What happens when we try to divide by 0? Well, we break the fabric of reality and the universe implodes into itself. Okay, that’s a bit extreme, but we should note that this is something we can’t do. But why can’t we divide by zero? What if we really want to, can we just do it and see what happens?
I just wanted to start by looking at the definition of division. We often take this idea as if it is a trivial topic, but we are so used to it working. That is, if I take 10/5 I get 2. My calculator agrees, so I’m done. As usual, there is quite a bit more going on here than is readily apparent.
In order to look at this hidden information, we will be a little more precise with our definitions. We say that a binary operation, *, is defined on a set A, if for every a and b in A, we have that * sends a*b to exactly one element. We can otherwise state this as * is a well defined function
In particular, I wanted to look at this definition of an operation, because it is important to realize that an operation is just a function from A2 to A. Therefore, each input needs to have exactly one output. When defining an operation on any set, not just the real numbers, we still use this property.
While I would like to discuss operations other than the usual operations and sets other than real numbers, I will save that for another time. Instead, I’d like to take a moment to focus on subtraction. In order to do so, lets first look at addition. Note that 0+x=x for all real numbers x, so we call 0 the additive identity. We would then note that if we add a number and add the same number with the opposite sign, we would get back to 0. In fact, we would think of the negative numbers as the positive with an opposite direction assigned. That is x+(-x)=0 for all real numbers x. Therefore, we would say that -x is the additive inverse of x. Now, to define subtraction, we say that a-b=a+(-b). That is, we add a and the additive inverse of b together.
Note that, we could just define subtraction as a distinct operation, but it doesn’t behave very nicely. In particular, it doesn’t satisfy the associate law (a*b)*c=a*(b*c) that we would love to be able to use. Therefore, instead of defining subtraction as a distinct operation, we define subtraction as the inverse operation of addition. This works out pretty well for us, because every real number has a unique additive inverse. That is -x is well defined.
If we follow this example for division, then we are going to define division as the inverse of the multiplication operation. That is, we have the identity 1*x=x for all real x. Therefore, when we say x/y, we mean x*z where z is the multiplicative inverse of y. That is, z*y=1, or z=1/y. Now, this works well for almost all real numbers. In particular, if we take 1/y for all reals, except 0, we get a unique answer. However, if we try to solve z*0=1, we end up with 0=1, which is a contradiction, so there are no solutions to this equation. Hence, there is no multiplicative inverse of 0, so division is not an operation on the real numbers.
Dividing by 0.
We saw above that there was no multiplicative inverse of 0, so we can’t divide by 0. While we can just stop there, we will continue to look at the problem because of the following example. Suppose that you want to find the velocity of an object over a given time frame, then the velocity is given by the change in position divided by the change in time. That is, if v is velocity, s is position and t is time, then
However, what happens if you want to find the velocity at one instant in time? That is, if I want to know how fast a car, person, ball or any other object is moving at a given time, how would I find that?
Following the above formula we would get that over a single instant, there would be no change in position and no change in time, therefore, the velocity would be 0/0. However, since we can’t divide by 0, this would mean that velocity is indeed an illusion and nothing actually moves.
While this sounds like an absurd thought, it is one that has occurred in people’s minds before. For example, you can look at Zeno’s Paradox for such an example. However, how do we resolve the difference between the inability to divide by zero and the observation that things do indeed move? Well, we have to examine 0/0.
Therefore, let’s just start with what 0/0 means. To use the formulation above, this would be equivalent to saying, 0/0=y if 0y=0, solve for y. Above, we had that when we tried to find 1/0, we couldn’t find any solutions to the equation, because such a solution would imply 1=0. In this case, however, we note that there is a solution to 0y=0, in fact any real number is a solution. Therefore, we might be tempted to say, well 0*1=0, so 0/0=0*1/0=1. However, this leads to the conclusion that 1=1/0*0=1/0*(0*2)=(1/0*0)*2=2. Furthermore, we can use this to show that every number is in fact equal to every other number, and, as stated in the introduction, the universe would implode into a single point.
What we start to see is that if we stopped time, there would indeed be no motion. However, since we don’t want to stop time, what we can do instead is ask what will happen if time is changed by an arbitrarily small amount? This is the idea of a limit. That is, if we want to find the instantaneous velocity, we would instead find what happens if we have an arbitrary small change and we write
As an example, suppose that our position is given by the function s(t)=t where time is in seconds and s(t) is in feet. Then we can find a velocity as
Note that we are able to say that Δt/Δt is equal to 1, because this is true as long as Δt is not 0, which is our assumption when finding a limit. If we generalize this to arbitrary functions, instead of just position, we would say that the instantaneous rate of change of a function, f(x), with respect to x is the limit as the change in x goes to 0 of the average rates of change. Since this is a mouth full, we would normally just call this the derivative of f(x) with respect to x and write,
Note that explaining motion was indeed one of the primary motivations of Newton when he developed his works on Calculus. That is, in trying to find a way to do the impossible, divide by 0, he developed ideas of both a limit and a derivative giving rise the first two major topics covered in a Calculus course.
We have seen that, while we would really like to divide by 0 in some cases, doing so ends up causing serious issues with mathematical models. While we can’t ever divide by 0, we can look at what happens if we instead divide by some really small number. In certain situations, we are able to do so in a meaningful way. In particular, when trying to find instantaneous velocity, we can look at what happens to an object over small changes in time to find how it is moving. In trying to describe this velocity we were indeed able to motivate defining both a limit and a derivative. Now that we have the limit and derivative, we can use these to look at the power rule for derivatives next time.
In my Calculus class, instead of using velocity as a motivation, I use marginal profit. Here is a preview I use with my students of the Pizza and Marginal Profit lab. The full lab is also for sale on the Products page.