FROM 1 TO A USEFUL INFINITE FRACTION

 Hello! In this blog post, I will try to explain about a class of infinite fractions(which I call the Cha Fractions, but I don't know the technical term). Before answering the question, "What are they?",  I have to answer the question, "Why are they important in the first place?" 

The thing that makes the Cha Fractions special is their really nice property of going closer and closer to 1 as the input goes closer and closer to infinity. This means that the output stabilizes at 1, i.e., it neither grows nor decays once the output reaches 1. Let me show what this looks like in a graph:

Graph of the Cha Fraction

Now compare it with this:

A typical curve used for logistic regression

The curves are similar in the sense that they stabilize once they reach a particular output. The output will never be more than 1 for any input( It doesn't necessarily have to be 1, you could just scale it up or down to whatever number you wish). This stabilizing property is the defining feature of what is technically called Logistic Growth (growth which stops once a particular point is reached, as opposed to exponential growth which just never stops). 

When mathematicians find this kind of stabilizing behaviour in data sets, it is called Logistic Regression. Logistic Regression is a very important tool for humanity, as you might already know. For those who don't, Logistic Regression models are used pretty much everywhere, ranging from COVID-19 Pandemic Spread Prediction Models(check out 3b1b's video) to population growth prediction models(not only humans, but other living things too) to xG models in football/soccer to name a few. 

 The point is that we could use the Cha fraction as an alternative in logistic regression, if needed. The way both the curves grow are slightly different, as you can see. Therefore, some data sets might fit better with the Cha Fraction, which will obviously be helpful.  

Now that we've learnt about potential uses, let's move on to the subject. Before showing the actual Cha fractions, however, I'd like to discuss about a much more popular infinite fraction, which is this:

Image Credit: Mathologer

Now, as some of you might already know, there is a relatively easy way to solve this. You could call the value of this fraction as x and using the fact that it extends to infinity, you could write x=2/(3-x) and solve for x. However, I think there is a much more pleasing way to look at this which offers a little bit more intuition into what is actually going on here.

Hopefully the image is clear. We start with the common f(x)=x function and write it as x/1. Now comes the magic. 1 is written as (x+1)-x. This allows us to create a loop which extends upto infinity because the expression for 1 has an x and the expression for x has 1. I suggest you to pause and ponder about this wonderful looping idea.

Now let me show you another way to arrive at the question posed above.

The idea is pretty much the same, except that we start with the 1 instead of x. Again, I would highly recommend you to think about what is going on here. 

Solving the specific question by putting x=2

Now, if you recollect, the idea we used in f(x)=x is to write 1 as (x+1)-x. Instead, what if we write 1 as (x+2)-(x+1)? Well, one of the two values we get will still remain the same(f(x)=x). However, unlike the previous method, this time the other value will not be 1. It will be some random number, rational or irrational between 0 and 1. This is the Cha Fraction.

The idea is to have consecutive numbers on both sides of the negative sign. To give a clearer picture of what I mean by that, look at this:

The values aren't important(and possibly aren't accurate either). You can see that to the right side of the -ve sign there is 2,3,4,5,6..... whereas to the left side you have 3,4,5,6,7....

You could extend this idea even further and write 1 as (x+3)-(x+2) or (x+1000)-(x+999) all of which yield slightly different variations of the Cha fraction.

So yeah, that is all I have to share for today. Thank You!

NOTE: For those who are math experts/keen on math, please let me know if anything I've written is inaccurate. Also, if you know the technical term for the Cha Fractions, please let me know. Suggestions of any kind are welcome.(Please keep it civil)