Maths π project School Project Numbers Numbers

PI NUMBER

1. WORLD'S MOST MYSTERIOUS NUMBER

2. π WHAT IS π ? The symbol π is the 16th letter of Greek alphabets. In the old Greek texts, π was used to represent the number 80. Later on, the letter π was chosen by mathematicians to represent a very important constant value related to a circle. Specifically π was chosen to represent the ratio of the circumference of a circle to its diameter.

3. According to the well-known mathematics historian Florian Cajori (1859-1930), the symbol π was first used in mathematics by William Oughtred (1575-1660) in 1652 when he referred to the ratio of the circumference of a circle to its diameter as π/δ , where π represented the periphery of a circle and δ represented the diameter. Where the symbol π in Mathematics came from?

4. Florian Cajori William Oughtred

5. William Jones was a Welsh mathematician, who is most noted for because he published his book “Synopsis Palmoriorum Matheseos” , in which he used π to represent the ratio of the circumference of a circle to its diameter. William Jones

6. Leonhard Euler Among others, Swiss mathematician Leonhard Euler also began using π to represent the ratio of circumference of a circle to its diameter.

7. Value Of Pi π

8. History behind PI (π) In ancient time, after a wheel was invented, the circumference was probably measured for the sake of comparison.
It was important to measure how far a wheel would travel in one revolution. To measure this distance, it was convenient to measure it by placing the wheel on the distance being measured showing that its length is slightly more than three times the

9. diameter. This type of activity repeated with different wheels showed that each time the circumference was just a bit more than three times as long as the diameter. This showed that the value of π is slightly more than 3. Frequent measurement also showed that the part exceeding three times the diameter was very close to 1/9 of the diameters.

10. Approximate value of Pi (π) In Rhind Papyrus, written by Ahmes -an Egyptian in about 1650 B.C., it is said to have been mentioned that if a square is drawn with a side whose length is eight-ninths of the diameter of the circle, then the area of the square so formed and the area of circle would be the same. Ahmes

11. Rhind Papyrus, written by Ahmes

12. Area of circle = (π d/2 ) 2 .. Area of square ABCD = d 2 *8/9 = d 2 *64/81 So, (π d/2 ) 2 = d 2 *64/81 ⇒ π = 256/81 =3.1604938271604938271 ∴ This gives a reasonably close approximated value of π.

13. ARCHIMEDES CONTRIBUTIONS In finding value of

14. Archimedes was born in Syracuse about 287 B.C. and was a Greek mathematician, physicist, engineer, inventor, and astronomer. We all know that Archimedes is credited with the discovery of the principle of buoyancy. He gave the following proposition regarding the circle that had a role in the historical development of the value of π . Archimedes

15. The ratio of the area of a circle to that of a square with side equal to the circle's diameter is close to 11:14. r 2r

16. i.e. π r 2 /4 π 2 = 44/14 = 22/7 This is again a familiar approximation of π which we often use in the problems related to mensuration.

17. Hexagones Inscribed and circumscribe in circle Archimedes perform one more method to find out the value of π . He i nscribe a regular polygon ( a square, a regular pentagon,a regul ar hexagon etc. ) in a give n circle. A nd also circumscribe the polygon about the same circle. In both the cases, the perimeter of the polygon gets closer and closer to the circumference of the circle.

18. He then repeated this process with 12 sided regular polygon, 24 sided regular polygon, 48 sided regular polygon, 96 sided regular polygon, each t ime getting perimeter closer and closer to circumference of the circle. 96 sided regular polygon inscribed perimeter = 3.141 circumscribed perimeter = 3.14 27

19. 3 = 3.14084507042253521126760563380281690 and 3 = 3.142857 Thus, Archimedes gave the value of π which is consistent with what weknow as the value of π today.   Archimedes finally concluded that the value of π is more than 3 but less than 3 . We know that  

20. Chinese Contributions π

21. Liu Hui was a Chinese mathematician and writer who lived in the state of Cao Wei during the Three Kingdoms period of China. Born: Zibo, China Died: 295 AD, China Liu Hui

22. Liu Hui in 263 also used regular polygons with increasing number of sides to approximate the circle. He used only inscribed circles while Archimedes used both inscribed and circumscribed circles. Liu's approximation of π was  

23. Zu Chongzhi, courtesy name Wenyuan, was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415927, a record which would not be surpassed for 800 years. Born: 429 AD, Jiankang Died: 500 AD, China Zu Chongzhi

24. Contribution by others

25. 1. John Wallis (1616-1703), a professor of mathematics at Cambridge and Oxford Universities gave the following formula for π : = 2. Brouncker (1620-1684) obtained the following value of :  

26. 3. Aryabhata (499) gave the value of π as 62832/20,000 = 3.14156 4. Brahmagupta (640) gave the value of π as √ 10 = 3.162277

27. 5. Al-Khowarizmi (800) gave the value of π as 3.1416. 6. Yasumasa Kanada and his team at the University of Tokyo calculated the value of π to 1.24 trillion decimal places.

28. 7. Babylonian used the value of π as 3+ = 3.1256.   8. French mathematician Francois Viete (1540-1603) calculated π correct to nine decimal places. He calculated the value of π to be between the numbers 3.1415926535 and 3.1415926537. 9. S. Ramanujan (1887-1920) calculated the value of π as = 3.14592652 which is correct to eight decimal places.  

29. 10. Leonhard Euler came up with an interesting expression for obtaining the value of π as- = 1- 1- 1- 1- 1-   -Leonhard Euler

30. A Paradox

31. Lenght of the arc = π r ------ (1) A B

32. A B Lenght of the arc = 2 × × π r = π r  

33. A B Lenght of the arc = 4 × × π r = π r  

34. So, even if we go on increasing the number of semicircles the total length of arc will be π r,

35. When we go on increasing the no. of semi circles at some point, it will be equal to the horizontal line in semicircle which is equal to d = 2r ------ ( 2 ) ∴By equating eq n 1 & eq n 2, we get π r = 2r ⇒ π = r And here we get the contradiction as we know that the value of pi is not equal to 2.

36. Made By- Atharva Bhokare Mahesh Chavhan Atharva Kule Atharva Tekale Yash Yadav Tanmay Deshmukh Vedant Rajput Thank You