# Why we can’t divide by 0

Quite rarely do I ever do anything about mathematics on Tweaked for your Pleasure but today I got e-mailed a question and although I rarely do requests, I figured I might as well ask this.

Basically, the e-mail was as follows:

Hi Matt

just wondered as you are good at maths why i am told 0 divided by anything is not possible. Mind explaing ?

Thanks

The simple answer is because we simply can not define the outcome as we can not decide on any rules that will stick for all inputs. So for example we could have;

• $\frac 1 0 =5$

There is a rule within arithmetic that means $(\frac b a) = b$ and following this rule and what we defined a minute ago that if we do;

• $\frac 1 0 = 5 \\ 0(\frac 1 0) = 0*5 = 0$

By looking at the above, you will notice that it simply does not work out. You could think we could just change the rules to say that it does not work for zero when it is the denominator but then, what’s the point of even doing the first step in the first place. I am sure you might be wondering that $\frac 1 0 =\infty$ which then comes to another question what is “infinity” and how will it work with our equations above. First we need to decide two things which are:

• $1 + \infty =\infty ?$
• $\infty - \infty = 0?$

If those sounds like infinity to you, the follow does not work – $(a+b)+c = a+(b+c)$ as you can see below;

• $1 + (\infty - \infty) = 1 + 0 = 1$ which does but next one does not.
• $(1 + \infty) - \infty = \infty - \infty = 0$ which does not and thus it will not always work.

While we can keep coming up with a good set of rules to counter this, it always just leads to nonsense that is not useable and thus we just say it makes no sense to divide by zero and therefore not possible and thus it is “undefined”. I can see blinking eyes looking at the screen so lets explain this is another way to explain it.

Lets say you are filling a big box with apples and this box is able to hold exactly 100 apples. Now if you filled up that box with half the size of those apples, you would be able to put in 200 apples. Now if we kept going half and half you would get up to 400, 800 apples and so for. Now if we imagine an apple that was magical (I can see the irony in talking about magic in a logical reasoning) that took up no space at all.

Now, how many apples could you fit into that box ?

Exactly, an unlimited or infinite amount and thus there is no answer to this. Get it now ?

No ?

Okay one last way to explain it. You can not divide by 0 because the calculator tells you that you can not.