Infinity

There was a young fellow from Trinity

Who took square root of infinity

Square root of

But the number of digits

Gave him the fidgets;

He dropped Math and took up Divinity.

-- George Gamow (One Two Three...Infinity) 

So what really is infinity? Infinity is an idea. It isn't really a concrete number. It is something that never ends, it goes on forever. 

Transfinite Numbers

As kids, all of us have baffled over the idea of the largest number there is. That’s when we came across “infinity” and we were told that it is the biggest “number”. Nevertheless, our curiosity wouldn't settle. Is there anything beyond that? If so, what is it? What if it told you there is. Well... sort of.

Some infinities are bigger than others. 

This idea can help you say that there are more points on a 1cm line than there are rational numbers in the universe or that there are an equal number of even numbers and even and odd numbers combined. 

There is an infinite number of natural numbers (1,2,3 and so on). The natural numbers theoretically can be listed...even though it will take you a very very long time. The number of natural numbers is a “countable” or “listable” infinity. An infinity that can be listed in order without missing any in between...if you are willing to spend that much time.

And then you have the transcendental numbers (which are numbers that cannot be expressed as an algebraic number or simply put - numbers that cannot be the answer to an equation). There is an infinite number of such numbers. Some examples are 𝜋 (pi) and e (Euler’s number). Transcendental numbers are impossible to list. You can start off with 1.0001... but what will the next number be? Will it be 1.0002... or will it be 1.00011...? That's the problem, there isn't a proper order with which you can list them, you will always miss some in between! You can’t express them as accurate fractions either. That means - the number of transcendental numbers is an “uncountable” or “unlistable” infinity. 

There is no way you can draw a one to one correspondence between natural and transcendental numbers. 

We can say that an uncountable infinity is larger or stronger than a countable infinity.

This also means that some infinities are bigger than others!

With all these interesting paradoxes, does infinity have any real-life applications?

Yes, it does! From computer science to physics (more specifically quantum mechanics and cosmology is filled with ideas regarding the concept of infinity. 

Additionally, many theorems that concern only finitistic mathematics have been proved using infinitistic logic like Andrew Wiles’ proof of Fermat's last theorem. 

So infinity is definitely something powerful, interesting and useful. It is something we shouldn’t neglect. It has the potential to expand our understanding of mathematics to infinity and beyond (quite literally).