Hi Ethan
Great question. Infinity has no end and you cannot quantify it. It is completely mindbogglingly big. No matter how big a number you can think of, it is tiny compared to infinity.
But there are different infinities. There is an infinity of positive numbers that is the same infinity as the number of negative numbers.
But the infinity of irrational numbers, which includes things like pi or square root of 2 is a bigger infinity thsn the infinity of positive numbers.
So it is all a bit crazy but fascinating.
Georg Cantor was the first to study infinity and now in maths it is used routinely.
M

As Michael says, there are lots of different infinities. One way to get a feeling of how counterintuitive the concept of infinity of the Natural numbers (1,2,3,4,…) is by the following example:

Imagine a hotel (Let’s call it the Hilbert Hotel) with an infinite number of rooms, each of which are occupied by a person. Using our intuition of _finite_ numbers, we might be led to believe that, since each room has a person in it, we would not be able to take on another guest… but we would be wrong! How is this possible — our hotel is fully-booked isn’t it?

Well this is where our ideas of counting break down. We are treating our infinity hotel as if it were a hotel with a _finite_ number of rooms. So let’s see how the Hilbert Hotel can take in new guests…

So say a new guest arrives and is looking for a room, the receptionist would look at the register and say

“I’m sorry all our rooms are taken… :(”

but then he would say,

“Oh wait, I can ask our guest in room 1 to move to room 2 and our guest in room 2 to move to room 3 and so on, which means that you can take room 1 …. :)”

And the guest will have a room in a fully booked hotel! Such a hotel simply couldn’t exist if it had a finite number of rooms.

We can extend the Hilbert Hotel to investigate all sorts of other weird things that happen when we have an infinite number of rooms — fascinating question, and one that has only been answered relatively recently as Michael says! ðŸ™‚

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