Meme assassination: Division by zero
One of my previous Mathematics teachers liked to recall an occasion when he gave a guest talk at an elementary school, where he found that the students would always respond to problems which could be stated within the scope of what they already knew, but couldn't be answered within that scope, with the same phrase:

"You can't do that."

This is understandable, of course; I don't expect a young child to look at the equation x + 1 = 0 and conclude that there must exist numbers below zero for the purpose of solving it. I certainly don't expect anyone not studying a maths-powered subject at sixth-form college to look at x² + 1 = 0 and conclude that there must exist something beyond the 1-dimensional 'number line'1, though he did have a point in that it would be a step forward to teach the kids to say something else such as "That has no solution"2.

The same guy also liked to deploy a figurative bombshell for students (perhaps around late elementary school or early high school) when asking for solutions to the friendlier equation x² - 1 = 0, by awarding only a half-point for answers such as x = 1. This answer isn't wrong of course, but it does neglect to mention that x = -1 is also a solution, and the real purpose was to stop people from thinking in terms of finding 'the' solution and instead start by considering that there could be many solutions (or indeed none!) before deciding that there is exactly one.

Where am I going with this? Well, as the classic chant goes, division by zero apparently isn't possible because the calculator says it isn't - or is it just saying that because it doesn't have a single definitive solution to give you? Is it a case of 'no solution', or are there many solutions? Or, if it wasn't already obvious that the entire purpose of the above anecdote was for pure foreshadowing, does it depend on the other values in the equation?

Begin by considering what 1 divided by 0 would be. To make at least some sense of this, it's best to think of division as the reverse of multiplication. We're looking for 'a number which, when multiplied by 0, gives 1', so we can write that as an equation:

0x = 1

I didn't want to rely on having to say that this is basically x = 1/0 rearranged, because it would be a bit unsatisfactory to take it on trust that one can just multiply each side of an equation by zero like that and still have 1 on the right-hand side, but by all means accept that alternative route if it helps.

At this point, the correct 'answer' should become obvious by simply plugging in values for x - no such value will be able to make the left side become 1, or indeed anything other than 0, so the answer is that there are no solutions3. 'Infinity' is not a solution here, since it isn't actually a number.

Unfortunately I've basically just given away the next part, but on the plus side, if you made the mental hop as a result of reading this article then you can celebrate your general analytical skill. Consider dividing 0 by 0:

0x = 0

This time, just by changing the other number involved, we can choose values for x which solve the equation. Clearly any choice works, so we can conclude that there are not only many solutions (as with x² - 1 = 0), but infinitely many solutions. One usually makes a statement about all numbers within some agreed scope, e.g. "everything in the set of real numbers is a solution". Here, 'the set of real numbers' is a formal way to say 'the number line'.


Of course, it sounds a bit pointless when literally any number is a solution, however this is only because we're looking at a single equation in a single dimension. Multiple dimensions do involve vectors, matrices, and all sorts of things with which I shall not burden anyone here, however it will suffice to note that something interesting happens - when one has a set of equations, it becomes possible for only some of those equations to degenerate (actual technical term!). We call this the nullity, and it is equivalent to the idea of 'degrees of freedom'. If a linear system has dimension 3 (such as 3D space) and the equation set has nullity 1, the set of solutions will form a straight line through the 3D space, where the location and direction is forced by the other equations.

On its own, any attempt to solve that 'null' equation would involve a division by zero; the notion that it can be solved to a single value answer is simply wrong. Instead, we do something very much equivalent to above - rewrite it in a way which gives us a free variable with which we can choose particular solutions, then run that through the rest of the system to produce its entire nature. Perhaps a 3D system has nullity 2, in which case the possible solutions may lie on a flat plane, slicing through space. A non-linear 3D system with nullity 1 may represent a curve of solutions, and nullity 2 may be some sort of surface; these are legitimate results, yet none of this happens without some equation at some point facing good old Division By Zero.

1 If you can't see why regular numbers won't work, try calculating x² + 1 using some different input values for x, including positive/negative/zero. You should soon find the true problem, that there's no way to make x² become negative. This issue is no different to attempting to solve x + 1 = 0 when you have no notion of what a negative number is; in both cases, the problem is written in terms of numbers which one does know, but cannot be solved without looking beyond that.

2 Obviously the uncropped statement would be "That has no solution in the set of real numbers", but the whole point is to set a cutoff for the scope of the material being taught, not subtly hint that the scope being taught isn't complete (or 'closed', as maths-people would say).

3 Nor, this time, are there any solutions in any useful extension of the number system. The number zero does not screw around.
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