By Samaira Kalia
Different people think of infinity in different ways. Some say it’s a number. Others say it’s another word for “endless”. The concept of infinity has been discussed for centuries. But why is it so interesting? It may as well have an infinite number of reasons, but this article only attempts to provide a few…
The earliest recordings of which the concept of infinity was first pondered are from 450 B.C., by a Greek philosopher named Zeno of Elea. Much later, in 1655, the common symbol for infinity (∞), the ouroboros, was invented by mathematician John Wallis. The Ouroboros was an ancient symbol of a snake biting its own tail. This was chosen because it was meant to symbolize a cycle, that the snake will keep moving in circles to eat its own tail.
Normally, when we try to define infinity, we usually talk about the mathematical concept, and how infinity is the largest number, with no actual known value. Sometimes people try to describe very large distances in the universe using the term “infinite distances”. But since nothing in this universe is never-ending, physical infinity doesn’t really exist.
To help understand infinity better, mathematicians come up with ideas where it would be easy to clarify infinity in the context of things we know.
One of the most well-known ideas is Hilbert’s Infinite Hotel, first considered by David Hilbert. The idea is that of a hotel with an infinite number of rooms, each room is assigned to a guest. Thus, we have an infinite number of guests staying at this hotel. The problem Hilbert tried to solve was what would happen when a new guest would arrive, and had to be accommodated. At first, he thought that moving everyone up by one room would make sense; so the people in Room 1 went to Room 2, Room 2 went to Room 3, and so on. But then, what if there was an infinite number of new guests arriving to stay at the hotel? Every single guest wouldn’t be able to move up an infinite number of rooms. What Hilbert decided to do instead was to request each guest to move to the double of their previous room. So Room 1 went to Room 2, Room 2 went to Room 4, and so on. As you can imagine, it’s easy to calculate the new room numbers initially, but it starts getting much more difficult for the rooms at higher numbers.
Hilbert’s hotel helps to clarify several very interesting and peculiar things about the nature of Infinity, and how it operates. Thus, for example, in the first scenario, we sent every guest up one room. So this means that if you add one to infinity, you’ll still get infinity. Similarly, once we remove the new guest, we’re still at infinity guests. Therefore, one subtracted from infinity gives us infinity! In the second scenario, we managed to accommodate an infinite number of new guests without a problem. So, using the same logic, we’ve discovered that infinity added to infinity would also equal infinity.
Now, one would assume that if we subtract infinity from itself, the result should be zero. But what if we remove the guests from each room with odd numbers, such as Room 1, Room 3, Room 5 and so on? There’s an infinite number of rooms with odd numbers – so we’ve subtracted an infinite number of guests. Yet, we will still be left with an infinite number of guests staying in rooms with even numbers! So, this implies that “infinity minus infinity is infinity”.
Another puzzle is the infinity-sided shape. The shape’s actual name is an apeirogon, coming from the Greek root of infinite or endless. Now, what would this shape actually look like, if it was a regular polygon? Let’s start with a triangle – a three-sided shape. If we add one side, we can make it a square. If we keep adding one side, we’ll get a pentagon, a hexagon, a heptagon, an octagon, a nonagon, and so on. You’ll start to notice that every time we add one side, the shape gets rounder and rounder, and more like a circle. So you can say a circle is an infinite-sided shape. But since a circle does not have any straight sides or line segments, instead a looping curved line, it can also be said to have zero sides.
Seeing how infinity is different compared to normal numbers is interesting since we don’t know its exact value. Describing exactly what infinity is is hard. It requires not just mathematics, but also arts, humanities, and science. And if we have to include all of it in one article, this page will never end.