Mathematical constants

Министерство образования и науки РФ

Государственное образовательное учреждение высшего

профессионального образования

Тульский государственный университет

Кафедра Иностранных языков

 

 

Реферат

 

” Mathematical constants “

 

 

 

 

 

 

Выполнил: ст. гр. 220531                                                         Житенёв Д.А.

 

Проверила:                                                                                Клименко Л.С.

 

 

 

 

2014

 

Contents

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

In computer culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Number theory  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

 

         Antiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Computer era and iterative algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Adoption of the symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Motivations for computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Infinite series  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Spigot algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

 

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

История . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 В компьютерной культуре . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 Теория чисел . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

 

         Античность . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Определение . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

Свойства . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Компьютерная эра и итерационные алгоритмы . . . . . . . . . . . . . . . . . . . .24

Принятие символа  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Мотивы для вычисления . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Бесконечные ряды . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Spigot алгоритмы . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Список литературы . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

 

 

 

e

The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828. And is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series.

The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function. Its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k. In which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.

Sometimes called Euler's number after the Swiss mathematical Leonhard Euler. e is not to be confused with γ—the Euler – Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant. But Euler's choice of the symbol e is said to have been retained in his honor. The number e is of eminent importance in mathematics, alongside 0, 1,  and i. All five of these numbers play important and recurring roles across mathematics. And are the five constants appearing in one formulation of Euler’s identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e):

.

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christian Huygens in 1690 and 1691.Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Euler’s Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

 

 

 

 

 

 

 

 

 

In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e.

For instance, in the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is e billion dollars. Google was also responsible for a billboard that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of e}.com". Solving this problem and visiting the advertised (now defunct) web site led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a resume. The first 10-digit prime in e is 7427466391, which starts at the 99th digit.

In another instance, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth. Similarly, the version numbers of his TeX program approach π.

 

 

 

 

 

 

 

 

 

 

 

 

Number theory

The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite.

Furthermore, by the Lindemann-Weierstrass theorem, e is transcendental. Meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose the proof was given by Charles Hermite in 1873.

It is conjectured that e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Antiquity

 

The Great Pyramid at Giza, constructed c. 2589–2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2 ≈ 6.2832. Based on this ratio, some Egyptologists concluded that the pyramid builders had knowledge of  and deliberately designed the pyramid to incorporate the proportions of a circle. Others maintain that the suggested relationship to  is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of , and because the dimensions of the pyramid are based on other factors.

The earliest written approximations of  are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats  as 25/8 = 3.1250. In Egypt, the Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats  as (16/9)2 ≈ 3.1605.

In India around 600 BC, the Shulba Sutras (Sanskrit texts that are rich in mathematical contents) treat  as (9785/5568)2 ≈ 3.088. In 150 BC, or perhaps earlier, Indian sources treat  as   ≈ 3.1622.

Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of ten cubits and a circumference of thirty cubits; the verses imply  is about three if the pool is circular.

 

 

Definition

 

 

 is commonly defined as the ratio of a circle's circumference C to its diameter d:

 

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula = C/d. There are also other definitions of that do not mention circles at all. For example,  is twice the smallest positive x for which cos(x) equals 0.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Properties

 

 is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as 22/7 are commonly used to approximate ). Since  is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that is irrational; they generally require calculus and rely on the reductio ad absurdum technique.

Because  is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge.

 

 is a transcendental number, which means that it is not the solution of any non-constant polynomial with rational coefficients, such as .  The transcendence of  has two important consequences: First,  cannot be expressed using any combination of rational numbers and square roots or n-th roots such as   or   Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is impossible.

 

 

 

 

 

Computer era and iterative algorithms

 

The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.

Two additional developments around 1980 once again accelerated the ability to compute . First, the discovery of new iterative algorithms for computing , which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly. Such algorithms are particularly important in modern  computations, because most of the computer's time is devoted to multiplication. They include the Karatsuba algorithm.

The iterative algorithms were independently published in 1975–1976 by American physicist Eugene Salamin and Australian scientist Richard Brent. These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing  between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Adoption of the symbol 

 

The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics. The Greek letter first appears there in the phrase "1/2 Periphery ()" in the discussion of a circle with radius one. Jones may have chosen π because it was the first letter in the Greek spelling of the word periphery. However, he writes that his equations for  are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones. It had indeed been used earlier for geometric concepts. William Oughtred used  and δ, the Greek letter equivalents of p and d, to express ratios of periphery and diameter in the 1647 and later editions of Clavis Mathematicae.

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler started using it, beginning with his 1736 work Mechanica. Before then, mathematicians sometimes used letters such as c or p instead. Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly. In 1748, Euler used  in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as ; thus  is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.

 

 

 

 

 

Motivations for computing 

 

For most numerical calculations involving , a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations. Despite this, people have worked strenuously to compute to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with  often make headlines around the world. The extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Infinite series

The calculation of was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD. The first written description of an infinite series that could be used to compute was laid out in Sanskrit. Verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD. The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory–Leibniz series. Madhava used infinite series to estimate to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.

 

 

 

 

 

 

 

 

 

Spigot algorithms

Two algorithms were discovered in 1995 that opened up new avenues of research into . They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of  that are not reused after they are calculated. This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.