Number Pi

Mathematical constants. Radius. Circumference. History. Ancient Egypt. Classical Antiquity. China. Approximations. Modern Times

  • Enviado por: Karo1
  • Idioma: inglés
  • País: España España
  • 7 páginas
publicidad
publicidad

Index

  • Summary

  • Keywords

  • Contents

  • Origin of number π and of its name.

  • Features of number π.

  • The approximations of number π in Ancient Times.

    • Mathematics of Ancient Egypt

    • Mathematics of Mesopotamia

    • Mathematics of Classical Antiquity

    • Chinese Mathematics

    • Indian Mathematics

    • Islamic Mathematics

    • Table of approximations

    • The approximations of number π in the Modern Times

    • Value of π with 100 places.

      • Bibliography

      • Personal Opinion

      Summary

      Pi or π is a mathematical constant whose value is the ratio of any circle's circumference respect to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π. π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats.

      Keywords

      • Radius - is any line segment from its center to its perimeter.

      • Decimal places - they are numbers followings the point.

      • Circumference - is the distance around a closed curve.

      Contents

    • Origin of number π and of its name.

    • History

      The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π. Egyptologists such as Professors Flinders Petrie  and I.E.S Edwards have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects.

      The name

      The name of the Greek letter π  is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available, or its usage could be problematic. It is not capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced /'paɪ/, "pie" inEnglish, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced [pi].

      The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.

    • Features of number π.

    • Definition

      Euclid was the first to show that the ratio of a circle to its diameter is a constant quantity. However, there are various definitions of the number π, but the most common is: π is the ratio between the length of a circle to its diameter.

      Irrational and transcendental number

      π is an irrational number, which means it can not be expressed as a fraction of two integers, as demonstrated by Johann Heinrich Lambert in 1761 (or 1767). It is also a transcendental number, which means it is not the root of any polynomial with integer coefficients. In the nineteenth century German mathematician Ferdinand Lindemann proved this fact and permanently closing the problem of squaring the circle indicating that there isn't solution.

    • The approximations of number π in

    • The Ancient Times.

        • Mathematics of Ancient Egypt

      The approximate value of π in ancient Egypt was wrote by scribe Ahmes in 1800 a. C., in the Rhind papyrus, where he used an approximate value of π by saying that: the area of a circle is similar to a square whose side equals the diameter of the circle decreased in 1 / 9, that's equal to 8 / 9 of the diameter. In modern notation:

      'Number Pi'

      'Number Pi'

        • Mathematics of Mesopotamia

      Some mathematicians of of Mesopotamia used in the calculation of segments, values of π equal to 3, reaching in some cases approximate values, such as 3 + 1 / 8.

        • Mathematics of Classical Antiquity

      The Greek mathematician Archimedes (III century BC) was able to determine the value of π between the interval by 3 10/71, as the minimum value, and 3 1 / 7, the maximum value. With this approximation of Archimedes gets a value with an error of between 0.024% and 0.040% on the actual value. The method used by Arquímedes was simple and consisted circumscribe and inscribe regular polygons of n-sided circles and calculate the perimeter of these polygons. Archimedes started with hexagons circumscribed and inscribed, and was doubling the number of sides to reach 96-sided polygons.

        • Chinese Mathematics

      In 263, the mathematician Liu Hui was the first to suggest that 3.14 was a good approximation using a polygon of 968 or 1926 sides. Later  he estimated the value of π as: 3,14159 using a polygon of 3,072 sides.
      In finals of V century, the Chinese mathematician and astronomer Zu Chongzhi calculated the value of π as: 3.1415926 and which he called "default" and 3.1415927 "excess value" and gave two rational approximations of π: 22 / 7 and 355/113, both well-known, the second one being so good and precise that wassn't equaled until more than nine centuries later, in the XV century.

        • Indian Mathematics

      Indian mathematician, Aryabhata estimated the value of π as 3.1416 using a regular polygon of 384 sides inscribed. Around 1400 Madhava get an accurate approximation to 11 digits (3.14159265359), being the first to use series to estimate.

        • Islamic Mathematics

      In the ninth century Al-jwarizmi in his "Algebra" (Hisab al yabr ua al muqabala) notes that the practical man used 22 / 7 as the value of π, the geometer uses 3, and the astronomer 3.1416. In the fifteenth century, the Persian mathematician al-Kashi Ghiyath was able to calculate the approximate value of π with nine digits, using a sexagesimal numerical basis, equivalent to a 16 digit decimal approximation: 2π = 6.2831853071795865.

        • Table of approximations

      Year

      Mathematician or

      document

      Culture

      Approximation

      Error
      (parts per million)

      ~1900 b. C.

      Papyrus of Ahmes

      Egyptian

      28/34 ~ 3,1605

      6016 ppm

      ~1600 b. C.

      Tablet of Susa

      Babylonian

      25/8 = 3,125

      5282 ppm

      ~600 b. C.

      The Bible (Reyes I, 7,23)

      Jewish

      3

      45070 ppm

      ~500 b. C.

      Bandhayana

      Indian

      3,09

      16422 ppm

      ~250 b. C.

      Archimedes from Siracusa

      Greek

      beetwen 3 10/71 y 3 1/7

      he used 211875/67441 ~ 3,14163

      <402 ppm

      13,45 ppm

      ~150

      Claudius Ptolemy

      Greek-Egyptian

      377/120 = 3,141666...

      23,56 ppm

      263

      Liu Hui

      Chinese

      3,14159

      0,84 ppm

      263

      Wang Fan

      Chinese

      157/50 = 3,14

      507 ppm

      ~300

      Chang Hong

      Chinese

      101/2 ~ 3,1623

      6584 ppm

      ~500

      Zu Chongzhi

      Chinese

      entre 3,1415926 y 3,1415929
      empleó 355/113 ~ 3,1415929

      <0,078 ppm
      0,085 ppm

      ~500

      Aryabhata

      Indian

      3,1416

      2,34 ppm

      ~600

      Brahmagupta

      Indian

      101/2 ~ 3,1623

      6584 ppm

      ~800

      Al-Juarismi

      Persian

      3,1416

      2,34 ppm

      1220

      Fibonacci

      Italian

      3,141818

      72,73 ppm

      1400

      Madhava

      Indian

      3,14159265359

      0,085 ppm

      1424

      Al-Kashi

      Persian

      2π = 6,2831853071795865

      0,1 ppm

    • The approximations of number π in

    • The Modern Times

      When the first computers were designed, there began to appear programs for calculating the number π with figures as much as possible. In 1949, ENIAC was able to break all records, earning 2037 decimal places in 70 hours, a few years later (1954) a NORAC arrived to 3092 figures. During the decade of the 1960s IBM were breaking records, until a 7030 IBM was able to reach in 1966 to 250,000 decimal places (8 h 23 min). During this time computers were being tested new algorithms for generating sets of numbers from π.

      In 2009 they found more than two and a half billion decimals using a supercomputer T2K Tsukuba System, formed by 640 high-performance computers, which together get processing speeds of 95 teraflops. To do this it took 73 hours and 36 minutes.

      Year

      Discoverer

      Computer used

      Number of decimal places

      1949

      G.W. Reitwiesner and others

      ENIAC

      2.037

      1954

      NORAC

      3.092

      1959

      Guilloud

      IBM 704

      16.167

      1967

      CDC 6600

      500.000

      1973

      Guillord y Bouyer

      CDC 7600

      1.001.250

      1981

      Miyoshi y Kanada

      FACOM M-200

      2.000.036

      1982

      Guilloud

      2.000.050

      1986

      Bailey

      CRAY-2

      29.360.111

      1986

      Kanada y Tamura

      HITAC S-810/20

      67.108.839

      1987

      Kanada, Tamura, Kobo and others

      NEC SX-2

      134.217.700

      1988

      Kanada y Tamura

      Hitachi S-820

      201.326.000

      1989

      Brothers Chudnovsky

      CRAY-2 y IBM-3090/VF

      480.000.000

      1989

      Brothers Chudnovsky

      IBM 3090

      1.011.196.691

      1991

      Brothers Chudnovsky

      2.260.000.000

      1994

      Brothers Chudnovsky

      4.044.000.000

      1995

      Kanada y Takahashi

      HITAC S-3800/480

      6.442.450.000

      1997

      Kanada y Takahashi

      Hitachi SR2201

      51.539.600.000

      1999

      Kanada y Takahashi

      Hitachi SR8000

      68.719.470.000

      1999

      Kanada y Takahashi

      Hitachi SR8000

      206.158.430.000

      2002

      Kanada and others

      Hitachi SR8000/MP

      1.241.100.000.000

      2004

      Hitachi

      1.351.100.000.000

      2009

      Daisuke Takahashi

      T2K Tsukuba System

      2.576.980.370.000

      5. Value of π with 100 places.

      This is the value of π with its first 100 places:

      3, 1415926535 8979323846 2643383279 5028841971 6939937510
      5820974944 5923078164 0628620899 8628034825 3421170679

      Bibliography

      Internet, especially Wikipedia but not at all, I used other websites too.

      Personal Opinion

      I think this work is very interesting because I could to learn things about very specific number, the number π.

    Vídeos relacionados