NCERT Class 9 Maths Lab Manual – Story of π

NCERT Class 9 Maths Lab Manual – Story of π

TITLE
Story of π.

PRE-REQUISITE KNOWLEDGE

1. Irrational numbers
   Concept of π (pi)
   Area of circle and square

THEORY

1. Irrational numbers are those numbers which cannot be expressed in the form of \(\frac { p }{ q }\) , where p and q are integers and q ≠ 0. e.g. √2, √3, √5, √7, etc.
2. 1. The number π (pi) is an irrational number, which means that its value cannot be expressed in the form of \(\frac { p }{ q }\), where p and q are integers and q ≠ 0.
   2. It is also a transcendental number, i.e. no finite sequence of algebraic operations on integers.
   3. The constant π is the first Greek letter, which is used to find the circumference or perimeter of a circle.
   4. The number π is the ratio of the circumference of a circle to its diameter.
      i. e. π = \(\frac { c }{ d }\) , where c is the circumference of a circle and d is the diameter of a circle.
      
   5. The first scientist Archimedes Syracure (287 BC) gave the first theoretical calculated value of pi (π).
3. 1. Area of circle = πr²  = \(\frac { \pi { d }^{ 2 } }{ q } \)
      and r = \(\frac {d}{2}\).
      where r and d are the radius and diameter of circle respectively
   2. Area of square = Side x Side

VALUES OF π
Different mathematicians calculated value of π in different manners, some of them are given below.

1. Ahmes in Rhind Papyrus
   Ahmes states that, if a square is drawn with a side whose length is \(\frac {8}{9}\)th of the diameter of the circle, then the area of the square so formed and area of the circle would be same.
   
2. Archimedes
   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.
      
      This approximate value is generally used in the problems related to mensuration.
   2. He inscribed a regular polygon (equilateral triangle, square, regular pentagon, regular hexagon, etc.) in a given circle and also circumscribe the polygon about the same circle.
      
      In both the cases, the perimeter of the polygon gets closer to the circumference of the circle. This process is also repeated with 12, 24, 48, etc., sided regular polygon.
      In each time, perimeter is closer to the circumference of the circle.
      
3. Liu Hui (Chinese mathematician) Liu Hui used regular polygon 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 = \(\frac { 3927 }{ 1250 }\)  =3.1416
4. Zu Chongzhi (Chinese astronomer and mathematician)
   
5. John Wallis (Professor of Mathematics at Cambridge and Oxford Universities)
   He gave the following formula for π
   
6. Brouncker He obtained the following value of \(\frac{4}{\pi}\) ,
   
7. Aryabhata He gave the value of π as = \(\frac{62832}{20000}\) = 3.14156
8. Brahmagupta He gave the value of π as √10 = 3.162277
9. Al-Khwarizmi He gave the value of π as 3.1416.
10. Babylonian He used the value of π as 3 + \(\frac { 1 }{ 8 }\) = 3.125.
11. Yasurnasa Kanada and his team at University of Tokyo, calculated the value of π to 1.24 trillion decimal places.
12. Francois Viete calculated n correct to nine decimal places. He calculated the value of π to be between the numbers 3.1415926535 and 3.1415926537.
13. Ramanujan calculated the value of π as \({\left(\sqrt {{9}^{2}+\frac{{19}^{2}}{22}}\right)}^{1/4}\) = 3.14592652..which is correct to eight decimal places.
14. Leonhard Euler came up with an interesting expression for obtaining the value of π as
    \(\frac {2}{\pi}\) =\({\left( 1-\frac {1}{4}\right)}\quad\)\({\left(1-\frac {1}{4}\right)}\quad\)\({\left( 1-\frac {1}{16}\right)}\quad\)\({ \left(1-\frac{1}{36}\right)}\quad \)\({\left(1-\frac {1}{64}\right)}\quad\)\({ \left( 1-\frac {1}{100}\right)}\quad\)….

A π PARADOX

In the above figure, perimeter of the semi-circle with diameter AB =\(\frac {\pi }{ 2}\) (AB)Sum of the perimeters of smaller semi-circles =\(\frac { \pi a }{2}\)+\(\frac { \pi b }{ 2 }\)+\(\frac { \pi c }{ 2 }\)+\(\frac { \pi d }{ 2 }\)+\(\frac { \pi e }{ 2 }\)  =\(\frac {\pi }{ 2}\) (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 AS say of 2 units.

In the above figures, the sum of the lengths of the perimeters of smaller semi-circles appears to be approaching the length of the diameter AS but infact it is not, because the lengths of the perimeters of the smaller semi-circles is \(\frac { \pi \times 2 }{ 2 }\)= π, while length of AS is 2 units. So, both cannot be the same.

CONCLUSION
Here, we conclude that generally the value of π is used as \(\frac{22}{7}\) or 3.141 in different areas of field.

APPLICATION
The 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.