Sunday, 11 August 2013

·      Interesting Pi

            March  14 is celebrated as international pi day 
    that is pi is 3.14 and march 14 is 3/14
·        Pi day is also Albert Einstiens birthday along with        birthdays of Appolo 8 ,commander Frank borman, astronomer   Giovanni schiparreli and last man to moon Gene cernan
·        In 1768 Johann lambert proved that the value of pi is an    irrational number and in 1882 ferdinand lindemann a  renowned mathematician proved that pi is transcendal
·        If you were to print 1billion decimal values of pi in  ordinary font then, it would stretch from newyork to kanas
·        The record of calculating pi as of 2010 is to 5 trillion
·        The first million decimals of pi consists of 99758 ones    100026 twos 100229 threes 100230 fours 100359 fives 99548    sixes 9980 sevens 99985 eights 100106 nines
·        At position 763 there are six 9s in a row which is known as   frymans point
·        In greek alphabet pi[piwas] is the sixteenth letter in English alphabet p is also the the sixteenth letter.

                                                                               By Anupama.m- IX A
Arya bhata


Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta
The works of Aryabhata dealt with mainly mathematics and astronomy. He also worked on the approximation for pi.
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the “bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Bhramagupta’s references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" does not fit the meter either.
Time and place of birth
Aryabhata mentions in the Aryabhatiya that it was composed 3,630 years into the Kali Yuga when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476.
Aryabhata's birthplace is uncertain, but it may have been in the area known in ancient texts as Ashmaka India which may have been Maharashtra or Dhaka.
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. A verse mentions that Aryabhata was the head of an institution (kulapati) at Kusumapura, and, because the University of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.
Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.
Aryabhata is the author of several treatises on mathematics and Astronomy, some of which are lost.
His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya coversalgebra , plane trignometry, and spherical trignometry..
The Arya-siddhanta, a lot work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators,. This work appears to be based on the older Surya siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.
A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known.
Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara 1 calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:
The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) 

It is no doubt that the world today is greatly indebted to the contributions made by Indian mathematicians. One of the most important contribution made by them was the introduction of decimal system as well as the invention of zero. Here are some the famous Indian mathematicians dating back from Indus Valley civilization and Vedas to Modern times. -
 Aryabhata worked on the place value system using letters to signify numbers and stating qualities. He discovered the position of nine planets and stated that these planets revolve around the sun. He also statedthe correct numberof days in a year that is 365. -
 The most significant contribution of Brahmagupta was the introduction of zero(0) to the mathematics which stood for “nothing”. -
Srinivasa Ramanujan
Srinivasa Ramanujan is one of the celebrated Indian mathematicians. His important contributions to the field include Hardy-Ramanujan-Littlewood circle method in number theory, Roger-Ramanujan’s identities in partition of numbers, work on algebra of inequalities, elliptic functions, continued fractions, partial sums and products of hypergeometric series. -
P.C. Mahalanobis
 Prasanta Chandra Mahalanobis is the founder of Indian Statistical Institute as well as the National Sample Surveys for which he gained international recognition. -
 C.R. Rao
 Calyampudi Radhakrishna Rao, popularly known as C R Rao is a well known statistician, famous for his “theory of estimation”. -
DR. Kaprekar
DR. Kaprekar discovered several results in number theory, including a class of numbers and a constant named after him. Without any formal mathematical education he published extensively and was very well known in recreational mathematics cricle. -
Harish Chandra

Harish Chandra is famously known for infinite dimensional group representation theory. - Satyendranath Bose Known for his collaboration with Albert Einstein. He is best known for his work on quantum mechanics in the early 1920s, providing the foundation for Bose–Einstein statistics and the theory of the Bose–Einstein condensate. 
                                                                                                                                                                         AKHILA- IX A

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 

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 

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

Oh I think Mathematics is the most amazing subject in the world.
 Here are some facts that will make you feel math is amazing
   Much as with people, there are irrational, perfect and complex numbers.
   1260 is called vampire number because 21 X 60 fangs vampires have.[21 X 60=1260]
   It takes 10 days for your heart to beat one million beats and 27 years for it to beat one billion beats.
   The Fibonacci sequence are numbers where each following number is the sum of the previous two:
         0 1, 1, 2, 3,5 ,8, 13, 21, 34, 55 ,89 ...

                                                                                                IX B