexample to T. H. Huxley's famous statement:
Mathematics is that study which knows nothing of observation, nothing of
experiment, nothing of induction, nothing of causation.9
Such an idea was strongly refuted at the 1869 meeting of the British Association. If they needed a dramatic example they could have used the extensive calculations made to discover the factorization of a range of numbers. When no factors are found, then of course the number is a prime and can be added to the list. The story of these calculations is most remarkable. Here is the 1980 summary by the eminent mathematical historian Howard Eves:
Extensive factor tables are invaluable for research on prime numbers. Such a table for all numbers up to 24,000 was published by J. H. Rahn in 1659…In 1668 John Pell of England extended this table up to 100,000. As a result of appeals by the German mathematician J. H. Lambert, an extensive and ill-fated factor table was computed by a Viennese schoolmaster named Antonio Felkel. The first volume of Felkel's computations, giving factors of numbers up to 408,000, was published in 1776 at the expense of the Austrian imperial treasury. But, as there were very few subscribers to the volume, the treasury recalled almost the entire edition and converted the paper into cartridges to be used in a war for killing Turks. In the nineteenth century, the combined efforts of Chernac, Burckhardt, Crelle, Glaisher, and the lightning mental calculator Dase, led to a factor table covering numbers up to 10,000,000 and published in ten volumes. The greatest achievement of this sort, however, is the table calculated by J. P. Kulik (1773–1863), of the University of Prague. His as yet unpublished manuscript is the result of a twenty-year hobby, and covers all numbers up to 100,000,000. The best available factor table is that of the American mathematician D. N. Lehmer (1867–1938); it is a cleverly assembled one-volume table covering numbers up to 10,000,000. Lehmer has pointed out that Kulik's table contains errors.10
Lehmer himself wrote a history of these calculations (see the bibliography).
3.4.1 Using the Data
Gauss was involved with mathematical tables throughout his life. At fourteen he was given mathematics books, including tables of logarithms, by the Duke of Brunswick who recognized his talents and supported him during the early part of his life. Around the age of sixteen, Gauss wrote down his conjecture for the behavior of π(x) after studying tables of primes. Other people, like Legendre, also came close to giving the full story.
Based on the evidence, Gauss concluded that the nature of π(x) is summarized in the prime number theorem:
as x becomes large, π(x) asymptotically approaches x/log(x),
thus approaches 1 as x tends to infinity.
Two technical points: the log here is the natural log with base e; Gauss later gave a more accurate result using the logarithmic integral Li(x):
as x becomes large, π(x) asymptotically approaches
Since Li(x) is asymptotically equal to x/log(x) we can use either formula; it is just that for smaller values of x, the integral is more accurate. For example, if x = 1,000,000,000, the number of primes π(x) is 50,847,478; the prime number theorem gives 48,254,942, and Gauss's Li(x) gives 50,849,235. If x is taken ever larger, the values of π(x), x/log(x), and Li(x) become closer and closer. (Details of Gauss's personal calculations and samples of the tables he produced are given in the papers by Goldstein and Tschinkel, who also reproduce a letter from Gauss to his student Encke containing some interesting historical points.)
The prime number theorem at last reveals some sort of order in the set of primes. It is a beautiful example of the way calculations provide insight and inspiration, and for that reason it is my important calculation 7.
3.4.2 Beyond the Calculations
The link between analysis and number theory pioneered by Euler eventually led to independent analytical proofs of the prime number theorem in 1896 by J. Hadamard and C. J. de la Vallée-Poussin.
Research into the primes has produced an enormous number of fascinating results. (See the book by Wells for a simple introduction.) For example, Euler showed that if we keep only the terms involving primes in the famous harmonic series, equation (3.3), the series still diverges. However, if we use only the twin primes, Viggio Bruns showed in 1919 that
Thus this infinite series does converge, and by 2002 the sum was known to even greater accuracy: 1.902160583104. Such are the charms and delightful results of prime number theory.
It is impossible not to mention the most famous unsolved problem involving the primes. In 1742, Christian Goldbach wrote to Euler asking about numbers and the sum of primes. The result is the famous Goldbach Conjecture: every even number, except 2, can be written as the sum of two primes. For example, 16 = 5 + 11 and 160 = 59 + 101. The conjecture has been verified by calculations with numbers up to the staggeringly large 4 × 1014, but to this time, no analytical proof has been found to back up the calculations (and if you find one, there is a $1 million prize you can collect).
Prime numbers offer challenges and lots of fun for professional and amateur mathematicians alike. The factorization problem has also become important in a practical way. The difficulty of factoring a very large number into a product of primes is the basis of the RSA algorithm for public key encryption of data for transmission in military and business contexts. (Wells give more details and references.)
3.5 WHAT IS LEFT OUT
The choice of calculations for this chapter is almost limitless, and I am sure every reader will have some favorite work that I have excluded in my desire to limit the size of this book. Some of you may wonder how I could ignore Ramanujan, for instance. Here are just five topics that ended up in my extensive near-misses category.
Pascal's triangle contains the numbers used in binomial theorem expansions and is also used in combinatorial problems and in the theory of probability. Like the primes, it leads to many fascinating mathematical properties and uses (see the book by Edwards). This is also an example of a topic that was explored by early Chinese mathematicians (see the books by Martzloff and by Li Yan and Du Shiran in the chapter 2 bibliography).
Rafael Bombelli (about 1526–1573) solved a cubic equation using a standard procedure, but as part of his working, he bravely manipulated expressions involving the square root of minus one. This led people to believe such things could enter mathematics, and so began the use of complex numbers. It is almost impossible to imagine modern pure or applied mathematics without the use of complex numbers.
Linear equations and sets of such equations are ubiquitous in mathematics, and few techniques in mathematics could be more famous than Gaussian elimination as a solution method for them. Gauss made some very important calculations using this method. This is another example where early Chinese mathematicians also showed how to systematically tackle relevant problems (see references to Martzloff, and Li Yan and Du Shiran, as mentioned above).
The classification of groups resulted in some mammoth calculations carried out by many mathematicians. The book by Marcus du Sautoy tells the wonderful story, including how group classification involves integers eighteen digits long.
As a quirky example, I offer Benford's law, also known as the first digit phenomenon, which states: in many collections of numbers, including tables of various kinds, the first digit is most commonly 1; if not 1, then 2; and so on in a logarithmic distribution. There is something quite captivating about this and it even has applications. To go further, try the paper by Hill and Berger.
in which we see how mathematics helped to answer questions about our world and sometimes stirred up major controversies in so doing; learn more about how calculations are linked to experiments and the difficulties involved; and see some of the different approaches used when making calculations.
As science gradually developed man began to ask questions about the world in which he lived, something that continues today. I have chosen four examples for discussion in this chapter, and a fifth appears in the next chapter. As always, there are many more that I might have included, and at the end I will discuss some of them and make a few general comments about calculations in this area. The four examples will illustrate the theory-experiment link and show diffe
rent facets of physical calculation work.
4.1 HOW BIG IS THE EARTH?
We naturally appreciate the size of things around us and start to measure them in terms of convenient units—millimeters for small things, meters for larger everyday objects, and kilometers for longer journeys. Interestingly, the very first letter Leonard Euler wrote for his wonderful Letters to a German Princess is titled On Magnitude. Measuring the size of things is one of the most basic tasks of science. Once it was accepted that the earth can be taken as a sphere, it was natural to ask how big it is. Sailors knew about the curved horizon and vanishing landmarks as they sailed out to sea, and various schemes were used to estimate the size of planet Earth. Aristotle (of course) gives a value, although not a very accurate one.
Eratosthenes devised a different approach: he showed how data from an experiment could be used in a simple calculation to relate the circumference of Earth to a given known distance in Egypt. Eratosthenes was born around 285 BCE in Cyrene, a Greek city in the part of North Africa we know as Libya, and lived to be about eighty. He was educated in Athens, and in 245 BCE, he moved to Egypt to become head librarian at the illustrious Musaeum in Alexandria. He was what today we would call a polymath and made contributions across many fields of science, mathematics, and the arts. It is a tragedy that none of his original work survives today, but there are good descriptions of his life and achievements written by his contemporaries and other ancient scholars. The book by Nicastro gives a readable introduction to the man and his work.
4.1.1 Eratosthenes's Calculation
Eratosthenes knew that at the summer solstice, the sun at its zenith in Syênê was directly overhead as indicated (reputedly) by the fact that it shone down to the bottom of wells. If the angle θº to the vertical made by the sun at the same time at Alexandria was measured, the geometry shown in figure 4.1 could be used.