Class 09 PROJECT 04 π - WORLD'S MOST MYSTERIOUS NUMBER

 PROJECT 04

π - WORLD'S MOST MYSTERIOUS NUMBER

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. 

Symbolically π= c d , where c represents the circumference and d represents the length of the diameter of the circle. 

Since the diameter of a circle is twice the radius, d = 2r, where r is the radius. So, π 2 = c r . Where the symbol π in Mathematics came from? 

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. In 1706, William Jones (1675-1749) published his book Synopsis Palmoriorum Matheseos, in which he used π to represent the ratio of the circumference of a circle to its diameter. 

This is believed to have been the first time π was used as it is defined/used today. Among others, Swiss mathematician Leonhard Euler also began using π to represent the ratio of circumference of a circle to its diameter.

VALUE OF π 

It is said that after a wheel was invented, the circumference was probably measured for the sake of comparison. Perhaps in the early days, 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 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. Fig. 1 AB: 

Circumference of the wheel 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 diameter. 

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. 

Area of circle = 2 2 π π 2 4 = d d

Area of square ABCD = 2 8 64 2 9 81 = d d So, 2 π 64 2 4 81 = d d implies 256 π 3.1604938271604938271 81 = = This gives a reasonably close approximated value of π 

ARCHIMEDES CONTRIBUTIONS 

Archimedes, born in Syracuse about 287 B.C. gave the following proposition regarding the circle that had a role in the historical development of the value of π. 1. 

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. Fig. 3 2 2 11 4 14 π = r r i.e., 44 22 14 7 π = = 

This is again a familiar approximation of π which we often use in the problems related to mensuration. 

The circumference of a circle is less than 1 3 7 times of its diameter but more than 10 3 71 times the diameter, i.e., 10 3 71 < π < 1 3 7 . 

Let us see how Archimedes actually arrived at this conclusion. 

What he did was to inscribe a regular polygon (an equilateral triangle, a square, a regular pentagon, a regular hexagon etc.) in a given circle [see Fig. (4)] and 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. 

He then repeated this process with 12 sided regular polygon, 24 sided regular polygon, 48 sided regular polygon, 96 sided regular polygon, each time getting perimeter closer and closer to circumference of the circle. 

Archimedes finally concluded that the value of π is more than 10 3 71 but less than 1 3 7 . We know that 10 3 3.14084507042253521126760563380281690 71 = and 1 3 3.142857 7 = Thus, Archimedes gave the value of π which is consistent with what we know as the value of π today.

CHINESE CONTRIBUTIONS 

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 3927 3.1416 1250 = Zu Chongzhi (429-500), a Chinese astronomer and mathematician found that 355 π= 113 = 3 . 141592920353982300884 955752212389380530973 451327433628318584070796 460176991150442477876106 1946902654867256637168 

CONTRIBUTION BY OTHERS 

1. John Wallis (1616-1703), a professor of mathematics at Cambridge and Oxford Universities gave the following formula for π : ( )( ) 2 2 4 4 6 6 2 2 ... ... 2 1 3 3 5 5 7 2 –1 2 1 π × × × × = × × × × × × × × + n n n n

 2. Brouncker (1620-1684) obtained the following value of 4 π : 4 π = 2 2 2 2 2 1 1 3 2 5 2 7 2 9 2 2 ...

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 

5. Al-Khowarizmi (800) gave the value of π as 3.1416 

6. Babylonian used the value of π as 1 3 3.125 8 + = 

7. Yasumasa Kanada and his team at University of Tokyo calculated the value of π to 1.24 trillion decimal places. 

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 2 2 4 19 9 22 + = 3.14592652 ... which is correct to eight decimal places. 10. Leonhard Euler came up with an interesting expression for obtaining the value of π as 2 1 1 1 1 1 1– 1– 1– 1– 1– ..., π 4 16 36 64 100 = A π Paradox

In the above figure, perimeter of the semi-circle with diameter AB = (AB) 2 π

 Sum of the perimeters of smallar semi-circles 2 2 2 2 2 2 π π π π π π = + + + + = a b c d e (a+b+c+d+e) This may not 'appear' to be true but it is! Let us now proceed in the following way. 

Increase the number of smaller semi-circles along the fixed line segment AB say of 2 units. Fig. 6 Fig. 7 Fig. 8 Fig. 9

 In the above figures, the sum of the lengths of the perimeters of smaller semicircles "appears" to be approaching the length of the diameter AB but in fact it is not! 

because the lengths of the perimeters of the smallar semi-circles is 2 2 π × = π while length of AB is 2 units. So, both cannot be the same.

In the above figure, perimeter of the semi-circle with diameter AB = (AB) 2 π 

Sum of the perimeters of smallar semi-circles 2 2 2 2 2 2 π π π π π π = + + + + = a b c d e (a+b+c+d+e) This may not 'appear' to be true but it is! 

Let us now proceed in the following way. Increase the number of smaller semi-circles along the fixed line segment AB say of 2 units. Fig. 6 Fig. 7 Fig. 8 Fig. 9 

In the above figures, the sum of the lengths of the perimeters of smaller semicircles "appears" to be approaching the length of the diameter AB but in fact it is not! because the lengths of the perimeters of the smallar semi-circles is 2 2 π × = π while length of AB is 2 units. So, both cannot be the same. 

CONCLUSION 

π can be seen as a number with unusual properties. It has wide variety of applications in real life.

 APPLICATION 

Value of π is used in finding areas and perimeters of designs related to circles and sector of circles. It has applications in the construction of racetracks and engineering equipments.