What Really Happens When You Divide By Zero?

When you divide by zero using a calculator, you’ll likely get some form of error message. You won’t get a rational answer. If you look into math textbooks, you’ll get the term ‘undefined’. As advanced as we are as a species, not everyone has a liking for math, and not everyone is good at math. So, it is understandable that one just takes such a result for granted. But if you are the curious kind, you’ll note that when zero is divided by another non-zero number, we do have a result. When zero gets divided by a non-zero number, we get a rational result (zero). At this point, if you are a human being with a modicum of curiosity, you have to ask THE question. WHY?! Let’s try and unravel this puzzle.

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What is Zero?

Zero is a special number as far as the philosophy and the history behind it is concerned. But logically, it is very simple. If you are counting anything, the absence of entities is represented by zero. In simple terms, it is used to denote the fact that ‘no’ entities are present to count. Zero added to any number results in the same number (example: 3 + 0 = 3). Zero subtracted from any number also results in the same number example: (3 – 0 = 3). Any number subtracted from zero results in the negative of that number (example: 0 – 3 = -3); this is no different from adding a negative number to zero, which results in the same number (i.e., (-3) + 0 = -3). So far, so good. Now that we have covered the first part of the puzzle, let’s move onto the next part.

What is Multiplication?

Multiplication is the repeated addition of a number by a certain number of times. For example, if 3 is multiplied 8 times, we add ‘3’ to itself 8 times to give 24 as the results (3 x 8 = 24). Multiplication was invented for us to simplify complicated addition. It could also be seen as a way of simplified representation of a long string of numbers being added. Imagine writing this: 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 24. I literally had to triple check if I had written the correct number of 3’s there. I hope you see the complication here, and why it makes sense to write ‘3 * 8’ instead. Furthermore, multiplication has advantages such as the property of commutativity, which is a fancy way of saying 3 * 8 = 8 * 3. In other words, the result is going to be the same if I add 3 to itself 8 times or if I add 8 to itself 3 times. Cool, right? But what does this have to do with division by zero? Right, we’ll get to that next.

What is Division?

Now we arrive at a key part of our puzzle: Division. Division can be seen as the inverse of multiplication. Instead of looking at it theoretically, it helps if we look at a practical example. Consider the following: 30 / 10 = 3 while 3 * 10 = 30. So, in short, we could say that when we consider a division example such as 30 / 10 = 3, the answer (3) multiplied by the denominator (10) should be equal to the numerator (30). You can feel free to try a few other examples of integers, and you’ll see this is how division and multiplication work. They work this way because we designed them to work this way; they are our inventions.

Now, let’s venture a little further towards our question and look at another interesting example. Consider the following: 0 / 4 = 0. Now I will try to inverse this like I did before: 0 * 4 = 0. All right. That looks good. I see no problems there. Fine.

Now, let’s venture further right to the core of our problem. Consider the following example: 4 / 0 = X. Here I’m using ‘X’ to represent the answer because I currently don’t know what it could be. Let me nonetheless try to inverse it again like I’ve done before: X * 0 = 4. Oops! I think we’ve hit a road block here. We know that any number multiplied by 0 is, by definition, equal to zero (because you are adding the number to itself zero times). The equation X * 0 = 4 actually doesn’t have a solution that conforms to our other axioms (the most fundamental assumed truths) of mathematics.  You can replace 4 with any other number, and this problem still persists.

What Happens When You Divide By Zero?

We’ve essentially hit a paradox here. Instead of bending our entire set of inventions to solve this equation, we choose to accept that it is an exception that cannot be solved, and choose to keep the rest of the mathematical systems we invented. This is because, our systems work just fine almost everywhere else. So, this is the reason why you ger an ‘error’ or come across the term ‘undefined’ when you divide by zero.

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