I wanted to understand complex numbers $\mathbb{C}$. I was told, that $i$ is the value of $\sqrt{-1}$. I understood why I cannot calculate the root of a negative number. I thought, to be able to understand imaginary numbers, I need to understand negative numbers better. And here is the account of what I thought about negative numbers.

My goal was to see if I can find out a way to show that physical entities could exist in negative amount.

• There can be 5 pages

• There can be 0 pages (no pages)

• But how can there be -5 pages?

May be, while considering 3 dimensions, things cannot be in negative. Let’s take time into account.

We are in time “now” which is 0sec. We can go forward in time 5sec, we can go backward in time 5sec. Going forward would mean positive movement and going backward would mean negative movement. Oh! Did we really figure it out? Nah, this way, movement in any dimension can be thought of as being negative and positive.

Let’s look around more.

Suppose we are in another universe. After all, mathematics is not confined in this single, boring universe! In this new universe, we have an apple lying in front of us. We go backwards in time -1sec, take the apple. We go -2sec take the apple away, we go -3sec take the apple away. After this procedure, how many apples are there in “now”? Of course, there will be -3 apples at time 0 (now), or will there be? The order here is important - first go -1, then -2 and then -3. If we traveled -3,-2,-1 we could argue that removing the apple from -3sec will make it disappear from -2 and -1. This might help us understand existence of things in negative amount, but then we will have to go through a lot of things to make sure the new universe of ours is consistent.

Let’s look around more.

Initially, we thought of an object going forward and backward in time. What if we think of time itself, and not of some object? At any point, time is “now”. If we take 1sec away from it, it would become 1 sec less than “now”. Do you see how we can go negative here? The assumption here is that we will have to consider time = 0 at some point to be able to go in negative realm.

Let’s look around more.

We have a standard case that should be looked at. I find it a boring one, but important one, nonetheless. It is that of accounting. Suppose I get 1 apple a day. I have 5 apples right now. Someone asks for 6 apples in 2 batches of 3 apples/batch. Surely, I can give them 6 apples if I spread the batches over the period of 2 days. So I accept the deal. How many apples do I now have? Yes, -1. I think this is just for convenience that we thought of apples being negative. We don’t have -1 amount of actual physical apples. It is an abstract way of thinking about amounts.

If you observe, in all the cases we examined so far, our calculations have been relative. We took the point of origin to be 0 and then traveled on the number line. May be it is not possible for physical entities to be in negative existence, but when we look at things relatively, negative entities can exist - or at least, we can think of them as being in negative amount.

It seems, that we would not be able to progress anywhere unless we think of things as being in negative amount because relativity is how the universe operates.

I now feel equipped enough to embark on a study of complex numbers. A post on that will be coming in future.