Maths blog: The number 0

We all have probably learnt in primary school that dividing by 0 is not allowed. But we have not got told why so. In the primary school, we were taught division simply is how many times does one number go into another. Using this method, we can see that any number divided by 0 will yield infinity, if we define anything divided by 0 as infinity we will get in trouble. Say if 1 divided 0 is infinity, and 2 divided by 0 is infinity, doesn’t this mean that 1 equal to 2? This goes against the foundation of maths. Later on, in secondary school we learnt that we could have functions to find the limits. With this knowledge, we can set y=1/x lim->0. The graph 1/x looks like this:

We can see that x tends to 0, y tends to infinity. So surely, we can then deduce that 1/x is infinity, right? Wrong, we can also see that if x tends to 0 in the negative region, then y would tend to negative infinity. So, depends on the contexts of the problem you are trying to solve, your answer can be very different. When we added another axis of complex numbers on to the graph, it will become very complicated very quick. So mathematicians long time ago decided that anything divided by 0 would be undefined, and our calculators are programmed in a way if a division by 0 is inputted an error will be outputted so the calculator won’t be crashed trying to do an infinite amount division.

We probably also heard of another forbidden calculation involving 0, that is to put 0 to the power of zero. We got taught that 0 to any power would be 0, but also anything to the power of 0 is 1. So can use the previous trick of limits to find it right? Unfortunately, it is not that simple. Here is a graph of x^x, we can see that as x tends to 0, the result tends to 1. However, this is only the real number line, as we introduce the complex number to the graph, it will become a 3-dimensional graph. Now we have an entire surface of number we got real number in 1 only on 1 line and everywhere else is complex. Now using this we can form complex equations; this creates a whole load of different ways x tends to 0. It doesn’t simply just tend to 1 anymore, so on the surface it might look like that 0^0 is tending to 1. But once you dig deeper, thus entire hypothesis falls apart. This is the reason why if you say 0^0 has a value, your maths teacher gets very upset. Because there are multiple different answers, thus being is undefined as well.

Finally, what about 0/0? If we draw a graph of y=x, we will get a straight line through the origin. Rearrange this we will get y/x = 1. This mean every x coordinate on the line divide it by y coordinates equalling to 1, and our origin (0,0) is on the line. So, doesn’t this mean that 0/0 is equal to 1? Unfortunately, this is wrong again if we draw a graph of y=-x, and rearrange, y/x = -1. Every Y coordinate divided by x coordinates would equals to 0 on that line and that goes through the origin as well. In fact, 0/0 can equals to anything, so mathematicians agreed that it is best to leave it as undefined.