What Really Happens When You Divide Zero By Zero?

In real life or mathematics, you rarely have the requirement to divide zero by zero. Yet, most recently, I have been put in a position where I need to answer this question. This journey started when I originally investigated: What really happens when you divide by zero? Following this article, I received quite a few comments from math enthusiasts trying to prove that division by zero leads to infinity. So, I decided to entertain that line of thought and investigated: Does division by zero really lead to infinity? At the end of this article, one of the big questions that remained unanswered was:

“What is the result of 0/0?”

And this is what I’ll be investigating in this article.

This essay is supported by Generatebg

Zero is a Special Number

At first glance, 0/0 looks like a harmless problem. But it is trickier than it looks on the surface. The challenge lies in the origins of zero. You see, zero is a special number. Unlike other numbers, zero has a very specific meaning, and it is easy to misinterpret its meaning. We as humans intuitively understand zero as ‘nothing’ or ‘empty’ or ‘void’ or any other synonymous word in the English language.

However, what zero really means is ‘the lack of something’. To be more specific, zero is a context-specific number.  To define the context, the ‘something’ has to be defined first. Only then does zero start to make logical sense. If we use words like ‘nothing’ or ‘empty’ to represent zero, the meaning is too general, and it leads to logical contradictions, which is bad news in mathematics. For example, let us say that you are hanging out with your friend at your home. She comes over to you in the living room and says, “There is zero left in the fridge!” You look perplexed. You reply with a question: “Zero What?” She then replies, “Zero apples!” Only then does it start making sense to you. The context here was ‘apples’.

What is 0/0 in real life?

To answer this question, we need to review division in real life first. We use division to split things between other things or people. Let us say that we have 20 apples and 4 people. Let’s further say that we need to split the apples equally between the people. To achieve this, we do the division operation: 20/4, and as a result, each person gets 5 apples. In other words, we keep subtracting 4 from 20 until we reach zero. And the number of times we subtracted 4 from 20 is the result of the division operation.

Now, let us try and answer the important question. If we split 20 apples between 4 people using 20/4, what does 0/0 mean in this context? That’s right, we are trying to split zero apples between zero people. This makes no real sense. In other words, it is an illogical question. So, it is safe to say that in our day-to-day lives, 0/0 fails to register any logical context. It ends up being an absurd question.

What is 0/0 in Mathematics?

Mathematics is a strange world. It allows for scenarios that we don’t see in day-to-day life. In the mathematical world, 0/0 is not as uncommon as in the real world. I received a comment once from a math enthusiast who claimed that 0/0 leads to 1. This person was probably treating zero as ‘just another number’. However, zero is a special number that cannot be treated just like any other number.

To start understanding what is going on here, let’s look at scenarios that could lead to 0/0. Consider the equation x/y, where x and y are arbitrary numbers (including zero). Now, x and y could be any number respectively, which makes our lives a tad bit difficult. So, let us consider the subset where x and y are directly and linearly related (not to worry; I’ll explain this later). More specifically, I will be considering 4 relations of x and y that will be leading to 0/0. Based on those results, we would be able to start figuring out what’s going on.

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Lines That Go Through the Origin (0,0)

To begin, consider the following graphic x-y-plane. At the centre, you have the origin, where both and y are zero. Remember that I told you about considering the subset where x and y are directly and linearly related? This is a fancy way of saying that I’m going to consider 4 lines. The only special requirement that I impose here is that I need these lines to go through the origin. This way, we can see what happens along those lines when we divide x/y.

For the first line, consider x=y. This line clearly passes through the origin when x=0 and y=0. Whenever we divide x and y along this line, the result is 1. So, by assuming continuity, we could say that at the origin, the result of x/y = 0/0 = 1. Let us not worry about whether this is correct or not for now. Let us just proceed with the next line.

Consider the line x=-y. This line also passes through the origin, when x=0 and y=0. If we divide x and y along this line, the result is -1. So, by assuming continuity, we could say that at the origin, the result of x/y = 0/0 = -1. This is a different result from before, which is intriguing. But let’s just move along to the next line for now.

Consider the line y=0. This is nothing but the horizontal line of the graph passing through the origin. If we divide x and y along this line, the result is always ‘undefined’ (because of the zero in the denominator) as I argued in this article. So, by assuming continuity, we could say that the result at the origin is x/y = 0/0 = ‘undefined’ as well. Okay, now things are getting a little bit confusing. We have had three lines that lead to three different results for 0/0. Let us see what the fourth line leads to.

Consider the line x=0. This is nothing but the vertical line of the graph passing through the origin. If we divide x and y along this line, the result is 0 (because of the zero in the numerator). So, by assuming continuity, we could say that the result at the origin is x/y = 0/0 = 0. Well, call me batman, and paint me red! The results for 0/0 appear to be all over the place.

What Really Happens When You Divide Zero by Zero?

We have had 4 different lines lead to four different results for 0/0. If we keep continuing this analysis, we would find out that 0/0 takes any value that fulfils the equation of the line we consider. In other words, 0/0 takes any value that we wish for it to take, mathematically speaking. There are an infinite number of lines (and curves) that go through the origin in the x-y-plane. So, 0/0 can take an infinite number of values depending upon the context (the line or the curve).

Considering all of this, the only logical conclusion that we can come to is that 0/0 is logically ‘undefined’. You might note from my previous article that any rational number divided by zero also leads to an ‘undefined’ result (as for the line y=0 above). But 0/0 is logically ‘undefined’ for the different set of reasons that I just covered in this article. They both lead to the same result but are fundamentally different problems.

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Credit: My work on the proof in this article was inspired by the work done by James Grime.

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Further reading that might interest you: The Fascinating Reason Why Temperature Has No Upper Limit? and How To Use Science To Win At Rock-Paper-Scissors?