## Tuesday, 6 August 2013

Amazing facts on Pi

Ø  Pi is the most recognized mathematical constant in the world. Scholars often consider Pi the most important and intriguing number in all of mathematics.

Ø  The symbol for pi (π) has been used regularly in its mathematical sense only for the past 250 years.

Ø    We can never truly measure the circumference or the area of a circle because we
can never truly know the value of pi. Pi is an irrational number, meaning its digits go on forever in a seemingly random sequence.

Ø  In the Greek alphabet, π (piwas) is the sixteenth letter. In the English alphabet, p is
also the sixteenth letter.

Ø  The letter π is the first letter of the Greek word “periphery” and “perimeter.” The
symbol π in mathematics represents the ratio of a circle’s circumference to its
diameter. In other words, π is the number of times a circle’s diameter will fit around its
circumference.

Ø  Egyptologists and followers of mysticism have been fascinated for centuries by the fact that the Great Pyramid at Giza seems to approximate pi. The vertical height of the pyramid has the same relationship to the perimeter of its base as the radius of a circle has to its circumference.

Ø  The first 144 digits of pi add up to 666 (which many scholars say is “the mark of the Beast”). And 144 = (6+6) x (6+6).d
Ø  If the circumference of the earth were calculated using π rounded to only the ninth
decimal place, an error of no more than one quarter of an inch in 25,000 miles would
result.

Ø     In 1995, Hiroyoki Gotu memorized 42,195 places of pi and is considered the current pi champion. Some scholars speculate that Japanese is better suited than other languages for memorizing sequences of numbers.

Ø      Ludolph van Ceulen (1540-1610) spent most of his life calculating the first 36 digits of pi (which were named the Ludolphine Number). According to legend, these numbers were engraved on his now lost tombstone
Ø     William Shanks (1812-1882) worked for years by hand to find the first 707 digits of pi. Unfortunately, he made a mistake after the 527th place and, consequently, the following digits were all wrong.

Ø  In 2002, a Japanese scientist found 1.24 trillion digits of pi using a powerful computer called the Hitachi SR 8000, breaking all previous records.
Ø  Since there are 360 degrees in a circle and pi is intimately connected with the circle, some mathematicians were delighted to discover that the number 360 is at the 359th digit position of pi.
Ø  Computing pi is a stress test for a computer—a kind of “digital cardiogram.”

Ø  Pi has been studied by the human race for almost 4,000 years. By 2000 B.C., Babylonians established the constant circle ratio as 3-1/8 or 3.125. The ancient Egyptians arrived at a slightly different value of 3-1/7 or 3.143.a

Ø  One of the earliest known records of pi was written by an Egyptian scribe named Ahmes (c. 1650 B.C.) on what is now known as the Rhind Papyrus. He was off by less than 1% of the modern approximation of pi (3.141592).l

Ø  The Rhind Papyrus was the first attempt to calculate pi by “squaring the circle,” which is to measure the diameter of a circle by building a square inside the circle.

Ø  The “squaring the circle” method of understanding pi has fascinated mathematicians because traditionally the circle represents the infinite, immeasurable, and even spiritual world while the square represents the manifest, measurable, and comprehensive world.

Ø  The first million decimal places of pi consist of 99,959 zeros, 99,758 1s, 100,026 2s, 100,229 3s, 100,230 4s, 100,359 5s, 99,548 6s, 99,800 7s, 99,985 8s, and 100,106 9s.

Ø  ”Pi Day” is celebrated on March 14 (which was chosen because it resembles 3.14). The official celebration begins at 1:59 p.m., to make an appropriate 3.14159 when combined with the date.

Ø  The Bible alludes to pi in 1 Kings 7:23 where it describes the altar inside Solomon’s temple: “And he made a molten sea of ten cubits from brim to brim . . . and a line of thirty cubits did compass it round about.” These measurements procure the following equation: 333/106 = 3.141509.k

Ø     Pi was first rigorously calculated by one of the greatest mathematicians of the ancient world, Archimedes of Syracuse (287-212 B.C.). Archimedes was so engrossed in his work that he did not notice that Roman soldiers had taken the Greek city of Syracuse. When a Roman soldier approached him, he yelled in Greek “Do not touch my circles!” The Roman soldier simply cut off his head and went on his business.

Ø  A refined value of pi was obtained by the Chinese much earlier than in the West. The Chinese had two advantages over most of the world: they used decimal notations and they used a symbol for zero. European mathematicians would not use a symbolic zero until the late Middle Ages through contact with Indian and Arabic thinkers.

Ø           Ancient mathematicians tried to compute pi by inscribing polygons with more and more sides that would more closely approach the area of a circle. Archimedes used a 96-sided polygon. Chinese mathematicians Liu Hui inscribed a 192-sided polygon and then a 3,072-sided polygon to calculate pi to 3.14159. Tsu Ch’ung and his son inscribed polygons with as many as 24,576 sides to calculate pi (the result had only an 8-millionth of 1% difference from the now accepted value of pi).

Ø          William Jones (1675-1749) introduced the symbol “π” in the 1706, and it was later popularized by Leonhard Euler (1707-1783) in 1737.

Ø  The π symbol came into standard use in the 1700s, the Arabs invented the decimal system in A.D. 1000, and the equal sign (=) appeared in 1557.e

Before the π symbol was used, mathematicians would describe pi in round-about ways such as “quantitas, in quam cum multipliectur diameter, proveniet circumferential,” which means “the quantity which, when the diameter is multiplied by it, yields the circumference.”

 Leonardo da Vinci briefly worked on ”squaring the circle” or approximating pi
Ø  Leonardo da Vinci (1452-1519) and artist Albrecht Durer (1471-1528) both briefly worked on “squaring the circle or approximating pi.

Ø  There are no occurrences of the sequence 123456 in the first million digits of pi—but of the eight 12345s that do occur; three are followed by another 5. The sequence 012345 occurs twice and, in both cases, it is followed by another 5.

Ø   The father of calculus (meaning “pebble used in counting” from calx or “limestone”), Isaac Newton calculated pi to at least 16 decimal places.

Ø   Pi is also referred to as the “circular constant,” “Archimedes’ constant,” or “Ludolph’s number.
In the seventeenth century, pi was freed from the circle and applied also to curves, such as arches and hypocycloids, when it was found that their areas could also be expressed in terms of pi. In the twentieth century, pi has been used in many areas, such as number theory, probability, and chaos theory.

Ø  The first six digits of pi (314159) appear in order at least six times among the first 10 million decimal places of pi.

Ø   Thirty-nine decimal places of pi suffice for computing the circumference of a circle girding the known universe with an error no greater than the radius of a hydrogen atom
Plato (427-348 B.C.) supposedly obtained for his day a fairly accurate value for pi: √2 + √3 = 3.146.a
Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:
0 occurs 599,963,005 times,
1 occurs 600,033,260 times,
2 occurs 599,999,169 times,
3 occurs 600,000,243 times,
4 occurs 599,957,439 times,
5 occurs 600,017,176 times,
6 occurs 600,016,588 times,
7 occurs 600,009,044 times,
8 occurs 599,987,038 times,
9 occur 600,017,038 times.

AJMAL ROSHAN--IX A