How Crazy The Concept of Infinity is

When we talk about infinity ∞, we talk about the existence or something so large or so tiny, it's impossible to imagine. But ironically, we've done just that.

Imagine you take a bag of salt and tried to count how many grains of salt there is in it, you'd say it's impossible because it feels so infinite. Well, that analogy hasn't even begun to scratch the surface of the concept that is infinity. 

Infinity is imaginable yet so complex to understand. It took years before mathematicians could first understand what infinity is. It is not a number! It's is more like an abstraction. Some people define it as something that is so large and limitless. But that's not all true. You'll see why later on.

We Have Infinities

Are you a science enthusiast? If so, you'd have probably heard of a theory of multiple universes that makes up the multiverse. If not, click here. The theory that makes room for the possibility of a multiverse infers we might have a countless number of universes. But our universe is already infinite, isn't it?

We can say the exact same thing for ∞. We have many kinds of infinity. You may say it's a little nonsensical because earlier I said infinity is a concept. Well, it still is and we have different kinds of it but they're all still infinity.

For clarification, let's pick it apart at a time.

We have the infinity of natural numbers and it's quite a big one. We all know our positive integers (1, 2, 3,…). They are called so because they are naturally occurring whole number integers. Now, the series can go on indefinitely as we have an infinite amount of positive integers by simply adding 1 to the previous. So the series of natural numbers tend to ∞. Also, natural numbers consist of an infinite amount of even numbers which is just as large as the infinite amount of odd numbers which also is just as large as the number of natural numbers. Quite amazing, isn't it?

If we could recall our number line, we'd also see that we do not only have positive integers but negative ones as well. And they are just as big as that of their positive counterpart. You could say that the negative series tend to -∞. So if both the positive side and the negative side of the number line tend towards ∞, we could say that the whole number line is twice as big as infinity but it is not 2∞. It is still infinite. 

Let's take another example. We have 1 and then 2 but we also have numbers between 1 and 2. These are called rational and irrational numbers (irrational being the bigger of the two) and believe it or not, their series is quite bigger than that of the number line. You might say after 1 comes 1.1 but that's is conceptually incorrect. Between 1 and 1.1, there's another series of rational numbers which consists of 1.01 all the way to 1.1 and the same can also be said for the series between 1 and 1.01. It's like the series expands upon itself so that after one, we can never reach the next rational number. It's mind-blowing but what do you use to qualify this series? Some say it's "∞^∞" (that's infinity raised to the power of infinity) and that's quite a misconception. It is just infinity. This is why infinity is not a number but an abstraction. 

Some Weird Things With Infinity

Physics is about precision. We do not expect to gamble in calculations with the concept (the exception being quantum mechanics). So some weird things happen when arithmetic operations are done using infinity ( or an infinite series to be precise).

The concept of infinity was first misconstrued by a scientist who decided to add up a series of continually halved numbers starting at one (1 + ½ + ¼ +…). He thought the sum would be infinite but it was seen eventually that it is not the case. Rather the sum adds up to 2. How? The numbers get smaller and hence less consequential to the sum. The issue with this solution is that the series never actually add up to 2 in a real sense. 2 is considered as the limit of the s of the series through the method of "Differential Calculus".

Secondly, let's look at this alternating series: 1-1+1-1+1-1+…. Judging from the alternation, if it's an odd number of digits, the sum is 1 and if its an even number of digits, the sum is zero. Easy enough right? But no. Going to infinity, it can be both an odd and even number of digits. So we find the average and we get ½. So the sum of the alternation 1-1+1-1+1… to ∞ is ½. Such a small number for an infinite sum but it's not even the most surprising. 

Another alternating series is 1-2+3-4+…. This sum all the way to ∞ gives us ¼ which is quite interesting and smaller.

The most fascinating of all is the simple addition of all natural numbers (1+2+3+4+...). Looking at the previous series, you could say you want to account for the solution by making a case for the inclusion of the negative integers (still incorrect) but no such thing can be done for this series. It is a sum of positive numbers to infinity. One may think since it's positive, the sum to infinity is probably a very large number (maybe even infinity itself) but that is totally incorrect. The answer is a menial -1/12. What is quite tricky is the fact that if you stop the sum at any number before infinity (which basically means any number at all because infinity can never be reached), the answer is then an incredibly huge number but once you get to a point that is infinity, it becomes -1/12.

How this is gotten is explained below:

How can a sum of all positive numbers to infinity be a negative fraction? Well, infinity is truly something weird. These results have been applied in many areas of physics (maybe most notably string theory). 

We as humans also constitute to the abstraction that is infinity. We make decisions in every second of our lives. There are endless actions we could take to engage in an activity but it all reduces to a simple action (similar to the uncertainty in quantum physics). Space is infinite and is always expanding upon itself but our observable universe is at least a comparably menial 13.7billion light years across. 

To end, the concept of infinity still houses some of the most important yet unanswerable questions in Mathematics (most notably the Continuum Hypothesis). Even if Mathematics is widely conceived, and rightly so, as the pinnacle of human reasoning, it does have its limitations. The insane concept of infinity showed us that.