User:Anshuman trip/sandbox

Mathematical Pattern
Mathematics requires patience and patience comes with hard work. The time was 1666, Sir Isaac Newton was busy with his expressions like of binomial theorem. He was at home due to the plague. He was wondering about the expansion of the expressions

as the constants which are numbers of Pascal's triangle. The way everyone was following was by finding the perimeter of a much-sided figure and dividing it by the diameter which is the approximate value of pi. Even Archimedes followed the same method and found pi by a 96-sided polygon. At last, this method was followed by Ludolf van Ceulen  who calculated pi with 2^62 sides which is

4,611,686,018,427,387,904 sides '''spending 25 years on that. ￼'''

Curiosity and formulas
The binomial theorem states that if (a + b)^n is there then n always belongs to positive numbers. Newton thought that what if we blindly apply the theorem by putting n as suppose (-1) on (1 + a)^n as he plugs in that, he gets the series of constants as 1,-1,1,-1 ...and so on. Newton thought that it could be wrong but multiplying 1 + a gives us 1 which satisfies that it is correct and he started to extend Pascal's triangle and made that. For that, he placed other constants of minus power which he gets a triangle, and neglecting

the negative signs, that is the same triangle series just tilted right. He tried it for n is 1/2 as he knows x² + y² = 1 which he can solve for 1 and get there as y = (1 - x²)½. So he tried it and as the radius of the circle is 1 unit the area of it will be just the same as pi. He integrated on both sides from 0 to 1 so he would be getting the area of a quarter circle so pi/4 is equal to the integrated series.

So pi is 4 times it and the higher the value you calculate, the much you get the accurate value of pi with more precision. Putting some effort into extending, observing patterns looking for new ideas, and trying them can make them more easy and efficient and states that the common way couldn't be the best every time. Anshuman's