# Dividing by zero

I was having a discussion with a friend and fellow mathematics teacher the other day when the topic of dividing by zero came up. She is the department head and had this in a questionnaire she gave to prospective teachers at her school. She was shocked by the number of incorrect responses provided by mathematics teachers.

This leads me to my question: "How do/would you explain why division by zero does not produce a result." I would imagine that answers would vary depending on the age/level of the student. Please indicate for which level you think your explanation would be appropriate.

Edit - 10/28/14 In response to many of the comments that address the fact that it "does produce a result" or "is defined"

The discussion was initially set in, and meant to address division with the real numbers. I recognize that an answer of "undefined" is a "result," but it is not a result in the image set in this scenario. Extending the reals or defining division in some other less-well known (and by less-well known I mean by the non-mathematical world) is certainly a valuable thing for mathematicians to be able to do. I feel like the discussion ultimately breaks down into one in Group Theory with different sets and operators. Perhaps this question should have initially been split as such:

1. How do/would you explain why division by zero does not produce a result for Elementary age children?

2. How do/would you explain why division by zero is undefined for Algebra/Calculus students?

3. How do/would you explain division by zero to the advanced mathematics student?

This, too, may not be partitioned enough. That is why I have additionally edited my original question by highlighting the last sentence!

• I would go back to the meaning of division. Let us think about division as defined using multiplication: For (e.g., real) numbers $a$ and $b$, write $a \div b = c$ iff there is a unique (e.g., real) number $c$ such that $a = b \times c$. Now, writing $a \div 0$ means there is a unique $c$ for which $a = 0 \times c$. This cannot happen when $a$ is non-zero, for $0 \times c$ for any $c$ is zero. It also cannot happen for $a = 0$, because then $c$ would not be unique: $0 \times 1$ and $0 \times 2$ both give $0$. So there is no (e.g., real) number $a$ for which dividing by zero is meaningful. Commented Oct 26, 2014 at 8:37
• Be sure you define 'result'; 'infinity' could be a result. Commented Oct 26, 2014 at 15:08
• The thing that has always confused me is why so many people think the answer is zero. Commented Oct 26, 2014 at 18:34
• I protected the question not to get too many answers. Please notify me if there is reason to undo it.
– quid
Commented Oct 28, 2014 at 18:08
• regarding lying: I wouldn't say it is "lying" since we are teaching them division in the real number system, not division in the extended reals. Commented Jan 7, 2015 at 8:34

"How do/would you explain why division by zero does not produce a result."

Any such explanation that is not rooted in student understanding would be talking to ourselves, not to students. Therefore both meaning and student understanding are important. Otherwise, what's the point? So I have grounded my response there.

Young students (elementary level) have different understandings of division situations, prior to generalizing to the understanding of division that more advanced learners have formed.

For such students there are two meanings of division: the partitive (or "fair sharing") meaning and the quotative (or "measurement") meaning of division.

To discuss division by zero to these students (or any students) you have to be aware of their understanding of division. In the case of students who see quotative and partitive situations differently, if you do not address both then you are not giving them an explanation (or allowing a discussion) that covers all of the situations that we would like them to see as division.

The box has 10 cookies. If 10 friends are at a party, how many cookies does each friend get, if everything is to be fair? [After some discussion] Now, what if there are no friends -- no people at the party. What does partitioning mean if you're partitioning cookies among no people? Is there even a party? How do we make sense of this?

That may be a puzzler to them, and worth thinking about. However, the quotative or measurement division example may be even more fruitful.

Willy Wonka has determined that a serving of gum is 2 pieces. If you arrive in class with a bag of 10 gumballs, how many students in class can you give a full serving to? [After consideration of this problem that verifies understanding] OK, now the Surgeon General doesn't think sugary gum should be in anyone's diet, so she has determined that 0 pieces of gum is a serving. With your bag of 10 gumballs, how many people can you distribute servings of gum to now?

For more on quotative and partitive division, including seeing students work with these understandings, see the Annenberg Learner site.

EDIT: There is no reason to skip to the answer for them. It's a worthwhile problem for them to consider on their own and if the object is to get them to think about it the surest way to prevent it is to simply tell them that it is defined a certain way. That authoritative approach is likely to remove the need and impetus for them to actually consider, if indeed they are curious about it.

My recommendation at this point, then, would be to continue to mostly give them questions instead of answers. "Is it possible to give everyone a 0 serving of gum? Or is it that you can give nobody a 0 serving of gum?" "Does it even make sense?" "How should we handle this?"

You can always tell them later that it is undefined, and have a discussion about that. Having puzzled over it for a time (and possibly frustrated themselves productively), that revelation might actually make sense to them.

• +1; note that I left my earlier comment before reading this answer! I agree that beginning by unpacking the partitive/quotative (equal sharing vs. measurement) approaches is a good option for elementary school students (and elementary school PSTs); I, too, think answering the OP requires this sort of pinning-down of division's meaning before any progress can be made. Commented Oct 26, 2014 at 21:39
• @BenjaminDickman - I rather figured that you were talking about definition, and I know that you know the difference because of your background. I asked because I think readers here can benefit from thinking about the role of meaning in mathematics education. Cheerio! Commented Oct 26, 2014 at 23:37
• @JasonC - I would respond with more questions. "What if the number of students in the class is different? Does the answer depend on the number of students?" Also, it's great they came up with an answer, but we still need to reconcile the different answers other people might give. "Why is this a question that people can come up with different answers for?" I don't see where any answer kills reasoning. It's just another possibility to be considered. Commented Oct 28, 2014 at 3:20
• "Why is this a question that people can come up with different answers for?" is brilliant. Thanks! Commented Oct 28, 2014 at 3:21
• @JasonC I know that, traditionally, math class is a place where students (and even teachers) often think that coming up with an answer is the end of thinking about a problem. I think that coming up with an answer is just one part of the process of learning to reasoning mathematically. Commented Oct 28, 2014 at 3:22

From the viewpoint of a mathematician and a formalist, you're asking the wrong question: the right question is not "why does division not produce a result", but "why do we define division so that it's not defined for division by zero?"

Arithmetic operations aren't "god-given" or properties of "nature": we define arithmetic so that we can describe things and solve problems. In the context of arithmetic like what students learn in primary school, the point can be roughly summarized as we want to solve equations $ax=b$ for $x$ (or any equivalent 'visualization' of this idea), and define division so that $x = b/a$ is the solution. This doesn't make sense when $a=0$, so we define division so that we forbid zero denominators.

It is worth noting that in other contexts, division by zero is not only allowed, but it would be silly to forbid it aside from the case of $0/0$ (and there are even a handful of contexts where even that should be defined): e.g. in one-dimensional projective geometry, or in complex analysis. In both cases, for $x \neq 0$, the value of $x/0$ would be the "point at infinity".

• Welcome to the site! I find it very interesting that you do not believe arithmetic operations are properties of nature, and further, that you believe this to be an accepted fact! Commented Oct 26, 2014 at 20:54
• @ChrisCunningham: Note the Hurkyl begins by limiting the viewpoint to that of a "formalist." They indeed "do not believe arithmetic operations are properties of nature." One may disagree with Hurkyl, but he did explain himself well. Commented Oct 26, 2014 at 21:02
• @Chris: I could go at lengths, but it's somewhat off topic, so I'll just say something brief and limited. A loose analogy is the map versus terrain argument; a map of the United States doesn't really resemble the United States at all: it's small, it's flat, it's covered in ink, and has all sorts of strange colors. Nature didn't give us the map: we made it ourselves, and we made up the rules for playing games with the map. This doesn't diminish the fact that our map games, suitably interpreted, tell us useful things about the United States, like how to get from my house to the grocery store.
– user797
Commented Oct 27, 2014 at 0:18
• My apologies, I didn't see the note about being a formalist, and didn't know the name for this view. Cheers! Commented Oct 27, 2014 at 2:18
• I believe in this point of view too. I have had success by saying "Well, what do you think 3/0 (for example) should be?". Then going through the repercussions of that result (the bits of maths that seem sensible it would 'break'). So, we conclude that mathematicians leave it undefined because any value you give it makes other pieces of useful maths stop functioning (or require a special exception for 0). Commented Oct 27, 2014 at 10:51

Have the students tell you that division by zero is a non-sequitur. This is possible at any age where division is understood at all.

Teacher: If there are eight cookies and four children, how many cookies does each child get?
Student: Uh, two.
Teacher: Yes! This is a division problem. $\frac{8}{4} = 2$. Now, if there are 8 cookies shared by only two children, how many cookies would each child get?
Student: Four.
Teacher: Yes, right. Now tell me the division statement for that problem.
Student: Um, 8 divided by 2 is 4.
Teacher: Good! Now, suppose we have 8 cookies and there are no children. How many cookies does each child get?
Student: That question doesn't make any sense.
Teacher: Why not?

Things will vary considerably here depending on individual students, because it's a bit difficult to put your finger on the concept that it's impossible/meaningless to assign characteristics to things which don't exist.

The real point though is that when considering the physical and logical meaning of division, the absurdity of division by zero becomes obvious. Students have trouble getting here because they often don't have a strong sense of the physical and logical meaning of the operations that they are working with, but instead consider division as just one more button on the calculator, and just one more hoop to jump in math class.

• Good! Now, suppose we have 8 cookies and there are no children. How many cookies does each child get? ---- Each child gets twelve cookies (it's logically true)! Commented Oct 26, 2014 at 20:52
• @ChrisCunningham - about 40 years ago a math teacher asked a similar question. I answered "fish". She told me that made no sense. I said neither did dividing by zero. Commented Oct 27, 2014 at 0:25
• Fine. So now I have eight cookies, and there are 0.625 children. How many cookies does each child get? Commented Oct 27, 2014 at 23:04
• @GraphicsResearch 12.8... Do the (slightly dismembered) children accept IOUs? Commented Oct 28, 2014 at 9:30
• @GraphicsResearch That's a fun hole to punch in the 'sharing between people' understanding of division, but note that 'sharing between' works fine with fractional denominators. $\frac{8}{2}$: if 8 cookies fill 2 boxes, then each box gets 4 cookies. $\frac{8}{0.5}$: if 8 cookies fill half a box, then each box gets 16 cookies. Interestingly, this model is somewhat suggestive of infinity as an answer to $\frac{a}{0}$ - if 8 cookies can fit in no space whatsoever, then surely any number of cookies can fit into a box. Commented Oct 28, 2014 at 15:38

In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number line, where it takes 3 steps of 2 units to go from 6 to 0. Teaching the concept of division this way is just the inverse of what we have done for multiplication.

After we do a few simple examples and everyone has gotten the idea of division as repeated subtraction, I offer 8÷0. We start subtracting 0 from 8: 8-0=8 and we continue 8-0=8,8-0=8,8-0=8,etc. When the whole class is giggling, I point out that this isn't working. We can also try to go backwards from 8 on the number line by steps of 0 and somehow we never leave 8. At this point I tell them that you can't divide by 0 and they all "get it" at their level. We then try dividing by 0 on the calculator and everyone is excited to see E for error.

• this is very nice for third graders. Commented Jul 23, 2015 at 20:37
– user507
Commented Sep 6, 2015 at 17:22
• @BenCrowell You made my day - thanks. Commented Sep 7, 2015 at 22:18
• so 0:0 is 0 according to this explanation?
Commented Jan 8, 2020 at 9:37
• @RiaD It seems like that at first glance, but if you look deeper you will see that 0:0 is also 1, 2, 3, 4, 5, etc. for this reason mathematicians will tell you that 0:0 is undefined. Commented Feb 9, 2020 at 20:27

Dividing $1$ disk into $\frac{1}{n}$-ths ($\frac{1}{3}, \ldots$), leads to $\frac{1}{1/n} = n$ pieces ($3,\ldots$):

As $\frac{1}{n}$ approaches $0$, the number of pieces $n$ grows without bound. The result upon division by $0$, $1/0$, should be this limit. But there is no limit.

• @RoryDaulton: PieChart[] in Mathematica. Commented Oct 26, 2014 at 0:57
• This is a good, graphical, explanation of a divide by zero (in terms of division)...I have no qualms with this answer...although it would be nice to see why sometimes dividing zero by zero results in a finite amount. Commented Oct 26, 2014 at 6:32
• To me, this suggests that $1/0 = \infty$. Commented Oct 27, 2014 at 1:27
• @JosephO'Rourke Infinity is an actual number in the relevant number systems. In this case, $\mathbb{R}^*$. Incidentally, in real analysis, "diverges-to-$\infty$" is shorthand for something like "for-any-$\epsilon$-there-exists-no-$n_0$-such-that-for-any-$n>n_0$-the-function-is-$\epsilon$-bounded-to-some-constant-$L$", but it's perfectly possible to be a first-class-number as well. Commented Oct 27, 2014 at 23:12
• A very bad vicious-circle-style argument. Impossibility of resolving 1/0 in numbers must be understood before passing to limits, not explained with them, because it is an important piece of motivation for a (new) concept of limit. Commented Oct 28, 2014 at 9:08

This answer is intended for the second category of students:

1. How do/would you explain why division by zero is undefined for Algebra students?

Begin by introducing the reciprocal of a real number as that number that satisfies the property of reciprocals:

$$a⋅\dfrac 1{a} = 1$$ Note here that $$\dfrac1a$$ must be thought of as the result of dividing $$1$$ by $$a$$ in the same way as $$\dfrac a{1}$$ is thought of as equal to $$a$$, and emphasize that $$a≠0$$ because $$0$$ times any number is $$0$$.

Next state the definition:

Definition of Division

For every real number a and every nonzero real number b, the quotient a$$\div$$b,

or $$\dfrac{a}{b}$$, is defined by:

$$a\div b=a \cdot \frac{1}{b}.$$

Dividing by zero would mean multiplying by the reciprocal of 0.

But 0 has no reciprocal (because 0 times any number is 0, not 1.)

Therefore, division by 0 has no meaning in the set of real numbers.

To show that $$0$$ cannot satisfy the property of reciprocals we must prove the multiplicative property of $$0$$.

Multiplicative property of 0

Prove:

If $$a$$ is any real number, then $$a\cdot 0 = 0$$ and $$0\cdot a = 0$$.

Proof:

Statement _________________________Reason

1. $$0 = 0 + 0$$ ___________________1. Identity property of addition

2. $$a\cdot0 = a(0 + 0)$$ ______________2. Multiplication property of equality

3. $$a\cdot0 = a\cdot0 + a\cdot0$$ ___________3. Distributive property of mult. with respect to add.

4. But $$a\cdot0 = a\cdot0 + 0$$ __________4. Identity property of addition

5. $$\therefore$$ $$a\cdot0 + a\cdot0 = a\cdot0 + 0$$ _____5. Transitive property of equality

6. $$a\cdot0 = 0$$ ____________________6. Subtraction property of equality

7. $$0\cdot a = 0$$ ____________________7. Commutative property of multiplication

Therefore, 0 times any number is 0, not 1.

(Source: Algebra: Structure and Method Book 1)

The two cases are sometimes presented as follows:

1. Dividing a nonzero number by zero, violates the multiplicative property of zero and therefore the properties of the real numbers upon which it is proven, as shown above.

2. Dividing zero by zero, which does not violate the multiplicative property of zero, but multiplication by zero is an operation that results in zero for every real number.

The following argument is presented in an older edition of the above source:

If $$\dfrac{a}{0}$$ = c, then $$a = 0\cdot c$$. But $$0\cdot c = 0$$. Hence, if $$a$$ is not equal to $$0$$, no value of $$c$$ can make the statement $$a = 0\cdot c$$ true, while if $$a = 0$$, every value of $$c$$ will make the statement true.

Thus, $$\dfrac{a}{0}$$ either has no value or is indefinite in value.

This separation into two cases, one of which results in no value satisfying the multiplicative property of zero and the other resulting in an indefinite value satisfying it, gives the impression that $$\dfrac{0}{0}$$ is allowed.

The direct argument starts with the equation: $$a⋅\dfrac 1{a}= 1$$ and emphasises that $$a≠0$$ because $$0$$ times any number is $$0$$.

Thus, the product of $$0$$ and no real number equals $$1$$.

This further reinforces the importance of the central idea that $$0$$ has no number, called its reciprocal, that when multiplied by it equals $$1$$.

Since $$0$$ has no reciprocal, division by $$0$$ is not defined. I've found a similar detailed explanation here.

• While this answer is good and very thorough, I guarantee that it would not convince any high school student or enlighten them in the least bit. The target audience of this answer is an upper division college mathematics student (or someone who has a very good knowledge of mathematical proofs--perhaps physicists or computer scientists)--no one else would fully understand this argument--especially below the college level. Commented Oct 27, 2014 at 6:43
• Too abstract? I question whether or not you have taught high school to ask such a question. Of the high school students I have encountered, there are a handful (literally perhaps 4 or 5) that would understand a proof using things such the additive identity, etc. Again, I'm not saying your statements aren't correct (they are--under certain assumptions you did not state), only that they would confuse the vast majority of high school students. If you showed this in a high school classroom you would get zombie nods. Commented Oct 27, 2014 at 7:30
• @Jared I disagree completely, I think this is a great answer that's easy to follow and understand. While a regular high-school student might not know anything about transitiveness et al, the multiplying-dividing idea is easy to grasp. Commented Oct 27, 2014 at 8:12
• @IceBoy I agree that you answered the OP's question (hence why I upvoted it--and that hasn't changed)...I was only commenting that your explanation would fall on deaf ears for most high school students (perhaps that is not your targeted audience). Commented Oct 27, 2014 at 9:27
• This is mathematically the most-correct answer, and matches the standard definitions in any college algebra book I've ever seen. If there's any high school math teacher that is unfamiliar with this, then that is simply a catastrophe (go back and re-read your abstract algebra book and think about how it serves as a model/foundation to the courses you teach).... Commented Dec 22, 2017 at 16:43

Here is my answer to high school underclassmen.

If we let $\frac10=n$ for any $n$, we then get $$\frac10 \cdot 0 = n \cdot 0$$ $$1=0$$

This works for any non-zero value divided by zero. Allowing any value here leads to a contradiction.

If we let $\frac00=n$ we get $$\frac00 \cdot 0 = n \cdot 0$$ $$0=0$$

This works for any value of $n$, so which value of $n$ should we choose? We do not have a good choice, so we make no choice.

I admit the last argument is weaker than the first, but better arguments require more time or a higher level of math.

The above is my short answer. I have a longer answer, which I developed as a response to the related Algebra 2 question "Why can we take the square root of a negative number but not divide by zero?" Here is more (but not all) from that longer answer.

My short answer assumes we are talking about the real number system and also assumes that we keep all the usual properties of that number system. Jared's comments are valid but are applicable to another number system, such as the projective extended real number system (where we add the value $\infty$, let $1/0=\infty$, and say that $0 \cdot \infty$ is undefined or "indeterminate"). The main problem with such extended systems is that they lose some fundamental properties of the real numbers. Jared's comments assume that multiplication is not closed, disallowing $0 \cdot \infty$, which is much more drastic and un-intuitive than disallowing $1/0$.

In brief, I was keeping things simple and sticking to the real number system.

• I disagree with this logic (this is from a calculus/limit standpoint). If you are given that $\frac{1}{0} = n = \infty$, then you you have: $\frac{1}{0}\cdot 0 = \infty \cdot 0$--the right side is indeterminate, so you cannot unequivocally say that it's $0$...nor can you can logically conclude that $\frac{0}{0} = 1$ which it appears you did here. Commented Oct 26, 2014 at 4:33
• First, I don't disagree that $\frac{1}{0}$ is infinite, but it's not undefined--it's infinite. This argument is extremely naive because it's multiplying an infinity by zero which is not logically sound. As an example, I can do $\lim_{n\rightarrow 0}\frac{1}{n}\cdot n = 1$ You cannot talk about a divide by zero without talking about limits! Commented Oct 26, 2014 at 6:20
• @Jared: I can "talk about a divide by zero without talking about limits" by sticking to the real number system (as I did in my answer) or by using the projective extended real number system or the affine extended real number system. See the addition to my answer. Thanks for your comments: they are thought provoking. Commented Oct 26, 2014 at 11:06

Having recently covered this topic in a course for pre-service elementary school teachers, I thought I would write a bit about the somewhat subtle difficulties entailed in tackling this question. I am going to use language that may be at the level of undergraduate majors in mathematics or mathematics education, but I believe that the content can be scaled so that it can be implemented at the elementary level.

This answer is to some extent a re-hashing of other responses here; however, I am confident that multiple presentations of similar material can still be valuable in developing our own understandings.

As the OP writes:

I feel like the discussion ultimately breaks down into one in Group Theory with different sets and operators.

In a related spirit, I will list below four possible difficulties (phrased as questions).

1. When we use a binary operator (in this case, division) in elementary school, what are the associated sets under discussion (and what do we need to know about binary operators)?

2. If our ultimate goal (in elementary school mathematics) is to come to a consensus that $a\div0$ is undefined, then how will we explain what we mean by this word (undefined)?

3. If we are working with the rational numbers, then what background knowledge do students need?

4. What are the different interpretations of division, and how can we use them to make sense of an expression such as $a\div0$?

In thinking about 1, we might recall earlier discussions of the binary operator of addition and the set of whole numbers. In particular, we can take as input any two whole numbers (let us say order matters, though we would show at some point that addition is commutative), and give as output a unique whole number. The uniqueness here corresponds to the concept covered in an undergraduate mathematics course as well-definedness. For early mathematics education, even this feature is not obvious for addition: Consider an interpretation of addition that involves combining and counting discrete objects, and note that (cf. MESE 5866 on counting) "children sometimes believe the same collection can be characterized by two or more numbers; yes it has 14, and it also has 15!"

Subtraction presents problems of its own, because, unlike the above case, we may take as input two whole numbers in a particular order, but the output might not be a whole number. For example, if we are given the whole numbers $3$ and $4$, then we find that $3-4$ is not a whole number. How does one deal with such a scenario? The mathematical temptation might be to extend to the integers; early on in one's mathematics education, though, the simplest subtraction problems (i.e., those that involve an action: "take away/separate problems") are only posed if the minuend is greater than the subtrahend. We do not begin our discussion of subtraction by showing a child $3$ apples and asking that s/he take away $4$ apples.

These brief remarks about subtraction relate back to 2; namely, if we are talking about subtraction strictly using whole numbers, then an expression that cannot be evaluated as a whole number is said to be undefined. It is not that $3-4$ is secretly negative one; rather, we have not defined what happens when the minuend is less than the subtrahend, and so any such expression is (theretofore) literally undefined.

Briefly, with regard to 3: Doing justice to the (let us say positive) rational numbers is nontrivial, for it requires students to grasp the fractions (ordered pairs of a whole number and non-zero whole number) as well as their standard equivalence relation and the corresponding set of equivalence classes (cf. MESE 1447). Therefore, discussed below is only the case of why a whole number divided by zero is said to be undefined in elementary school mathematics; similar reasoning can be extended when one wishes to discuss more generally the division of rational numbers, but is not pursued here.

More precisely: Just as subtraction problems are initially presented only when the minuend is greater than the subtrahend, division problems for whole numbers are initially presented only when the remainder is zero. Sometimes a discussion of remainders is used as scaffolding in helping to deepen students' understanding of division; one must be careful, though, for the notation involved in this endeavor can be misleading. E.g., $4\div3 = 1R1$ and $3\div2 = 1R1$, but we would not wish to conclude that $4\div3 = 3\div2$.

As to 4, there are essentially three different interpretations of division. These are sometimes referred to as: the partitive (equal sharing), quotative (measurement AKA repeated subtraction), and missing factors interpretations. The former two interpretations are both discussed in JPB's answer; the missing factors interpretation states that $a\div b = c$ means there is a unique $c$ for which $a = b \times c$ (cf. my comment).

How do/would you explain why division by zero does not produce a result?

Recall that in the notation $a\div b = c$ we call $a$ the dividend, $b$ the divisor, and $c$ the quotient. In both the equal sharing and measurement interpretations, the dividend refers to the total amount of objects. In the equal sharing interpretation, the divisor is the number of (equal sized) groups of objects, and the quotient is the number of objects in each group. In the measurement interpretation, these meanings switch: The divisor is the number of objects in each group, and the quotient is the number of (equal sized) groups.

To resolve a question such as $a\div0$, it may be wise to begin with a few other questions:

Use each of the three interpretations to explain why $6\div2 = 3$. For the equal sharing and measurement interpretations, what might the corresponding pictures look like?

(Sketch: In equal sharing, we could begin by drawing two circles to represent the meaning of the divisor; then we would alternate putting one dot in each group until all six were used up. At the end of this process, there would be three dots in each group, which tells us that the quotient - i.e., the number of objects in each group - is three. Alternatively, using measurement, we could begin by drawing six dots to represent the dividend. Next, we note that the divisor tells us there are two dots in each group, so we begin to draw circles around the dots, two at a time. At the end of this process, there would be three groups of two dots, which tells us that the quotient - i.e., the number of equal sized groups - is three.)

Using the missing factors interpretation, $6\div2 = 3$ means that $6 = 2 \times 3$, which is a true number sentence. Moreover, we ought to observe that $3$ is the unique whole number that, when multiplied by $2$, gives $6$. A smaller number multiplied by $2$ will be less than $6$, and a bigger number multiplied by $2$ will be greater than $6$.

From here, we can segue into a discussion of $0\div6$ and $6\div0$; then, finally, $0\div0$.

I believe that you will find all three interpretations can be readily applied (as in the parenthetical example using $6\div2$ above) to each of the former two scenarios. The most difficult case to discuss is that of $0\div0$; again, pushing students to explain what is meant in each case (as suggested by JPB) will allow them to understand why dividing by zero (in the context of whole numbers) does not make sense.

In such a discussion, I would emphasize that the output is supposed to be a unique whole number. Without this attention to binary operators and underlying sets, one difficulty you might run into is that students believe, e.g., $6\div0$ is undefined (there is no whole number that, when multiplied by zero, gives six) whereas $0\div 0$ is "all numbers" (because any number, when multiplied by zero, gives zero). I believe that the latter remark ought not to be viewed simply as a misunderstanding, but rather as an authentic effort to make sense of the mathematics. However, it asserts that the answer is a(n infinite set) of whole numbers, whereas we have required (by definition) that any admissible answer be a single, unique whole number. Bearing this in mind, one should be able to convince elementary school students (better: have students convince themselves) why it does not make sense to divide a whole number by zero.

Division is multiplication, backwards. These two equations are exactly equivalent, by definition:

$$a / c = b$$ $$a = b \times c$$

It's easy to understand division by zero if we look at the equivalent multiplication.

$$a / 0 = b$$ $$a = b \times 0$$

For any real number $b$:

$$a = b \times 0 = 0$$ $$a = 0$$

There are two cases with division by zero:

1. If $a = 0$, then $a/0 = b$ is unconstrained, any real number $b$ satisfies the equation. You can discard such an equation which does not constraint the result.

2. If $a \ne 0$ then $a / 0 = b$ is contradictory. There is no real number $b$ which satisfies that equation. This is still useful to know; "there is no answer" can be a sort of meta-answer. For example if trying to solve a system of equations of static forces, "there is no answer" might mean you need to consider a different design for your bridge!

There is no need to consider advanced concepts such as limits in order to fully understand division.

In short, $a / 0 = b$ is true if and only if $a = 0$.

If you see such an equation $a / 0 = b$, you may simplify it to $a = 0$.

$a / 0 = b \iff a = b \times 0 \iff a = 0$

• The suggestion that $a/0 = b$ implies $a = 0$ is highly irregular (as is your closing sentence: $a/0 = b$ can be "simplified" to $a = 0$). No, in school mathematics, $a/0$ is undefined (including the case in which $a = 0$). Commented Dec 3, 2014 at 15:56
• @Benjamin, my answer is a complete and correct explanation of division by zero, which can be understood by any student of arithmetic. The conventional approach, to "give up" on seeing a division by zero, is conceptually weak and less useful in practice. The department head might learn something from this answer. If she is any good at mathematics, I doubt she would down-vote it. Commented Dec 5, 2014 at 1:11
• @SamWatkins, I'm not sure I agree with your statement that this "can be understood by any student of arithmetic." You do not state, as I asked originally and then bolded upon editing, for what level of student this is appropriate. Could you perhaps clarify your answer to incorporate this? To suggest that "$a=0$ implies $a/0=b$ is unconstrained" hints at the idea of a non-unique answer, but I don't think that means that it is totally unconstrained. Commented Dec 8, 2014 at 15:52
• How many zeros are there in 10? We can't make 10 from zeros. How many zeros are there in zero? As many as you like. If we can divide something by zero, it must be zero also. That "translation" in simple words is suitable for anyone who can understand division at all, I guess. Commented Dec 10, 2014 at 0:19

Division itself is defined like this: The result of division (the quotient) of "b divided by a" is defined as the number x in

ax = b.

For example, let's divide 8 by 4:

We look for the number x with: 4⋅x = 8

We find x = 2. So 8 divided by 4 equals 2.

Now, if we want to divide for example 8 by zero (!) we have to look for x with

0⋅x = 8.

But, you would agree it makes sense to say

0⋅x = 0 for any x.

So, there is no x that satisfies 0⋅x = 8 and 8/0 can't be found.

If you divide by numbers closer and closer to zero

0.1⋅x = 8

0.00001⋅x = 8

0.0000001⋅x = 8

etc. you see that x becomes greater and greater. So it could make sense to say that 8/0 = infinity. You could define it that way if it made sense to any calculation.

Let's try another thing and divide zero by zero. That means: let's find the x with

0⋅x = 0.

You will agree this is true for any number. So 0/0 is any number if you wanted it to be this way. But that's disappointing because we want division to always have exactly one result. But once again, you could define it that way if it made sense to any of your calculations.

I think it's important to explain students that formally it's not defined, but also give them motivation for why it is not defined. Yes, definitions are in some sense arbitrary, but we have motivations for defining things certain ways, and I think it is good for high school students to grasp some of those motivations and intuitions behind certain definitions. A couple of ideas, and I apologize if I am repeating what has already been said: 1, Ask them how they "check" division with multiplication: "why is 6 divided by 3 equal to 2? because 2 times 3 is equal to 6. Ok, so who has a proposal for what 6 divided by 0 is?" And they will realize that there is nothing that they can multiply zero by to get 6. Naturally your students might ask about 0/0. This is a good chance for a class discussion. It is possible that one student may say zero, since zero times zero is zero, and then another point out that it could be 3, since 0 times 3 is equal to 0 as well.

2, I think it will be hard to come up with a satisfying reason on the spot for why 0/0 is undefined. I suppose you could do something about wanting to preserve continuity. I'd suggest having students come up with definitions of 0/0 and explain and defend their results - hopefully some students will show what rules about multiplication and division will have to be amended to account for the 0 case. For example, distributivity.

• To clarify, this is an argument you would have with an Algebra level student, yes? Commented Jan 7, 2015 at 9:02
• I was envisioning an algebra level student, but I think it could apply to pretty much any level in high school or early college. Commented Jan 7, 2015 at 22:06

I just realised that this solution is similar to NiloCK but it is different.

1. Teacher: With these 6 pens if I put 2 in each bag how many bags will I have? Students: 3

2. Teacher: With these 6 pens if I put 1 in each bag how many bags will I have? Students: 6

3. Teacher: With these 6 pens if I put half a pen in each bag how many bags will I have? Students: 12

4. Teacher: With these 6 pens if I put 0 in each bag how many bags will I have? Students: You can't do it!

(YAY)

BUT some will say Students: infinite!

Teacher: But i do this a huge number of times and keep going on and on and on, will I ever FINISH grouping the 6 pens?

... and then explain to them how infinite is a concept and not a number. .. in senior classes I would link it to graphs and asymptotes and why some students believe that dividing by zero is infinity, and then explain why it tends to infinity

• I am not sure which of the comments I wrote will reach you so I summarize: I moved the answer to the correct place. No further action of you is needed (I initially did not think of the option to merge).
– quid
Commented Aug 29, 2015 at 19:50

The answer should be that if you divide a finite value by zero then it can be either $\pm \infty$ (it's not clear if it's positive or negative without further context). This doesn't mean that the value doesn't clearly exist--you can say something about it if you know it's infinite. On the other hand, if you try to divide $\frac{0}{0}$ then the answer is indeterminate--it could range from $\pm \infty$ to some finite value (including $0$) to the limit does not exist at all.

In answer to your second question about age/level of appropriateness. I would say that at the level of Algebra, dividing by zero is always undefined. The real level depends on whether or not you've introduced the idea of limits (which could be as early as pre-calculus and certainly by calculus). You cannot talk about dividing by zero without limits.

• I would appreciate some feedback on the downvote. I am fairly new to this section of stack overflow, so I'm sure I've given bad advice, but I do not see what is necessarily wrong here. Commented Oct 26, 2014 at 6:01
• A comment on your comment: This isn't StackOverflow at all, it's the Math Educators site on StackExchange. Don't know about the downvote, though Commented Oct 26, 2014 at 8:57
• I'm not responsible for the downvote, but I agree that this isn't a very good answer. It's possible to extend the real numbers in various ways, some of which make dividing by 0 meaningful (or sometimes meaningful); on the projective line (i.e. the one point compactification), dividing by 0 (other than 0/0) just gives the unique point at infinity. On the two-point compactification, some divisions by 0 meaningfully give either $+\infty$ or $-\infty$ as an answer. Commented Oct 26, 2014 at 14:52
• Your answer is incorrect, however, in suggesting that one or the other is always possible: plenty of limits of the form $\lim_{x\rightarrow a}\frac{n}{f(x)}$ where $\lim_{x\rightarrow a}f(x)=0$ can't meaningfully be assigned either $+\infty$ or $-\infty$ as an answer. Commented Oct 26, 2014 at 14:54
• More importantly, it isn't appropriate to assume that the two-point compactification is always the right extension to think about. The right way to extend the real line for working with division by 0 is situational; saying that it involves thinking about limits is assuming a context that wasn't given in the question. Commented Oct 26, 2014 at 14:56

Define division: $a ÷ b$, also written $a/b$, means a number $x$ such that $x ∙ b = a$.

A few examples: (a) 6 ÷ 3 = 2 because 2 ∙ 3 = 6. (b) 20 ÷ 5 = 4 because 4 ∙ 5 = 20. (c) 72 ÷ 9 = 8 because 8 ∙ 9 = 72.

Now, an expression like 6 ÷ 0 is not any real number because we cannot meet the definition of x ∙ 0 = 6; the product of any real number times 0 would be 0, never 6.

• I tried to provide this answer ("missing factors" -- viewing division in terms of multiplication) in a comment that was also referenced in my response... Commented Sep 6, 2015 at 5:53

"divided by $$x$$" means "times $$x$$ inverse". "$$x$$ inverse" is the number such that $$x$$ times $$x$$ inverse = 1.

0 times anything = 0. 0 times nothing = 1. Nothing is 0 inverse. "0 inverse" does not refer to anything. "$$x$$ times 0 inverse" does not refer to anything.

"x divided by 0" does not refer to anything.

You might explain, at any level, that for any real number $$x$$, we have $$0x=0$$. So the equation $$0x=0$$ has no unique solution. And $$0x=1$$, for example, has no solution whatsoever. For these reasons, neither $$0/0$$ (with no unique solution) nor $$1/0$$ (with no solution) can be real numbers.

The term "division" is actually applied to a few different operations in different context, depending upon whether the operands and intended result are whole numbers, integers, or real numbers. Although whole numbers are often thought of as simply being a superset of the integers, integers, whole numbers, and real numbers have some fundamental differences in nature which can be important when performing operations like multiplication or exponentiation, but become most apparent when performing division.

Fundamentally, the meaning of whole numbers is defined in terms of multiplying any kind of thing with a defined additive identity and associative addition operator by them. $ONE$ is the thing which, when multiplied by any x, will simply yield X. $TWO$ is the thing which, when multiplied by any x, will yield x+x. $ZERO$ is the thing which, when multiplied by any x, will yield $ZERO$; it may also be added to anything without affecting it. Note that whole numbers don't require "mathematical" multiplication. If a person has one rock and adds another rock, the person has "two" rocks. An important feature of whole numbers is that multiplying $ZERO$ by anything will yield $ZERO$.

Real numbers behave a lot like integers, but with a twist: although multiplying real-number zero by any finite number will yield real-number zero, some "problematic values" which would cancel out when multiplied by (whole-number) $ZERO$ do not cancel when multiplied by real-number zero but may yield anything--either a problematic value or a normal-seeming number.

All of this becomes important when trying to define division. Whole division entails finding the largest whole number which, when multiplied by the divisor, will yield something which is defined as being no "larger" than the dividend according to some defined means of comparison. The difference between the dividend and the result of that multiplication is called the remainder. Note that when dividing by any form of zero or additive identity, there is no defined quotient, since there is no limit as to how large a whole number one could multiply by the divisor without yielding a result larger than the dividend. Nonetheless, the equation $dividend = quotient * divisor + remainder$ holds just as well when dividing by zero as when dividing by anything else.

Division of an arbitrary thing by a whole number seeks to find a value which, when added to the dividend type's additive identity the number of times specified by the divisor, will yield the dividend precisely, without a remainder. If the whole number divisor is $ZERO$, however, the operation becomes one of finding a value which, when multiplied by $ZERO$, will yield the dividend. Unless the dividend is itself $ZERO$ (as opposed to merely being an additive identity like real-number zero), no such value will exist.

Division of a real number by another real number seeks to find a real number which, when multiplied by the divisor, will yield the dividend without a remainder. If the divisor is real-number zero and the dividend isn't, there won't be any real number which could satisfy the constraints, but the division may yield a "problematic value" which, when multiplied by real-number zero, might arbitrarily yield the dividend. Note that while the result of multiplying such a problematic divisor by a real number is not well-defined, the result of multiplying by $ZERO$ is well-defined--it yields $ZERO$.

The three styles of division have slightly different behavior when dividing by zero; the kinds of inferences one may draw in cases involving "division by zero" will thus differ. When performing whole division by zero or any additive identity, the "division equation" will be satisfied by making the remainder equal the dividend. Division of anything other than $ZERO$ by $ZERO$ is simply undefined. Division of a real number by zero may yield a "problematic value". In no situation is division by zero "nicely defined", but the exact meanings differ.

• Sorry for my ignorance: what is “whole numbers, a superset of integers”? Commented Oct 28, 2014 at 9:15
• @IncnisMrsi: "Whole numbers" describe how many of an indivisible and non-invertible object something can have. It can have one things, two things, etc. or it may have zero. Whole numbers can be defined in terms of the universal additive identity (adding anything to $ZERO$ yields that thing) and universal multiplicative identity (multiplying anything by $ONE$ yields that thing). The value $TWO$ is $ONE+ONE$; $THREE$ is $TWO+ONE$; every whole number may be defined likewise without reference to anything else. Commented Oct 28, 2014 at 15:39
• @IncnisMrsi: Integers are a supplement to the whole numbers, predicated on defining $MINUS_ONE$ as a value which, when added to $ONE$, will yield $ZERO$. Then $MINUS_TWO=MINUS_ONE+MINUS_ONE$, etc.; every integer may be defined likewise without reference to anything else. Note that unlike whole numbers and real numbers, which only have one defined method of addition, integers have three: one may choose a method which yields a the smallest quotient, the lowest quotient, or a non-negative remainder; a method which always yields one will not always yield either of the others. Commented Oct 28, 2014 at 15:48
• Could you rewrite all this text in standard numerals (note the ‹edit› link just below the posting) and then delete comments? Ī won’t dissociate my brain in censored UPPERCASE TeXed words interleaving with ridiculously rendered underscore characters. Or, if your theory depends on THICK_SNAKE_CASE, then remove TeX from it. Commented Oct 28, 2014 at 16:34
• The TeX came out a little over-emphasized; I'm not sure what markdown style would be best; my point was to make a clear typographical distinction between the value one gets by e.g. adding 1.25+0.75, versus the value that's twice the universal multiplicative identity. This distinction is important if one considers that whole-number multiplication is defined, and the distributive rule holds for, anything with an associative addition operator, but other forms of multiplication are not defined. For example, if one defines string addition as concatenation, one may multiply a string "xy" by... Commented Oct 28, 2014 at 16:54