Take 1+2+4+8 and continue on and on adding the next power of 2 up to infinity. T

Take 1+2+4+8 and continue on and on adding the next power of 2 up to infinity. This might seem crazy, but there’s a sense in which this infinite sum equals 1. If you’re like me, this feels strange or obviously false when you first see it but I promise you, by the end of this video you and I will make it make sense.
To do this, we need to back up, and you and I will walk through what it might feel like to discover convergent infinite sums, the ones that at least seem to make sense to define what they really mean, then to discover this crazy equation and stumble upon new forms of math where it makes sense.
Imagine that you are an early mathematician in the process of discovering that 1/2 + 1/4 + 1/8 + 1/16, on and on up to infinity, whatever that means, equals 1, and imagine you needed to define what it means to add infinitely many things for your friends to take you seriously. What would that feel like?
Frankly, I have no idea, and I imagine that more than anything it feels like being wrong or stuck most of the time but I’ll give my best guess at one way the successful parts of it might go.
One day you are pondering the nature of distances between objects, and how no matter how close two things are, it seems that they can always be brought a little bit closer together without touching. Fond of math as you are, you want to capture this paradoxical feeling with numbers so you imagine placing the two objects on the number line, the first at 0, the second at 1 then you march the first object towards the second such that with each step the distance between them is cut in half. You keep track of the numbers this object touches during its march writing down 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, and so on. That is, each number is naturally written as a slightly longer sum with one more power of 2 in it. As such, you are tempted to say that if these numbers approach anything, we should be able to write that thing as a sum that contains the reciprocals of every power of 2.
On the other hand, we can see geometrically that these numbers approach 1, so what you want to say is that 1 and some kind of infinite sum are the same thing. If your education was too formal, you’d write off this statement as ridiculous. Clearly, you can’t add infinitely many things. No human, computer, or physical thing ever could perform such a task. If, however, you approach math with a healthy irreverence, you’ll stand brave in the face of ridiculousness and try to make sense out of this nonsense that you wrote down, since it kind of feels like nature gave it to you.
So how exactly do you, dear mathematician, go about defining infinite sums? Well-practiced in math that you are, you know that finding the right definitions is less about generating new thoughts than it is about dissecting old thoughts, so you go back to how you came across this fuzzy discovery. At no point did you actually perform infinitely many operations. You had a list of numbers, a list that could keep going forever if you had the time, and each number came from a perfectly reasonable finite sum. You noticed that the numbers in this list approach 1, but what do you mean by “approach”? It’s not just that the distance between each number and 1 gets smaller, because for that matter the distance between each number and 2 also gets smaller.
After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1. Which is to say, no matter how small your desired distance, 1/100th, 1/1,000,000th, or one over the largest number you can write down, if you go down the list long enough, the numbers will eventually fall within that tiny tiny distance of 1. Retrospectively this might seem like the clear way to solidify what you mean by “approach”, but as a first time endeavor it’s actually incredibly clever.
Now you pull out your pen and scribble down the definition of what it means for an infinite sum to equal some number, say x. It means that when you generate a list of numbers by cutting off your sum at finite points, the numbers in this list approach x, in the sense that no matter how small a distance you choose, at some point down the list all the numbers start falling within that distance of x. In doing this, you just invented some math. But it never felt like you were pulling things out
When considering the formula for powers of two approaching zero, it is helpful to make the formula more general and applicable to all numbers. In order to do this, we must examine the definition of distance between two rational numbers. The traditional way of organizing numbers on a line may not be the only reasonable approach. A distance function should have certain properties, such as shift invariance, meaning that the distance between two numbers should not change if you shift them both by the same amount.
One way to imagine a notion of distance where powers of two approach zero is to organize numbers into rooms, subrooms, subsubrooms, and so on. For example, 0 can be seen as being in the same room as all the powers of 2 greater than 1, in the same subroom as all powers of 2 greater than 2, and so on. This hierarchy of rooms can be used to determine where numbers fall in relation to each other, including negative integers.
Turning this general idea of closeness based on rooms and subrooms into an actual distance function is a bit more complex. The size of the smallest room shared by two objects should be the determining factor in the distance between them. Using this concept, we can assign distances based on the sizes of the rooms and subrooms shared by the numbers.
This notion of distance, known as the 2adic metric, falls into a family of distance functions called the padic metrics. Each prime number (p) gives rise to a different padic metric, resulting in a new type of number that is neither real nor complex. The 2adic metric, for example, allows for the understanding of why the sum of all powers of 2 equals 1.
This process of discovering new concepts to make sense of ill-defined or nonsensical ideas is a recurring pattern in the discovery of mathematics. It is through these discoveries that rigorous terms are constructed, leading to useful math and expanding our understanding of traditional notions.