Does Division By Zero Really Lead To Infinity?

Division by zero is a strange concept in mathematics. When you try it on a calculator (for example 4/0), you usually get an ‘error‘ of some sort. Textbooks often come up with the term ‘undefined’. I originally tried to investigate this concept in the article: What Really Happens When You Divide By Zero?

After writing this article, I have received quite a few messages and comments from math enthusiasts. Some of these enthusiasts tried to provide alternative explanations, and some tried to prove that division by zero leads to infinity. In this article, I try to investigate the validity and merit of some of these approaches.

I see this is as more of a discussion amongst enthusiasts, and hope that we could all learn from it.

This essay is supported by Generatebg

Negative Denominators Lead To Negative Results

In my original article, I tried to argue that multiplication can be considered as repeated addition. For example, (3 * 4) is the same as adding 3 to itself 4 times (3+3+3+3). Following this, I argued that division can be treated as the inverse of multiplication. For example, let us consider the following: 30 / 10 = 3 while 3 * 10 = 30. From these two equations, I could say that for the division example of 30 / 10 = 3, the answer (3) multiplied by the denominator (10) should equal the numerator (30).

We could use the same algorithm to inverse division by non-zero numbers. But as soon as we have zero in the denominator, we end up with an equation with no solution. For example, consider: 4 / 0 = X (I’m using ‘X’ here because I don’t know the solution at this point). If I try to inverse this, I get X * 0 = 4. No matter which math fundamentals (axioms) we use, this equation does not have a solution. Hence the term ‘undefined’.

Fellow math enthusiast Nadzeya Hry tried to prove the same point a bit more elegantly, and shared the following view:

“Actually, we also can consider division as repetitive subtraction. And from this point of view, considering 4/0=X, no matter how many times you’ll try to subtract 0 from 4, you’ll never get X (where X is an arbitrary number)”

— Nadzeya Hry

Let us consider an example: 8/(-4) = -2. According to Nadzeya, the question here is “How many times do we subtract -4 from 8 (until it reaches 0). Here, we subtract (-4), (-2) times from 8 to reach zero. This is the same as adding (-4), 2 times to 8.

What we have so far does not directly address our original question, but the key point to note here is that introducing negative denominators leads to negative results. This knowledge will come in handy later.

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Using Limits for Division by Zero

A few other fellow math enthusiasts tried applying the concept of limits to answer the question of division by zero. If you are not familiar with limits, let me explain in simple terms. With limits, we are trying to approximate zero with some small number close to zero, so that we get a ‘feel’ for what is going on.

The argument shared amongst many enthusiasts is that if you consider a small number close to zero, the result of division goes to infinity. For example, let us consider the following small number: 0.0000000000000001. When we do the operation 1/0.0000000000000001, the result is 10,000,000,000,000,000. Similarly, the smaller the number we choose, the bigger the result we get. The argument follows that as the number approaches zero, the result approaches infinity. Therefore, we should approximate the result for 1/0 as infinity.

This view seems to be so popular, that even some well-known calculators give this as the result:

However, this approach leads to logical contradictions (a word with technical might in mathematics). We’ll look at that as the next step.

We Go Where Limits Do Not Exist

Our original approach was to approximate zero by using a number close to zero. Initially, our ‘feeling’ was that as the number gets close to zero, the solution approaches infinity. So now, the question becomes, why not approximate the solution to infinity (like the calculator above does)?

There is a hurdle here since we are looking at only one half of the story. We did indeed approximate zero by a very small number, but this number was positive. What about approximating zero with very small negative numbers? For example, when we do the operation 1/(-0.0000000000000001), the result is -10,000,000,000,000,000. And as we approach zero from the negative side, the result approaches negative infinity.

For the more advanced reader: If you consider f(x) = 1/x, and then set a limit of x->0, the form is not violating any mathematical axioms so far. But it leads to a situation where the limit does not exist, as x->0-, f(x) -> – infinity, and as x->0+, f(x) -> + infinity. The left-hand side and the right-hand side of the limit don’t converge (as zero can be approached from two directions).

Is Everything Equal to Everything Else?

Let’s say we do indeed agree upon the notion that 1/0 = infinity. Then, we run into a bigger problem, which I can illustrate below:

Similarly, we could prove that every number is equal to every other number. Now if this sounds absurd to you, it indeed is!

To overcome this challenge, we need to differentiate between infinities created between different numerators. For example, 1/0 leads to an infinity that is different from an infinity that results from 2/0, and so on. We end up with an infinite number of infinities, and it doesn’t look rosy for us at this point. So, what do we do next?

Why is Division by Zero Undefined?

We’ve essentially opened the pandora’s box here. By trying to solve the problem of division by zero, we have ended up with more problems that we originally started with. And I haven’t even mentioned the mother of all problems that we have invoked, which is 0/0. Going into that problem deserves an entire article on its own, so I’ll save that for another day.

Instead of burying ourselves in more problems and mathematical contradictions, it is far easier, and logically consistent if we accept and agree that division by zero is ‘undefined’. This is pretty much why ‘undefined’ is the logically popular result for division by zero in mathematics.

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Further reading that might interest you: How To Overcome Manipulative Statistics and 5 Reasons Not To Hate Mathematics.