Positive Infinity and Negative Infinity

Recently, I watched a BBC documentary named “Horizon: Infinity” that illustrates several concepts of infinity. These are some thoughts that have been perplexing humans for a really long time.

The most common facts about infinity are usually expressed using a concept of an infinitely large hotel – which has infinitely many rooms. Even when that hotel is full, it is possible to make room for 1, N or infinitely many new customers. How? If you need N rooms, tell all the existing customers to shift in room number i+N, where i is their current room number. You’ll get N free rooms (1 to N). If you need infinite rooms? Tell the existing customers to shift in room 2i. You’ll get all the odd rooms free. This is just an illustration of the fact that $\infty + N = \infty$ and $\infty + \infty = \infty$ . It is also possible to show that some infinity is larger than other infinity (e.g. set of real numbers is larger than the set of integer number). So, there are differences among infinities.

In this post, I want to talk about some of the thoughts that came to my mind after watching the documentary. At one point in the video, someone said, infinity minus infinity can be zero. For example, when all the customers from the hotel leave, no room is occupied anymore. So $\infty - \infty = 0$ . But that is not a unique answer. Infinity minus infinity can be infinity also ( $\infty - \infty = \infty$ ). For example, if you remove the set of odd numbers (count: infinity) from a set of integers (count: infinity), the remaining will be the set of even numbers (count: infinity). Similarly, infinity minus infinity can be any finite number n (using the same analogy, just remove all but n numbers). In fact, the cardinality (count) of the set of all answers that $\infty - \infty$ can take is also infinity.

Now let us consider what are the implications of signs of infinity. Is positive infinity something different than negative infinity? Till now, the mathematicians agree that they are different. Positive infinity is the rightmost point in the number line and negative infinity is the leftmost point.

Fig 1: A typical number line. The rightmost point is positive infinity and leftmost point negative infinity.

But I don’t like this number line. In my opinion, the number line is actually not a straight line. It looks like a ring, so the positive infinity and negative infinity is actually the same point. The benefit of this model is it allows us a continuous transition from the positive numbers to the negative numbers.

Fig 2: The conjectured circular number line

To understand why I like it better, let us think about a machine that takes any number and spites out the inverse of it. Mathematically, this machine is actually a function defined as below,

$f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{1}{x}$

Now, let us analyze the inputs and outputs of the function. The value of f(x) for 1, 0.5, 0.1, and 0.001 are respectively 1, 2, 10, and 1000. The more and more the input approaches to zero from the positive side, the more and more the output goes towards positive infinity. We can say mathematically, $\lim_{x \rightarrow 0^{+}} f(x) = \infty$ . Now, if the input of the function tries to approach zero from the negative side, it turns out that the output approaches negative infinity. $\lim_{x \rightarrow 0^{-}} f(x) = -\infty$ . Now think about it. Zero (0) is just a single number. Then why the output of the function should be two different values (positive and negative infinity) only based on the direction from which the input approached zero? What will be the value of f(0) in that case?

Fig 3: The output of the function approaches two different infinities based on from which side the input is approaching to zero

I think, it would be much nicer if we could think these two infinities just as “infinity”. In that case, the number line would look like the circular pattern as I showed in Fig 2. That is the reason why I said I like that circular number line — it makes this part of the math more beautiful. But, does it make some other part uglier than the current state? I have not analyzed that yet.

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