07 December 2017

Divider of Zeros

Back in school about 7 years ago or so, I always loved math. But a few things bothered me. Math always seemed logical to me. Of course, that's what it is supposed to be. However, there are some issues with it.

Math Was Created By Us

Math was created by us to make sense of the world we live in. This means math hasn't always been there. We needed a system to make sure to count things there are or to easily handle stacks of objects. Later we used it to explain the rules of the world, physics.

Math Is Not Perfect

Math tries it's best to logically represent real life in its system. However, it is not perfect. Certain things don't work in the system of mathematics and are thereby deemed unsolvable. One of these is the division by zero.

Let's Apply Logic

Let's try to solve this issue by using logic. If we have a cake (1) and we divide it for 5 people we get 1 / 5 per person. So if we have a cake (1) and divide it for 0 people we still have one cake. Resulting in:

1 : 0 = 1

So to say any certain number divided by zero equals the certain number:

x : 0 = x

However this doesn't work when we rearrange our equation:

x : 0 = x | *0
x = x * 0

Or does it? What does multiplication do? If I have a 3 and I copy it as often until I have 2 copies and the original (3 times the 3) I get 9. If I have a certain number like 8 and multiply it by zero I'm left with the original: 8. That means in our case above the result of multiplying by 0 is actually the number itself and thereby going with our logic.

We can also replace x with zero:

0 : 0 = 0
0 = 0 * 0

Our logic still applies.
Let's try something else.

The Mathematical Approach

What is multiplication? Multiplication was created to simplify multiple additions. For example if we calculate 3 + 3 + 3 + 3, we can say 3 * 4. So if we say we do no addition of our certain number. We can multiply it by 0: 3 * 0
And if we don't add up anything we have nothing: 3 * 0 = 0

Now we can rearrange our equation and we get:

3 * 0 = 0 | :3
0 : 3 = 0 (so far so good)

3 * 0 = 0 | :0
3 = 0 : 0

So zero divided by zero equals.. a specific number related to the problem in this case 3.
We can also say 3 = x:

x * 0 = 0 |:x
0 : x = 0
x * 0 = 0 | :0
x = 0 : 0

Starting With Division Instead

Let's look into this topic starting with division instead of multiplication we run into a problem.
In school at least I learned that division is a process in which you check how often one number matches into the other.

In the case of 12 divided by 3, we can subtract 3 from 12 and result in doing this 4 times until we reach zero. So 12 : 3 = 4. However, If you apply this logic to the division of zero you get infinity.
Why? Well, try to check how often you can subtract 0 from 12... stop! you won't be able even if you're immortal!

So we established:

x : 0 = infinity

We can rearrange this too:

x : 0 = infinity | *0
x = infinity * 0 | :infinity
x : infinity = 0 | *x
infinity = 0 * x | :

Totally different results. So what does that mean? Well, it means dividing a number through zero results in infinity. So what about zero divided by zero?

Exception! Exception!

Not talking about a thrown exception like in programming but yeah we come to an issue here.
So a number divided by zero is infinity. We've proven that in the last paragraph. However, there's another thing in math. When we divide a number like 4 by itself we get 1:

4 : 4 = 1

So technically applying this to 0 divided by 0 we get 1:

0 : 0 = 1

Though we already established a number divided by 0 equals infinity and back up we established a number divided by 0 equals the number itself.

When Looking at Graphs and Tendencies

When you're working in analysis in school or wherever you do. You may or may not know that looking at the behavior of a graph it tends to go towards positive or negative infinity at spots you divide a number that is not zero by zero. Additionally, when the graph comes to a point where 0 is divided by zero you'll see a jump in the graph, with multiple possible positions.

These anomalies make it hard to tell what the result of zero divided by zero is.

So What's The Solution?

don't know. There's a reason no one found a worldwide accepted solution to this. I mean of course you can say let's accept one of the systems above but neither seems flawless. You could also say let's apply all of the above with the mathematician's job to decide which to use when. I mean.. it depends on the case and definition.

So if you have an idea on how to solve this feel free to share maybe you get a Nobel-prize for it or something.
Good Luck

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I'm a B.Sc. Games Engineer and I created this blog to share my ideas, theorycrafting, thoughts and whatever I'm working on or doing.