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What is the Basel Problem

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  • Last Updated : 07 Jul, 2021
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What is the Basel Problem :
The Basel problem is an issue of Pietro Mengoli’s theory of numbers in 1644 and Leonhard Euler’s resolution in 1734. Since Euler’s solution stayed open for 90 years, at the age of 28 he was immediately renowned for discovering solution to this problem.
This problem basically asks for sum of inverse of the square number. Initially, it may sound somewhat confusing, but after reading this article you will be clear about concept and approach of this problem.

Let’s define k first, as it will help us initially with the problem. So here’s k…


Here k is an abbreviation. Now you need to find inverse sum of this ‘k’. This problem has five different ways of solving it.
There goes a certain probability that drags this problem in the complicated section.

Finding Inverse using Basel Problem :
For finding inverse, there are several different ways, and certainly, Basel way, is one of best possible ways to find inverse sum.

  • Just assume that you have abbreviation and then simply add a pi value to base of it. 
  • Then add it to the original question. 

Sounds Simple, Right? Well, indeed it is. Because the way of solving it is radically easy, some great mathematicians called it the fools way of solving the problem, however, the majority found it really fascinating as they did not have access to such powerful scientific calculators at that time.
The release of this problem garnered a lot of attention. People never had to do the advanced calculations neither had to invest in hiring metrics, all they had to do was to provide values, inverse them out and add to the original term! 
Euler could be called as the main person behind development of solution to this problem, thus he got immediate attention out of it.

A Short Back Story of Basel Problem :
The Basel Problem is straightforward. Despite this, it baffled mathematicians for 90 years. In 1734, a 28-year-old physics professor called Leonhard Euler made headlines when he published a solution. 
There have been several debates and concerns regarding the proof of this problem and, as well, some say that this is unsolvable, but there is evident proof that it is solvable. The trigonometric method of achieving the solution is mentioned below for the sake of it.

Proof :

We have,
sin x = x - x3/3! + x5/5! - x7/7! + ...
sin x /x = 1 - x2/3! + x4/5! - x6/7! + ...
ยป sinx/x (1-pi/x)

As you may see, that changing certain values and transposing the sin x we can quickly have the inverse of a function and that too without much calculation, something which was not easy back in the time when there were no computers.

This in turn proved that humans could calculate complex inverse functions without the need for specific hardware, something which was amazingly fantastic at that time!

Linkage to the Riemann Zeta Function :
The Riemann zeta function is one of the most significant functions in mathematics because of its relationship to distribution of prime numbers. And the solution for this Basel problem provided a conjunction point for connecting Riemann zeta function to this problem. It had major implications at that time. Eventually, mathematicians also had to admit its efficiency and they admitted it. This was not first time but, when Euler didn’t have a certain justification for a problem. Just like every other proof, he even proved existing solution to this problem!

Conclusion :
Euler was the first person who devised a method for actually solving this problem, and since back then it’s due to him that we have actually arrived at a possible solution to Basel Problem. The trick might sound absurd at this time because of technical advancements in science and technology, but it was a great discovery back at that time!

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