Have you ever wondered how pi was calculated before Newton

discovered it. Up until the 250BC, mathematicians knew for a fact that the

value of pi is between 3 and 4. And you may ask that how did they know that,

the answer to that would be that they did that with the help of a hexagon and

square.

A unit circle (area = Ï€)

was inscribed with a regular hexagon of unit sides. And the circle was inscribed in a

square. The hexagon was then further divided into 6 equilateral triangles with

unit sides. Then the perimeter of hexagon was found to be 6 units. Since the

hexagon was inside the circle the circumference of the circle must be a little more than

that of hexagon.

Circumference > perimeter

2 Ï€ > 6

Ï€ >3

On the other hand, the perimeter of square > circumference of the

circle.

2 Ï€<8

Ï€ <4

Hence, it was concluded to be between 3 and 4, i.e, 3< Ï€<4.

Later in the 250BC,** Archimedes** improved this calculation. He bisected

the hexagon into a dodecagon (12 sides) and inscribed one into the circle and inscribed

the circle in another 12-sided polygon. After calculation, it was found that Ï€

lies between 6.212 and 6.431. This process repeated for centuries until they

reached the 96-sided polygon and found a range of 3.1408< Ï€<3.1429.

In the late 16^{th} century, **Francois ViÃ¨te**

doubled a dozen times than Archimedes and computed the perimeters of polygon of

393,216 sides. And by the early 17^{th} century, **Ludolph van Caulen **surpassed

him by calculating perimeter of polygon of 2^62 sides. He spent **25 years** of his

life calculating Ï€ upto 35 correct decimal places.

People were used this method of computing pi until **Sir Newton** came along.

The first step to his discovery of Ï€ was building **Binomial theorem** from the**Pascals triangle**. Sooner after he also delt with the theorem being subjected to

incompetency when power of (1+x) was -1, since it would give an infinite

series. But he proved its accuracy by multiplying both sides by (1+x). The left

hand side gave 1 straightaway and when (1+x) was multiplied on the right hand

side, every term got cancelled leaving only one.

Afterward he went for (1+x) with power of ½ to solve for the area of a

unit circle by equation of the circle, which was given by x^2+y^2=1. Hence, the value y=(1-x^2)^1/2 can be calculated by substituting the value of x by -x^2.

Right after this, he also **invented calculus**. And the integration from 0 to 1

was used to find the area under y=(1-x^2)^1/2 which is the area of the quarter of

circle in the first quadrant. We also know that the area of quarter of circle

is (Ï€r^2)/4. In this case it will be Ï€/4 (since r=1). Upon equating them and calculating we get

the value of Ï€. To make things easier and accurate he decided to integrate from

0 to ½. It gives the area of the circle from the left of y axis to the straight line of

x=1/2. That area can be calculated as area of a sector of 30 degrees, which

will be Ï€/12, and a right triangle of sides 1 and ½ ( we can calculate the

third by Pythagoras).

And upon equating we once again can find the value of Ï€. Not only that

but when you solve up to 5 places of the integration, you get results Ï€ for up to 5 decimal places. And it

will take calculation of integration up to 35 terms to get what Ludolph

van Caulen got after 25 years. This is how years of calculation was simplified

by Sir Isaac in a way it took only a few days to learn the value of pi.

SOURCE- Veritasium on youtube

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