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Why You Can't Divide Nine By Zero

Asked by Lee Williams, student, Aldridge State High School on August 8, 1997:
What is the answer to 9 divided by 0 on a high school level and university level.
The answer to this question is that there is no answer. By this we simply mean that there is no number which, when multiplied by 0, gives you 9.

The question "what is 9 divided by 0" is simply another way of asking the question "which number, when multiplied by 0, gives you 9?" There is no such number, and therefore no answer to the question.

This is not simply a quirk of our usual number system: in any number system that satisfies certain basic properties such as distributivity and associativity, 0 multiplied by anything will give 0. So if you start with any non-zero number x, there cannot be any number which when multiplied by 0 gives you x, so there can be no answer to the question "what is x divided by 0". Mathematicians say that "division by 0 is undefined", meaning there is no way to define an answer to the question in any reasonable or consistent manner.

In general number systems, "0" is defined to be the unique number which has the property that a + 0 = 0 + a = 0 for all a in the number system. Also, for every number a there is a number -a called the additive inverse of a which satisfies the property a + (-a) = 0. Here is how to show that any number a times 0 gives you 0 (and thus that division by zero is always undefined):

(a)(0) = (a)(0 + 0) (since 0 = 0+0 by the definition of 0)

(a)(0) = (a)(0) + (a)(0) (distributivity is used)

(a)(0) + [-(a)(0)] = (a)(0) + (a)(0) + [-(a)(0)] (adding -(a)(0) to both sides)

0 = (a)(0) (since (a)(0) + [-(a)(0)] = 0 by the definition of -(a)(0))

There are some contexts in which it makes sense to talk about an "infinity" concept; see the pages on Does infinity exist? However, the above reasoning shows that you cannot simply include "infinity" in a number system and say that 9 divided by zero is infinity, at least not if you want to retain properties like distributivity which are so essential to the nature of numbers.

One final comment: the question "what is 0 divided by 0" is a little different from questions like "what is 9 divided by 0". Instead of there being no number which will work, now every number will work (every number, when multiplied by 0, gives 0). But there's still no answer to the question "what is 0 divided by 0", because this question is really asking "what is the one special and unique number which, when multiplied by 0, gives 0?" There isn't any such single, unique number, and hence there is no answer to the question.

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