Random Math Problems 1- Computing Nested Infinite Radicals

Compute

This is a problem containing an infinitely nested radicals of 20,where you have to compute the entire nested radical.It goes on forever which is a hurdle for our intuitive understanding.

Mathematical Steps (Algorithm)

First we need to assign this entire infinitely nested radicals to a variable,like s ,and declare it as equation (1).

We need to square both sides of the equation to separate one integer out of the infinitely nested radical so that it becomes an rational number+ infinitely nested radical. We are going to need this rational number later. We have to declare it as equation (2).

now we need to subtract equation (1) from equation (2)

Now we notice that the infinitely nested radicals cancel out.

(we can factor out s )

Now we look for two consecutive integers among the factors of 20 which can satisfy this equation

Factors of 20 = 1,2,4,5,10,20

4 and 5 are the consecutive integers. s has to the bigger one,as s>(s-1)

Therefore, s=5

Statement: One argument can be made against assigning variables to infinity, because infinity is not a number, it’s a concept. One can solve questions like is 0.999.. = 1? , using variables and prove them equal, but this confusion is the reason why I think limits in calculus are introduced.

Saikanam Siam
Junior
Brooklyn Technical High School

Comments

Using Python, Quadratic approximation of sine function without the idea of limits or infinitesimal quantities (Mini Research Paper)

The question that I actually I started out was, Is there any way to express transcendental functions using polynomials avoiding the idea of limits or infinitesimal quantities? Origin of the Question Nowadays,we learn about the Cauchy-Weierstrass interpretation of limits in our calculus classes.Although I genuinely think that it works and it’s more convenient but I thought there would be creative other ways to do this without the idea of estimating infinitesimal quantities.I found one paper named “ The Lost Calculus ”[1],it avoids the idea of abstract limits but fails when it comes to transcendental functions.I knew that if I broke down the sine function to be humble polynomials I could than use something like Descartes’s method[1] of Tangents to that polynomial to get the derivative without using the idea of limits. That’s when I was stumbled upon this question. First thing that came to my mind was a sine function,I was thinking if there was any way to represen...

Solving Linear Diophantine Equations for all integer solutions Using Python

What is a linear Diophantine Equation? The simplest linear Diophantine equation takes the form ax + by = d , where a , b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b . Moreover, if ( x , y ) is a solution, then the other solutions have the form ( x + kv , y − ku ) , where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b . (source: wiki) To know whether a linear diophantine equation has integer solutions, we...

Quantum Mechanics- A conceptual introduction

“ Anyone who thinks they can talk about Quantum Mechanics without getting dizzy hasn’t yet understood the first word of it. “ - Niels Bohr Quantum Mechanics is not something that can be predicted by using normal senses or intuition. From the dawn of civilization,we’ve been doing basic physics.Even in the stone ages,when someone threw a brick at us,we didn’t calculate the trajectory or the velocity vector of that brick to know where it’s path. We used our intuition,in a macroscopic world, we learned to predict as we became more and more evolved species.Everything follows our intuition when we do something in our day to day life,by using our classical mechanics and logical absolute way of thinking. But things are different in a microscopic level,those things are beyond our senses,and classical mechanics fails to describe it in an orderly manner. I think our leap into microscopic world started when we really tried to push classical mechanics to objects at a very small level,in...

Saik Anam Joaddar Siam

Search This Blog