You have probably heard the phrase "you are not allowed to divide by zero" often. I do not like it, though. It sounds as if mathematicians simply dictate that rule. It is not the case that the universe collapses if you write down 3/0. However there are reasons why it doesn´t make sense to divide by zero.
Let´s first make clear what a fraction actually is. Suggest a, b, x real numbers than a/b = x. In words "a/b is the number you have to multiply with x in order to get a". That means a/b is the solution for the equation a = b*x. For example we have a = 1 and b = 7 than x = 1/7 is the number you need. You might also think about fractions in terms of "how often is b part of a?".
What division by zero might be:
1. It might be zero that is 3/0 = 0:
Let´s look at our example of 3/0 now. If 3/0 exists there needs to be a number x so that 3/0 = x. Multiply both sides by 0 and we get 3 = 0 * x. But 0 * x = 0 for all possible numbers. That means 3 = 0 ? That is obvious non sense. Suggest we have 3$ and we want to distribute them. However if there is nobody to distribute we do not even have the chance to distribute non of them. In order to check if nothing was distributed there have to be persons that can get dollars in the first place. Therefor we see it doesn´t make sense to say 3/0 = 0.
We have a very special case left over however: 0/0. This case is interesting, because we can see that the equation 0 = 0* x is true for all x. So is at least 0/0 = 0? No, it isn´t, because 0/0 would represent all numbers. However the fraction of a number represents a single particular number and not a set of numbers. A fraction is simply another way to describe a particular number. If 0/0 would represent all numbers, than one could write Pi as 1 * 0/0 = Pi?! In the same sense the term 1 * |R = Pi would make no sense. (|R being the set of all real numbers). As a cheap trick you might solve every exercise by saying it is 0/0! ;-)
There is a second possible solution to 0/0: 0/0 = 1. Which sounds reasonable, because 2/2 = 1, -3/-3 = 1 and so on. In general any number c divided by c equals 1. But that is a direct contradiction to our findings above. We have seen that 0 = 0* x is true for all numbers, not just x = 1. Therefor it is arbitrary to say 0/0 = 1.
2. It might be infinity that is 3/0 = infinity:
This argument is based on the idea of a limit. It goes like this: We take 3/y and we make y smaller and smaller. So we get a list of numbers like 3/1, 3/0.1, 3/0.01, 3/0.001, 3/0.0001 and so on, that means our term gets bigger and bigger and "approaches infinity". Limits however describe a process, not a certain number, but 0 is a certain number. That means the limit of 3/y with y approaches 0 is not the same as simply dividing 3 by zero. Furthermore infinity is not a number - it is rather an abstract concept that stands for an increasing/decreasing process that has no bounds, but we made clear that a fraction is just another way to describe a number. We face another problem here. What if we let y not get smaller (y coming from the positive numbers), but we let y get bigger (y coming from the negative numbers). Than we would have a list like 3/-1, 3/-0.1, 3/-0.01, 3/-0.001, 3/-0.0001 etc than our term "approaches negative infinity". Does that mean in conclusion that 3/0 is positive infinity and negative infinity?
3. It doesn´t make sense therefor we don´t use it or in other words: it is undefined
We have seen that the explanations of dividing by zero equals either 0 or infinity do not make sense. So we should not define it. Simple as that.
If you find any errors post them in the comments please.
