The Beautiful and Mysterious Properties of Infinity

Infinity is very mysterious. It defies our common sense. Yet, it plays a very significant role in Calculus, particularly, differentiation and integration, which have a lot of real-life applications. To name a few, operation research and data science, where gradient descent and back-propagation in neural networks apply differentiation and Chain rule to achieve the desired outcomes. So, to understand Calculus better, it is natural that one would try to understand what infinity is.

In this article, we will look at the following questions and paradox.

1. Questions that break our common sense
2. What is infinity?
3. Hilbert Grand Hotel paradox

1. Questions that break our common sense:

To start with, let us ponder the following questions that break our common sense.

Round 1: Consider the following question: Is the sum below positive or negative?

Round 1: Finite sum

Your answer is a firm positive, since the sum involves positive numbers only. Great! Your answer is correct. Let’s go for round 2.

Round 2: Consider the following slightly modified question: Is the sum below positive or negative?

Round 2: infinite sum

You might instantly answer positive again, giving the same reason as above.

Pause right here! If you think your answer is certainly correct, please take a look at Ramanujan Summation. You might be astonished, as it suggests that the infinite sum above is actually negative. More precisely,

Ramanujan Summation

You might be thinking: How is this possible? Sum of positive numbers is negative? This can’t be true. This violates our common sense.

Don’t worry. We have one more round below for you to redeem yourself.

Round 3: Consider two boxes A and B such that A contains all whole numbers and B contains even whole numbers, that is,

Round 3: Which set has more numbers? A or B?

Which box contains more numbers? A or B?

Again, it is obvious that box A should have more numbers as it contains all numbers in box B. However, it is not correct, as the correct answer is neither. Both boxes A and B contain the same number of whole numbers.

Here is your last chance:

Round 4: Is it true that 0.999… with recurring 9 indefinitely is the same as 1? More precisely, does the following equality hold?

Are they equal?

Well, one might start from 0.9, 0.99, 0.999, etc and realize that all of them are not equal to 1. One way to observe this is that their difference is not zero. But, how about recurring 0.999. with recurring 9 indefinitely? If one comes across the geometric sum formula, he can straightaway obtain that the answer is true. Did you answer it correctly?

After all these rounds, you might realize that Rounds 2,3 and 4 have weird answers and contradict with our common sense, and the three rounds involve the notion of infinity.

Now, this raises the following question.

2. What is infinity?

Based on Wikipedia, infinity refers to something that is boundless or endless, or something that is larger than any real number.

We know that everything in our world is finite, everything must end eventually. For example, every living object has finite lifespan. This prompts the following question.

Does infinity exist in our world?

Based on our understanding of our world that everything is finite, one might think that infinity shouldn’t exist. In fact, infinity raises a lot of paradoxes such as

1. Zeno’s paradoxes, which asserts that when one attempts to travels from 0 to 1 by always travelling half a distance, the person will never reach 1.
2. Hilbert paradox of grand Hotel, where one assumes a hypothetical hotel without any vacant room but still can accommodate infinitely many new guests.

So, do you think that infinity exists in our world?

3. Hilbert Grand Hotel Paradox

Here we will look at the Hilbert paradox of grand Hotel. The setting goes as follows:

Imagine you are the manager of a hypothetical hotel with infinitely many rooms. Currently all rooms are occupied. In front of you are newly arrived guests looking to stay at your hotel. As a professional manager, can you find a way so that you don’t disappoint those new and existing guests?

One might think that it is impossible to achieve as all rooms are occupied. However, one can exploit the fact that there are infinitely many rooms. For simplicity, let us assume that all rooms are numbered as 1, 2, 3, 4,… If there are n new guests, you can simply ask all existing guests at room i to shift to room i+n. Doing so will free up the first n rooms and thus the n new guests will have rooms to stay. Since there are infinitely many rooms, all existing guests will have no trouble shifting to new rooms. Tada! This solves the problem.

Now, we have another problem.

Assume now that there are infinitely many new guests. How would you deal with this situation so that all new and existing guests are happy?

The same method above will not work as we cannot ask guest at room 1 to shift to room infinity.

However, we can perform a clever trick, again by exploiting infinity. We can ask existing guest at room i to move to room 2i. Observe that no room will contain 2 existing guests as the guest who was initially at room 2i will move to room 4i.

Now, we can ask new guest to fill in rooms with odd numbers. Since there are infinitely many odd numbers, so we have accommodated all new and existing guests.

Concluding remark

In summary, we have discuses that infinity refers to something that is boundless or endless, or something that is larger than any real number.

Lastly, we discussed the Hilbert Grand Hotel paradox which exploits infinity (of course, such hotel cannot exist in our world. Otherwise, the hotel will monopolise the world). Hope you enjoy the content.