Why We Don’t Divide By Zero in Arithmetic

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Division by zero is one of those basic concepts that confuses the poop out of people.

You’re taught, “You can’t divide by zero.” But are you taught why? Adequately? Nah. That’s one of the fundamental goobers of elementary school. They give you rules to memorize, but even the teachers are unclear of why those rules are rules.

It’s not too tough to understand why division by zero (in arithmetic) is “verboten.” You just have to get out of the mindset of “well, it doesn’t make sense.” It does make sense. It just doesn’t make sense if you only think about it with your brain-stem. You have to break out of the “intuitive” mindset. (Intuition is also not all it’s cracked up to be. Untrained intuition, that is.)

If you are a person who has a hard time letting go of the notion that division by zero (in arithmetic) cannot be done, consider this:

A long time ago, when you were very young, you learned the facts of life. And when you first learned where babies came from, the odds are you were shocked. (”My mommy and daddy never did thaaaat!” you probably cried.) It was unthinkable, and you immediately suspected evil of whoever told you that “lie.”

Eventually you got over it, I hope. When you look back, you probably cringe when you think of how you resisted the “fruit of the knowledge tree.” Well get ready, because as soon as you “see the light” about division by zero, you will be enlightened as to why your “intuition” and “common sense” and “but-everybody-knows…” mentality has held you back all your life.

So, although many people hold themselves back with immature (yet apparently reasonable) arguments like, “But zero goes into something an infinite amount of times, so anything divided by zero should be infinity.”  The simple rebuttal to that is that a) nothing is divided by zero and b) infinity is not a number, it is a concept. You can’t put it in an arithmetical equation. (For much more about this, see  The Zero Saga.)

For a good lesson on it, check out Division by Zero at MathMojo.com.

For those who prefer “plain english” or “common sense” (neither of which are as good as they’re cracked up to be) you might want to think of it like this:

To divide by nothing is like not dividing by something. It’s like not dividing at all. Therefore, when you divide by zero, you don’t divide by anything. So you are not dividing. It’s like “going nowhere.” You didn’t go anywhere. You stayed. Going nowhere is not going. You can’t go if you stay.  Dividing by zero is not dividing. You can’t divide if you don’t divide. That is why you “can’t” divide by zero. 

That’s not to say you can’t try. You can try. You can say you are dividing by zero, you can pretend you are dividing by zero, you can insist you are dividing by zero. You’re just not really doing it. 

Please realize that the above is just a “plain english” explanation. It is not the full monty. It’s only used to get you to try to see that the common “common sense” version has a more plausible uncommon “common sense” counter-argument. Now that you’ve read it and understand it, you’re ready for a more mathematical explanation. 

But don’t worry, the mathematical explanation is also explained in plain english. It is easy enough for a child to understand. Check it out at: Division by Zero at MathMojo.com.

If you have a desire to learn about the deep, intricate and wonderful properties of zero, expertly and clearly explained, do yourself a great favor and visit Dr. Hossein Arsham’s The Zero Saga.

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