Infinity is a concept that is strange to many of us. What really is this word? Why do we care about it so much? What are some interesting, thought-provoking questions that we can ask of infinity?
In this video we take a look at the answer to these questions, and try to clarify some misconceptions that people may have regarding this idea.
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Full transcript :
You have been drafted to participate in a strange race. The goal is to cover a distance of 2 units, but you can only run half of the remaining distance each time.
So you start at 0, then cover half of 2, which is 1.. From here, you move onto 1.5, and then onto 1.75 and so on and so forth. You are getting closer to 2 after each step. The question is, will you ever TECHNICALLY finish the race? According to this animation you can get very close but never finish, however, let us return to this later and move on to an even more strange concept.
What if I told you that the sum of all positive numbers is equal to -1/12?
Let’s start by looking at a different series. If we look carefully there are 2 ways we can arrange this sum. One of the ways gives us 0, and the other way leaves us with 1 as the answer. This is called a cesaro sum and we get 1/2 as our final answer.
Now, consider a different series. If we write the same series again, but in a slightly different way, moving all the terms to the right by one, we can add them up together and get the formula 2 times S_2 is equal to S_1. This gives S2 to be 1/4.
Now back to the main series which we will call S. If we consider S-S2, we end up getting the multiples of 4, which is exactly 4 times S. We do some rearranging and get S=-1/12. But does this make any intuitive sense?
There are more real numbers between 0 and 1 than there are integers between 1 and infinity. This is called the cantor diagonalization method, and what we do is write out a list of numbers between 0 and 1. Now, our new number is going to be the nth decimal place digit in the nth term. So for example we will take the first digit in the first number, second digit in the second, and so on and so forth. Our result will be a new number each time that was not in the list before.
Now looking at 1 to infinity, any number that you mention can be tracked on the number line, no matter how small or large. So, how does this make sense, and how does this relate to the previous two examples?
Going back to the first animation, using this logic is not a good idea. With this logic, you will have never ending chocolate, or never ending debt to repay.
What we do is we use limits. Now limits are a very simple concept to understand, but VERY important in maths. I will give you a simple example. If I ask you what number this expression is approaching as x goes towards 2, you will tell me 4. Now, approaching is a word that I use loosely here, and there are more rigorous ways to prove a limit, but you get the idea.
With the race we are talking about, we can express it as a limit in summation notation, and we will see that indeed using this logic now, it makes sense that the answer is 2, and that we will end up finishing the race. If you don’t believe me, you can search it up- it’s called Zeno’s Paradox.
The point of showing you this was to bring to your attention that sometimes we can be easily manipulated when working with infinity. So where is the mistake you might ask.
Well, there are convergent series, and then there are divergent series. Convergent series are those that approach a certain number as we keep on adding more terms, while divergent series are those that will never approach a specific constant. The mistake in this summation was to assume that this series even has a value. When we give numbers to things that cannot possibly be pinned down by one, we are asking for trouble. Instead, we call this a divergent series and continue on with our day.
Something that might surprise you is that there are different types of infinity. Not all infinites are the same, and this problem is the perfect example to showcase this.
Between 1 and infinity, no matter what number you choose, I will eventually count unto that number, even if it might take me a very very long time.
However, between 0 and 1, no matter what 2 numbers you choose, I can always give you a number that lies in between those numbers, and this is called uncountable infinity. There is nowhere to stop, and therefore in some senses this is a larger type of infinity than the countable infinity.
The idea behind this video was to clear some misconceptions about infinity, and to raise awareness that we cannot treat it like any other number. Let’s give infinity the respect it deserves by learning more about it, and doing our own research. Finally, please remember to like and subscribe if you haven’t done so already. Thank you for watching.
30 сен 2024
