Encounter with pi in physical problems examples
π - the story
1st chapter. First encounters
π is defined and calculated
The best way to approach the subject is to take a bold approach to the number π: We examine different definitions of π and look at the simplest methods that have been devised to calculate this number. Is the number π a mathematical or a physical constant? We are faced with a difficult question here, and we will carefully distinguish computational methods that depend on a physical premise from those that are independent of such a premise.
2nd chapter. Oddities and curiosities
Tricky and fun things to do with π
We are all fascinated by the number π, but for some of us this fascination goes so far that one has to speak of a “π fetishism” or even a “π mania”. If you spend several months of your life writing a book about π, don't you succumb to this addiction? In fact, there are quite a few people who are obsessed with π and who attach an almost mystical meaning to this number. Around π you are building a world that, although not always to be taken seriously, is nevertheless a pleasure to visit. There are people who memorize many decimal places of π; there are those who want to read in π and those who explore the decimals of π; and of course there have been fanatics for more than 2000 years who claim to have found a solution to the problem of squaring the circle; and finally there are contemporaries who simply have fun with the number π.
3rd chapter. The history of the number π in the age of geometry
The number π occurs directly or indirectly in all ancient mathematical texts that have been found. This also includes texts that are up to 4000 years old. The knowledge that these ancient civilizations had about the number π was sometimes limited to little more than nothing, but humanity's encounter with this number (always via geometry) posed difficult problems - such as, for example, in ancient Greece the famous problem of squaring the circle. This problem aroused the interest of the most eminent minds in mathematics and led to profound and subtle discoveries. The number π turned out to be the engine of science. We take a look at the "primitive life" of π in the West, in India, in China and in some other advanced cultures, where attempts were made to calculate this number.
4th chapter. The history of π in the time of analysis
The emergence of modern analysis (differential and integral calculus) gave rise to new definitions of π, which freed themselves from the shackles of geometry. From now on, the formulas found were purely arithmetic: they were infinite products, infinite sums and infinite fractions. Initially, these formulas, which were interpreted as numerical quadratures, were of no practical interest because they converge extremely slowly. Nevertheless, the profound advances a little later led to the powerful arctan formulas, which dominated until 1973. This was followed by other analytically derived formulas. The number π found in this way was a new, purely mathematical being, the geometric component of which had become secondary: for example, there is only a very indirect relationship between the circle and the sum 1 - 1/3 + 1/5 - 1/7 + ... which is equal to π / 4. In this chapter we meet again the names of the greats of the 17th and 18th centuries, Leibniz, Newton and Euler, all of whom at some point pondered π and succumbed to the magic of this number.
5th chapter. From handwritten bills to the computer age
The rule of the Arcus tangent
The level of knowledge about π obtained through analysis led to computational methods that sometimes turned out to be efficient; thanks to his formula, John Machin was the first to get to the hundredth decimal. The main merits of his successors were patience and perseverance. We briefly describe their somewhat monotonous history. They all used arctan formulas and wrote countless sheets of paper. We stop at the record of Jean Guilloud and Martine Bouyer, who were the first to reach a million decimal places in 1973. This ended a not-too-creative epoch in the history of n. Around 1945 the development of electronic calculating machines triggered a small revolution among job hunters. Contrary to the popular opinion that this is not the case, it made the competition more interesting and later assumed passionate forms, as we will see in the following chapters. Programming a computer is, as was already noted in the 1950s, a task that requires an ever deeper understanding of mathematics. Today we have the necessary prerequisites for this understanding.
6th chapter. The practical calculation of π
Examples of trickle algorithms
Here we discover the general principles needed to compute numbers with a long sequence of decimals. These calculations are carried out by hand or with the help of a computer. As an example, let's give a program that is as short (158 characters) as it is imaginative. We explain how this program works, which can calculate 2400 decimals of π. By reshaping one obtains a method for calculating π by hand, with which one can calculate more decimals within a few hours ... or a few days than Ludolph van Ceulen (35 digits) or Johann Dahse (200 digits) succeeded in doing. The chapter closes with brief remarks on the techniques of convergence acceleration.
7th chapter. Living math
How to get to a billion decimals
If we want to move forward, it pays to think a little before we reach for pencil and paper or turn on our computer. This thinking about π has proven useful over the past 25 years. Generally accepted things that are taken for granted must be revised. A more detailed investigation shows, contrary to all expectations, that the usual multiplication is not the most efficient method and that in the case of extensive calculations it has to be replaced by a more complex and powerful method that has to be newly designed and programmed. In addition, taking into account the ideas of the Indian mathematician Srinivasa Ramanujan, which were not understood for a long time, a noticeable improvement in the arctan formulas, which ruled π for almost 3 centuries, is achieved. In addition to the theoretical advances, there was the extraordinary improvement of the computers and the software. Through them, mathematicians and computer scientists gained knowledge about the number π, which goes beyond anything that would have been thought possible.
8th chapter. The computation of individual digits of π
A discovery of experimental mathematics
The new formula by David Bailey, Peter Borwein and Simon Plouffe offers the extraordinary possibility to calculate any digit from π to base 2 without having to determine the preceding digits. We explain the theoretical and practical consequences of this recently made and unexpected discovery.
9th chapter. Is π transcendent?
Irrational numbers, radical expressions and algebraic equations
Is the number π the ratio of two whole numbers, that is, is it rational? Can the number π be constructed with a compass and ruler, the ideal instruments of the geometer? In other words, can π be represented by a finite algebraic expression using only square roots? Is the number π the solution of an equation in which only whole numbers and elementary operations occur, is π therefore algebraic? It took more than 20 centuries to answer these questions, which consisted of increasingly refined forms of the following question: "Is π finally definable?" The very last answer was given in 1882 when Lindemann proved that the number π is transcendent (i.e. not algebraic). With that the puzzle of squaring the circle was solved. Today everything is clear and the relationship between geometry and numbers is very well understood. However, that does not mean that everything has become easy. Nor does it mean that all elementary questions have been answered. Because the abstract world in which mathematicians move is infinitely rich and complex and holds new puzzles in store that are even more profound and difficult.
10th chapter. Is π a random number?
Disorder and complexity
The transcendence of π implies practically nothing at all regarding the sequence of decimals. Whenever the job hunters break a record and are proud to be the first to open up a new parcel of the infinite universe of π, they subject their results to all kinds of statistical tests. Nothing remarkable has ever been found. If there were any peculiarities, they were unfortunately never confirmed - either because the decimals were wrong or because the peculiarity in question disappeared when the development was continued. The decimals of it appear - apart from the fact that they are the decimals of π - like random numbers, the occurrence of which can neither be proven nor understood! We have to ask ourselves what coincidence actually is, and we have to ask ourselves what the definition of a statistically arbitrary, complex, unpredictable, non-compressible, etc. sequence of decimals means. The theory of predictability proves to be an essential support, but despite the new insights it provides, we are still confronted with extremely simple, profound and unresolved questions. These questions certainly justify the fact that brilliant mathematicians, like the Chudnovsky brothers, should take part in the passionate search for the digits of π.
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