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Do You Know What Happens When You Divide By Zero | Touchstone Words

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What Happens When you Divide by Zero

By Shane Staret on 2017-01-13

It is probable that you have been told that it is impossible to divide by zero ever since the first day you learned about division. Maybe you never have thought too greatly about why dividing by zero is not possible...or maybe you have thought about to a great extent, however, you have had to stop yourself before having an existential crisis.

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Even modern computer programming languages are not built with the ability to handle a division by zero. In Java, it will give you an ArithmeticException, causing the program to not run entirely unless this error is purposely ignored by throwing it. But, why is dividing by zero impossible in the first place? To answer that question, one must first understand what dividing even is in the first place.

Even though division is one of the most basic mathematical principles, some people still get tripped up on the concept of division. If someone asks, “What is 32/4?”, someone can think of it like this: How many times does 4 go into 32? Well, 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 32, and we used 8 4’s, so therefore the answer to 32/4 is 8. However, what if someone were to ask, “What is 1/0?”. Using the same logic, we would then ask, “How many times does 0 go into 1? 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + … + 0 = 0. We might have just run into a bit of  a problem. Adding 0’s over and over again will never allow us to reach 1. Say instead of 1/0 someone asked what 2/0 was. The answer is similar, as adding 0’s over and over again never allow us to reach 2. Then what is the answer? Well, it seems that adding 0’s over and over again won’t get us anywhere, so maybe it would be best to look at a graph. Since we are looking at what 1/0 is equivalent to, maybe it would be best to view a 1/x graph:

 

It appears that as we approach zero from the left-hand side, the output gets exponentially smaller while approaching from the right-hand side leads to the output getting exponentially larger...with absolutely no bound in sight. That must mean that from the right-hand side, 1/0 = +∞, while from the left-hand side, 1/0 = -∞. Yet, we run into another problem. 1/0 cannot be equal to both +∞ and -∞. A fundamental property of limits, which we see with 1/x when x = 0, is that the left-hand and right-hand sides must approach the exact same number in order for an output to be defined at a given input. Here, when the input is 0, the output approaches +∞ from the right side, but -∞ from the left side. Since the left-hand side and right-hand side do not approach the same number, then 1/0 is not defined, which is also known as undefined. And that is the answer, 1/0 is simply undefined. It’s answer is that there is no answer.

...At least that is true if we are simply looking at the real numbers. I am definitely not qualified to speak about division by 0 when looking at commutative rings and fields, so if you want to go down that rabbit hole, here ya go. Also, this is the reason why computers typically don’t handle dividing by 0 very well, as there technically is not really an answer to and computers don’t like when they don’t know something.

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