To begin, perhaps it is appropriate to state that infinity is not a real number but rather an idea – a concept of something that is unlimited, endless, without bound. With this comes a few misconceptions that I will be addressing. In fact, this idea extends far beyond what you may have thought and as abstract as it may seem, it has many applications to the real world. It is worth noting that there are various types outside the realm of mathematics but for the purposes of this article I will only be focusing on those formally known as ‘mathematical infinities’.
The very idea of infinity is familiar to us all from as early as we learn to count. Naturally, we think to ourselves “What’s the biggest number ever?” Sooner or later we realise there is none since we could always add 1 to any number – this is what underpins the notion of infinity. Due to its nature it cannot be treated as a real number, i.e. the basic operations such as addition and subtraction cannot be performed on it to produce the results that are expected with real numbers. For example:
∞ + 1 = ∞ and ∞ – 1 = ∞.
This can be generalised to
∞ ± α = ∞, where α ≠ -∞.
This result can be relatively easy to understand by using our intuition because if you add an extremely large number to any positive or negative number, regardless of the magnitude, it will remain as an extremely large number.
To further illustrate this idea, I will use a famous thought experiment known as The Infinite Hotel Paradox.
Imagine a hotel with infinitely many rooms with all rooms occupied. Now, if a new customer came into the hotel one may assume that they will be sent away since all rooms are filled. However, the manager decides to let the customer in and gives them a room without anyone else leaving.
How is this possible? Well, he simply asks each person to shift to the next room – the person in room 1 moves to room 2, the person in room 2 moves to room 3 and so on and so forth. This leaves an empty room 1 which the new customer can take. This is one way of thinking about the equations above where the number of new customers can be considered α. This is not the whole thought experiment, but a similar idea follows in the different scenarios.
Now, to clear up a very common misconception that is 1 ÷ 0 = ∞.
There are various arguments that can be used to disprove this. Here, I will be explaining two different arguments one of which is far simpler to understand than the other. However, you are welcome to search up any other proofs if you wish to do so.
Graphically, we can look at the curve y=1/x and consider limits (a value that a graph approaches). From this we can see that if the x value approaches zero from the positive side the function tends to positive infinity, however, the problem arises if we were to apply the same logic but approach zero from the negative side.
The graph shows that the function tends to negative infinity instead. Both arguments have equal validity, thus proving that 1 ÷ 0 cannot equal positive infinity. It could be argued that 1 ÷ 0 = ±∞ which seems more reasonable but since zero has no sign (it is not positive or negative) its reciprocal must also have no sign so this also cannot be true. For this reason, the value is said to be ‘undefined’ since mathematicians have not yet come up with an appropriate answer.
The second argument is much simpler and requires only understanding of the fact that
α (∞) = ∞ where α ≠ 0 and is positive.
Assuming 1 ÷ 0 = ∞, by extension we could say that 2 ÷ 0 = ∞ if we were to multiply both sides of the equation by 2. With a little rearranging and substitution, it becomes apparent that our assumption must be false. This is because we arrive at the statement 1=2 which breaks one of the fundamental truths of mathematics, therefore 1 ÷ 0 ≠ ∞. This is an example of proof by contradiction.
Finally, another thought-provoking topic is the summation of a sequence with an infinite number of terms to return a finite value. This may seem illogical at first but what needs to be understood is that generally we can only do this with a certain type of sequence – one that is convergent. This means that the terms in the sequence are bounded by two values and seem to approach a certain value. Here the idea of limits mentioned earlier can be used to find what the sum to infinity would be.
written by Arjun Santosh