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Mathematics: On the "IP" PYRAMIDS OF THE WORLD

π name

The notation the Greek letter π comes from the initial words of Greek origin "περιφέρεια" (periphery) and "περίμετρον" (perimeter) of a circle. This notation was first used in 1706 by Welsh mathematician William Jones and popularized by mathematician Leonhard Euler in his book "Introduction to Calculus" in 1748. It was formerly known as Ludolph constant (after the mathematician Ludolph van Ceulaer) or as Archimedes' constant (not to be confused with Archimedes number).

rational and transcendental number

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, ie it is not the root of any polynomial with integer coefficients. In the nineteenth century German mathematician Ferdinand Lindemann showed this fact, thereby permanently closing the permanent and intensive research on the problem of squaring the circle indicating that no solution.

also know that π is not a Liouville number (Mahler, 1953), ie not only momentous but can not be approximated by a sequence of sound "fast converging" (Stoneham 1970). History

calculation of the value of π

Search the largest number of decimal places the number π has been a constant effort of many scientists throughout history. Some historical approaches π are as follows.

Ancient Egypt: (In modern notation)

Mesopotamia

Some mathematicians Mesopotamians used in the calculation of segments, values \u200b\u200bof π equal to 3, reaching some cases approximate values, as 3 + 1 / 8.

Biblical references

One of the oldest indirect references approximate value of π can be found in a verse from the Bible:

"He also cast a sea of \u200b\u200bten cubits from one side to another, perfectly round. He was five cubits high and around a line of thirty cubits. "

I Kings 7:23 (Reina-Valera 1995)

A similar quote can be found in II Chronicles 4:2. It appears in a list of requirements for the construction of the Great Temple of Solomon, built on the 950 a. C. Both appointments are 3 as the value of π is a significant loss of accuracy compared to previous estimates of Egyptian and Mesopotamian.

The Greek mathematician Archimedes (third century BC) was able to determine the value of π, including the range of 3 10/71, the minimum value, and 3 seventh, the maximum value. With this approach Archimedes is obtained with an error value ranging between 0.024% and 0.040% on the actual value. The method used by Archimedes was very 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 a 96-sided polygons.

Around 20 d. C., the Roman architect and engineer Vitruvius π calculated as the fractional value 25 / 8 by measuring the distance traveled in one revolution by a wheel of known diameter.

In the second century, Ptolemy provides a fractional value of approaches:

China
Mathematics

The calculation of pi was an attraction for the math experts from all cultures. By 120, the Chinese astrologer Chang Hong (78-139) was among the first to use the approximation that deduced from the ratio of the volume of a cube and sphere respective registered. A century later, the astronomer Wang Fang estimated at 142/45 (3.155555), although the method is unknown. A few years later, around 263, the mathematician Liu Hui was the first to suggest that 3.14 was a good approach, using a polygon de 96 o 192 lados. Posteriormente estimó π como 3,14159 empleando un polígono de 3.072 lados.

A finales del siglo V, el matemático y astrónomo chino Zu Chongzhi calculó el valor de π en 3,1415926 al que llamó «valor por defecto» y 3,1415927 «valor por exceso», y dio dos aproximaciones racionales de π: 22/7 y 355/113 muy conocidas ambas,siendo la última aproximación tan buena y precisa que no fue igualada hasta más de nueve siglos después, en el siglo XV.

Matemática india

Usando un polígono regular inscripto de 384 lados, a finales del siglo V Indian mathematician Aryabhata estimated value 3.1416. A mid-seventh century, estimating the approximation error of Aryabhata, Brahmagupta estimated π as , calculation much less precise than its predecessor. Around 1400 Madhava get an accurate approximation to 11 digits (3.14159265359), being the first series used for estimation.

Islamic Mathematics

IXAl In the century-Khwarizmi in his "Algebra" ( Hisab to ua to muqabala Jabr) 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 mathematician Persian Ghiyath al-Kashi was able to calculate the approximate value of π with nine digits, using a sexagesimal numerical basis, which equates to an accuracy of 16 decimal digits: 2π = 6.2831853071795865.

European Renaissance

From the twelfth century, with the use of Arabic numerals in the calculations are greatly facilitated the possibility of obtaining better estimates for π. The mathematician Fibonacci, in his "Practice Geometriae" amplifies the Archimedes method, providing a narrower range. Some mathematicians of the seventeenth century, as Viète, up to 393,216 polygons used to approximate sides with good accuracy at 3.141592653. In 1593 Adriaan van flamenco Room ( Adrianus Romanus) obtained an accuracy of 16 decimal digits using the Archimedes method. Interesting Facts

• On July 22 (22 / 7) is the day devoted to the approximation of π . The
• March 14 (3 / 14 date formats USA) is also marked as the day pi where fans are celebrating this issue with different actions. Curiously, it is Einstein's birthday .
• 355/113 (~ 3.1415929) is sometimes referred to as a quasi-perfect simulation! The
• A9.com search engine users who choose shop amazon.com as offer discounts (π / 2)% on purchases.
• John Squire (of the band The Stone Roses ) π mentioned in a song written for his second band, The Seahorses called "Something Tells Me." The song ends with a lyric like: "What's the secret of life? It's 3.14159265, yeah yeah!".
• The first million digits of π and its inverse 1 / π can be found at the Gutenberg Project or this link.
• numbering versions of word processing program TeX of Donald Knuth run as the digits of π. The version of 2002 was labeled with
3.141592
• this number is used in the series of signals from the earth in order to be identified by an extraterrestrial intelligent civilization.
• The probability that two positive integers are randomly chosen prime each is 6 / π 2
• online programs are looking for your phone number in the first digits of π 50,000,000
• In some programming languages \u200b\u200b can find many digits as you want by simply using expressions like: RealDigits [N [Pi, 105]] in ' Mathematica. "
• In 2002 Japanese Akira Haraguchi broke the world record for 13 hours reciting 83,431 digits of pi without stopping, doubling the previous record also held by the Japanese Hiroyuki Goto. The October 4 2006 of , at 1:30 am, and after 16 hours, Haraguchi again broke his own record by reciting pi digits 100,000, making a stop each two hours 10 minutes for air.
• The maximum number of digits of π required to find any sequence day-month-year with four digits in the decimal expansion of pi is 60,872.
• There is a Kate Bush song called "Pi" in which are recited over twenty-digit decimal number.
• In Argentina, the mobile phone number for emergencies in underground train stations and the number Pi: \u200b\u200b3.1416.
• The main value of the expression i ⁱ is a real number and is given by
• The website thinkgeek.com shirts and accessories can be purchased with π. In the link you can see a T-shirt is constructed π letter with his first 4493 digits.
• In England a crop circle that appeared to be investigated by scientists Statians State proved that his meaning was that of π (pi) A vehicle
• Mazda 3 amended, which was added 27 digits of π after 3. Srinivasa Ramanujan
• published an approximate solution, with ruler and compass, the square Circle in 1913 where he received a segment approximately equal to :

For further information please visit their main source:

Wikipedia.Org