Fun Fact Friday

June, 12 2025

To Infinity, and Beyond (Part One & Two, out of Infinity Parts)

What the fuck is infinity? Is it a number? Is it bigger than the biggest number? There’s some weird shit going on here and if you “do math” to it, things don’t make sense.

If it really is a number, then you should be able to add 1 to it. Then what do you get? Still infinity, right? Cool, let’s check the math:

Uh oh, algebra flashback. Remember how when you have an equation like this, you can do something to one side and as long as you do the same thing to the other, the equation still holds true? Let’s subtract infinity from each side and what we get:

That doesn’t look right. Maybe if we try adding infinity to infinity, we should still end up with infinity, you would think. So,

Which is the same as saying 2 times infinity equals infinity:

But then we can divide both sides by infinity and we get:

Clearly, we’ve got problems (literally “math problems” lol lol lol). Sorry to say, that it gets worse if we try to multiply infinities (which would still be infinity):

Divide both sides by infinity to cancel one out on each side and we get:

The obvious conclusion here is that infinity, if it is a number, is definitely not a normal number, and we need to be careful how we treat it. Even though it is the biggest “thing” we can try to wrap our heads around, it is still a delicate flower and needs to be handled like it’s the smallest thing we can think of…which is the last FF of this F: That infinity can also be the smallest thing we can think of.

Part Two: Infinity Small Things

Let’s do one more math thing to see this in action. Let’s say you want to go from your house to meet me at Little Al’s in Mokena, which is exactly 12km as the crow flies^[1]. Yes, I’m going to give you the power of flight (you’re welcome). Also, for simplicity’s sake, we’re inventing a new unit of distance: 12km = 1sm (samometer).

In order for you to fly from your house to Little Al’s, you first need to cover ½ the distance. Makes sense, you need to travel at least ½ a sm before you can continue any further. It then follows that you’d need to cover ½ of the remaining distance (which would be ¼ of a sm) before you can then cover ½ of that remaining distance (⅛ of a sm)…

Yay, you can fly! But Sam is making it evil for some reason and it’s literally going to take forever. Here’s what this looks like:

Since you can do this forever, the denominators will get infinitely larger, while the numbers get infinitely smaller^[2].

So will you actually ever get there? The only way to find out is to add up the infinite amount of numbers we just listed out. And since I am already being evil, I’m going to use some greek letters and a geometric series equation to make it more mathy:

FANCY! WTF does that mean? The big E thing means “add up all the numbers you get by plugging n into the 1 over 2^n thing, where n = 1, 2, 3, 4…until n = infinity.” (which is what we did above ending in yada yada yada).

Ok, so I wrote some Greek, now what? We’re not any closer to finding out what all these numbers add up to. Luckily, there is a simple equation (make that greek thing = to another thing) that we can use to plug in a value. And no, I didn’t just conjure up an equation out of thin air, there are multiple ways that it’s been proven using different logical methods. Here’s what the equation looks like and then we’ll plug in our known values to solve it^[3]:

for 

Trust me on this,   (the first term) and  , so:

Hence,

 1 sm ←YES!

YOU MADE IT! And I have a cold Miller Lite waiting for you after a REALLY LONG flight. Let’s reflect on all that scary Greek shit up there: We just added up infinity things and got 1.

Here’s the rub though…since it took you an infinite amount of time to get here, not only is the beer a little warm, but you also left me alone there for an infinite amount of time, which is too long to wait…and I finally played every song on the touch tunes a LOT of times.