Division by Zero

In studying mathematics, you have probably heard that division of zero is undefined. What does this mean?

Since we do not know exactly what is the answer when a number is divided by 0, it is probably reasonable for us to examine the quotient of a number that is divided by a number that is close to 0.

If we look at the number line, the numbers close to 0 are numbers numbers between –1 and 1.

Figure 1 – The number line showing the numbers close to 0.

For instance, several positive numbers close to 0 and less than 1 are 0.1, 0.01, 0.001 and so on. Similarly, negative numbers close to 0 but greater than – 1 are –0.1, -0.01, -0.001 and so on.

The table and the numbers below shows the quotient 1/x when 1 is divided by x, where the x’s are numbers close to 0.

Figure 2 – The value of 1/x as x approaches 0 from both sides.

In the graph, as x approaches 0 from the right (as x, where x are positive numbers, approach 0), the quotients of 1/x are getting larger and larger. On the other hand, as x approaches 0 from the left (as x, where x are negative numbers, approach 0), 1/x is getting smaller and smaller. Hence, there is no single number that 1/x approach as x approaches 0.  For this reason, we  say that 1/0 is undefined.

A simple analogy would also let us realize that allowing division by 0 will violate an important property of real numbers.  For example 8/4 = 2 because 2 x 4 = 8.  Assuming division of 0 is allowed. If 5/0 = n, then n x 0 = 5.  Now, that violates the property of a real number that any number multiplied by 0 is equal to 0.

Since division by 0 yields an answer which is not defined, the said operation is not allowed.

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11 thoughts on “Division by Zero

  1. “Because the universe will blow up,” was the usual answer I got when my teachers tried to explain why we couldn’t divide by zero. From a young age, I was a sort of anti-Pythagorean in that I believed people created numbers, not that the universe was ruled by them. So why then did we create the divide-by-zero bomb?

    The best way I’ve found to describe why dividing by zero will destroy everything is to go back to translating fractions. What does “1/2” really mean? “1/2” translates to “1 out of 2” or “I have one piece of candy out of the two pieces on the table, so I have half of what is on the table. My sister is a good sharer.”

    Now try this with “0/2”. This translates to “zero out of 2” or “I have zero pieces of the two that are on the table. My sister’s cheap!”

    Both of these situations are real. You can have one piece of candy out of two. You can have none of the pieces of candy. Even if the fraction is an improper fraction, like “3/2”, certainly you can’t have three out of two pieces of candy; this makes no sense at all. But then we remember that improper fractions can be written into mixed fractions, so “3/2” becomes “1 and ½”, and we sure can have 1 and a half of the pieces of candy on the table [leaving our cheap sister with just ½! Haha!]!

    So then comes “2/0”, which would translate to “2 out of zero” or “I have two pieces of candy out of the zero that are on the table.” HUH?? This obviously doesn’t make sense! Despite what Little Orphan Annie and Jay-Z may lead us to believe, you can’t make something out of nothing. It’s just basic physics.

    Once a student begins learning about slope and functions, the impossibility of “2/0” becomes even more obvious. Let’s think of a graph that measures your height against your age. “2/0” represents a rise (y-value or “height”) of 2 and a run (x-value or “time”) of 0. This is to say that, for example, at time 0 you are 2 feet tall. Ok, so maybe you were born 2 feet tall. That’s possible. Now let’s move up from coordinate (0, 2). The slope of “2/0” tells us to move up 2 and over 0. We move up two spaces to 4 feet tall and over to… over to nothing! We stay at zero! So a slope of “2/0” says that you can be 2 feet and 4 feet tall at the same point in time. This is impossible!

  2. I always found this concept difficult to explain if one is not thinking mathematical. I therefore explained this in terms of pizza slices. I first use the example of being sued but having no money. If someone sues you and you have no money can they still sue you. The answer is yes, but they will not receive anything since you have nothing to give, yet the action of suing (division) takes place. Therefore you can do the act with a zero outcome. The next example i use is the example of pizza slices. If there exists an eight slice pizza and there is no one around to consume it and you wnat to split that pizza up zero ways, what happens. The answer is nothing because there is no action of splitting up the pizza since no one is going to receive any. Since no action of dividing took place there is no definition of what happened since nothing happened. the operation of dividing did not take place.

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