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How do we explain that dividing a positive number by $0$ yields positive infinity to a 2nd grader? The way I intuitively understand this is $\lim_{x \to 0}{a/x}$ but that's asking too much of a child. There's got to be an easier way.

In response to the comments about it being undefined, granted, it is undefined, but it's undefined because of flipping around $0$ in positive or negative values and is in any case either positive or negative infinity.

Yet, $|\frac{2}{0}|$ equals positive infinity in my book. How do you convey this idea?

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Don't explain wrong things to a 2nd grader. Division by zero is not defined. Period. –  Hagen von Eitzen Nov 21 '12 at 21:33
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Perhaps it is OK to pretend to a little kid that there is a tooth fairy. But it is not OK to tell the kid that one can divide by $0$. –  André Nicolas Nov 21 '12 at 21:38
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Division by zero is not always defined - but I once explained to my daughter (10 years old at the time) how to wrap the real line around a circle and make the join using the point at infinity, and went onto the Riemann Sphere. Youngsters can understand these things before they have the mathematical sophistication to understand them precisely - which makes it really important that we guide their understanding accurately. And then there is projective geometry (including finite projective planes) rich territory to explore, just get it right. –  Mark Bennet Nov 21 '12 at 21:54
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Division by zero is not defined for many more reasons than just that particular limit issue. If it did exist then by definition of division you would need to have $1 = \frac{0}{0} = \frac{0 + 0}{0} = \frac{0}{0} + \frac{0}{0} = 2$. If, by the way, you don't mean that $\frac{0}{0} = 1$ then what do you mean by division by zero at all? Even when you work on the Riemann sphere or the one point compactification of $\mathbb{R}$ you lose a lot of the usual structure to get the definition of $\frac{1}{0} = \infty$. –  Chris Janjigian Nov 22 '12 at 0:23
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@Marcus My kid came home talking about his friends having a competition about "who can name the biggest number". One said "infinity!" Another said "infinity + 1!" My kid said "infinity to the infinity!" I gave him a time-out for that. Infinity is not a number. –  Fixee Nov 22 '12 at 0:34
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13 Answers 13

up vote 32 down vote accepted

The first thing to point out is that division by zero is not defined! You cannot divide by zero. Consider the number $1/x$ where $x$ is a negative number. You will find that $1/x$ is negative for all negative $x$. As $x$ gets closer and closer to zero, $1/x$ gets bigger and bigger in the negative direction: $1/x \to -\infty$ as $x \to 0$ from the negative side. Next, consider the number $1/x$ where $x$ is a positive number. You will find that $1/x$ is positive for all positive $x$. As $x$ gets closer and closer to zero, $1/x$ gets bigger and bigger in the positive direction: $1/x \to +\infty$ as $x \to 0$ from the positive side.

$$\lim_{x \to 0^-} \frac{1}{x} \neq \lim_{x \to 0^+} \frac{1}{x}$$

Informally: what does $6 \div 3$ mean? It means, how many times do you add $3$ together to get $6$, and the answer is $2$. What does $7 \div 2$ mean? It means, how many times do you add $2$ together to get $7$, and the answer is $3\frac{1}{2}.$ What does $1 \div 0$ mean? It means, how many times do you add $0$ together to get $1$? Well: $0 = 0+0 = 0+0+0$, etc. You have to keep adding zeros for all of eternity. In reality, you never get to $1$ and so there is no answer. It is not infinity: you can't have "infinitly many" zeros. But some people might say "You add $0$ together infinitely many times".

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Even if you could add $0$ infinite times, with any reasonable definition of infinite addition it still wouldn't be $1$. –  Javier Badia Nov 21 '12 at 22:24
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Ehh, Javier, to me that makes it sound like you're saying $0\cdot \infty=0$. I think we'd have to leave an infinite addition of zero as undefined. (NOT to be confused with the limit of infinite addition of zeros. That's definitely zero). –  NeuroFuzzy Nov 21 '12 at 23:01
    
What's stopping us from simply defining $1/0 = \infty$ just as we once defined $\sqrt{-1} = i$? Would that lead to "bad things"? –  JesperE Nov 22 '12 at 6:18
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@JesperE Once we have defined $1/0$ as you've proposed, what will we do with it? Can we ask about adding it to itself? Multiplying or dividing it by itself? This is not to say there is no way of making sense of such an idea, but a new definition like that will bring with it more questions than it answers. –  Benjamin Dickman Nov 22 '12 at 9:19
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@JesperE yes, it does lead to problems. The problem with $\infty$ is that things like $0\infty$, $\infty/\infty$, $\infty - \infty$, and $0/0$ all can't be consistently defined. For example, if you want $\infty - \infty=0$ and $\infty + 1 = \infty$, then you get $\infty + 1 - \infty = (\infty + 1) - \infty= \infty - \infty= 0$ or $\infty + 1 - \infty = (\infty - \infty) + 1 = 1$. So you lose even the most basic rules, like associativity. –  asmeurer Nov 30 '12 at 3:20
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When one works in the set of real numbers, division by $0$ does not yield infinity. It is undefined. The reason is this: What would $\frac{1}{0}$ be? It would be the number which when multiplied by $0$ gives you $1$, but there is no such number.

Your book saying that $|\frac{2}{0}|=+\infty$ without further qualification is incorrect. We have $\lim_{x\to 0^+}\frac{2}{x}=+\infty$ and $\lim_{x\to 0^-}\frac{2}{x}=-\infty$ and $\lim_{x\to 0}|\frac{2}{x}|=+\infty$, that is all.

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One does not simply divide by zero. –  000 Nov 22 '12 at 1:47
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Let's take another tack. Let $\varepsilon$ be the "special" number that results when you divide anything by zero; that is, $x/0=\varepsilon$ for any $x$. Unfortunately, if we want to do this, we will have to dispose of the convenient notion that $0\times x=0$ for any $x$... –  J. M. Nov 22 '12 at 5:50
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Take a glass jar/glass/something, and a bunch of small objects (ping pong balls, bouncy balls, marbles, whatever is the best size for this).

Suppose your jar holds ten balls, and it's easy to see it holds exactly 10 of these. Demonstrate that if you're dividing by one, you can put one ball in, 10 times. You divided the jar into 10 sections. If you're dividing by two, show that you can put two balls in 5 times. If you're dividing by five, you can put five balls in 10 times. Associate "divided by" as equal to "how many in my hand each time I put something in the jar".

Now ask "What's 10 divided by zero? How many times can I put zero balls in at a time until it's full?" Take an empty hand, pantomime dropping it in the jar, and repeat. Keep going frantically/comically for bonus points. You can keep doing this forever and never fill the jar up. That's infinity.

(I realize this may not pass peer reviewed journals for accuracy, but for the target audience of 2nd graders, I think this is going to be close enough)

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Or else how long will it take to get to school walking at $0$ miles per hour? Infinitely long? The commonsense answer is that we will not get there. –  André Nicolas Nov 22 '12 at 4:39
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I agree with the responses you received, but you might also want to read the following: http://www.merga.net.au/documents/MERJ_12_2_Tsamir%26Sheffer.pdf

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Very nice link; I hadn't seen this one before. $+ \left(\lim_{n \to \infty} \sqrt[\Large n]{n}\right)$ –  amWhy May 18 '13 at 0:55
    
@amWhy: That was a very nice write-up. I would love for someone to ask kids in kindergarten, 1st, 2nd, 3rd grade all the way through high school graduation and then compile the results of what they say. Wow, would that make for hours of laughter and maybe even some awesome nuggets! :-) –  Amzoti May 18 '13 at 1:04
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let us consider that any number divided by zero is undefined.

You can let the kid know in this way:

Division is actually splitting things, for example consider you have 4 chocolates and if u have to distribute those 4 chocolates among 2 of your friends, you would divide it(4) by 2(i.e : 4/2) = 2.

Now consider this, you have 4 chocolates and if u don't want to distribute among any of your friends, (that is like distributing to 0 friends) division does not even come into picture in such cases and also division(4/0) makes no sense. Hence in such cases its told UNDEFINED.

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+1 simply defining it to be what it is. –  skullpatrol Nov 23 '12 at 0:41
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One way to explain that division of $x$ by $0$ is undefined is by contradiction. Suppose $x/0 = a$ and suppose $x$ is a non zero value. Then, by cross multiplication, we get $0\cdot a = x$. At this point ask the child what number times $0$ equals a non zero number. After a little thought the child will most likely say that any number times zero is $0$ so that $0\cdot a = x$, $x$ a non zero number is not possible. Next consider $x = 0$ so you have $0/0$. Let $0/0 = b$, where $b$ is a non zero number. Then you cross multiply to get $0\cdot b = 0$. Now ask the child to come up with a number that satisfies this equation. The child will most likely realize that any number will do and pick one, say $5$. $0\cdot 5 = 0$, true. Now say, what about $0\cdot 6$? The child will say that equals zero too. So, going back to $x/0$, there is no solution and in the case of $0/0$, in effect, any solution will do. Neither of these are allowed in mathematics. The above is not a proof of course but it might help a little. Note: the explanation doesn't really work for the case where $x/0 = 0$ or $0/0 = 0$. I imagine this observation would have to be modified a lot to be useful but perhaps it would be a good starting point for explaining that division by $0$ is undefined.

Also, a way I use to think of limit is to imagine what happens when you place smaller and smaller number under, say $1$. $1/(1/2)$, $1/(1/100)$, $1/(1/1000000)$ etc. I imagine that any child will know that you flip the denominator to simplify these equations so you get $1\cdot (2/1) = 2$; $1\cdot (100/1) = 100$; $1\cdot (1000000/1) = 1000000$ etc.

These two approaches combined might be used to explain one reason for limit. One thing that limit allows is to go as close as you would like to something that is not possible, ie. division by zero.

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Dear Patrick, I think that this approach presupposes a fair amount of mathematical maturity for a second grader; cross-multplication is not something they will know at this age (unless they are quite unusually mathematically advanced); indeed, they will not be all that comfortable with fractions unless the denominators are quite small. Regards, –  Matt E Nov 30 '12 at 4:02
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Show It Visually

  • Draw a graph of 1/x.
  • Pick a point on the x axis (2, for example)
  • show that you can approach 2 from the left side and the right side
  • in both cases, you can hit x=2 exactly, and the y value is .5
  • i.e. approaching from the left or from the right, you get close to the same y value.
  • Now do the same thing for x=0.
  • Approach from the right, and you can see that the y value gets larger and larger the closer you are to zero.
  • Approaching from the left, the y values gets smaller and smaller the closer you are to zero.
  • Unlike all the other numbers on the number line, approaching from the left doesn't get you to the same value as approaching from the right.
  • So, we generally say division by zero is not allowed, because there's not one clear answer as to what that value might be.

This allows someone to get the idea without having to understand a lot of notation.

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When she's in 3rd grade, you can teach her about the indeterminate forms 0/0 and inf/inf. :-)

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I'm pretty sure a 2nd grader won't know what a graph is. –  asmeurer Nov 30 '12 at 3:07
    
@asmeurer, most second graders don't know what division and multipication are, lol. I interpreted the question as being "how to explain to someone not sophisticated in math or math notation." Apologies to all second graders on math.se, of course! –  Mark Harrison Nov 30 '12 at 7:44
    
I assumed from the question that they at least knew what division was (and thus they probably know what multiplication is as well). –  asmeurer Nov 30 '12 at 15:31
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Ok, I am assuming the 2nd grader knows the basic of division here i.e. when you divide 4 by 2.. you actually find out how many times you can add 2 to get to 4 in this case... answer is 2. But when you divide 4 by 0... you can keep on adding 0 as many times as you like but you will never reach 4. Hence its becomes infinity.....hope this helps.

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Does the child understand what it means to divide by a number smaller than 1? If so, just explain it like the following $$\frac{1}{1/2} =2,$$ $$\frac{1}{1/3} =3,$$ $$\frac{1}{1/4} =4,$$ $$\vdots$$

So as you divide by numbers that get smaller and smaller, you get numbers that get bigger and bigger. In fact, if you divide by $1/n$, you just get $n$, so it's hopefully easy to see that this can be as big as you want. As the numbers in the denominator go to 0 (also something that hopefully won't be too hard to see), their reciprocal goes to $\infty$.

I've always found that the concept of $\infty$ is best understood as "as big as you want" (or "arbitrarily large" or "unbounded", to use more mathematical terms).

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Dear asmeurer, This is the natural way to explain it, but unless the second grader is rather advanced, they may not be very comfortable with fractions such as $1/n$ when $n$ is large, and so $1/ (1/n)$ will be even trickier for them to compute with. (My experience is that what we consider trivial arithmetic is still in the non-trivial state for kids of this age. But for some reason, they love both infinity and zero, and seem to begin talking about them at a very early age!) Regards, –  Matt E Nov 30 '12 at 3:59
    
I seem to remember being able to understand such things, but then again, I was much better at math than my peers in 2nd grade. –  asmeurer Nov 30 '12 at 5:05
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The notion of division of numbers is pretty well understood, at least intuitively, from early in our mathematical educations. To say that the number $a$ divides the number $b$ means there is a number $q$ such that $b=qa$. Of course, if $b \neq 0$ and $a=0$, then no such $q$ exists, and in this sense, division by zero is undefined. However, if both $a$ and $b$ are zero, we have a different problem, as $b=qa$ is true for any number $q$. I suppose we could say that in this case, division by $0$ is over-defined. Since we usually want a quotient to be unique, if it exists, we usually say simply that division by zero is undefined. However, the $0/0$ case comes back to haunt us as an indeterminate form in limits, and we see the vestiges of our over-defined case of division by $0$. It seems to befuddle students somewhat that a limit that has the form $0/0$ can have any real number value, provided that the limit does, indeed, exist.

This discussion can be pursued in any algebraic system in which we have a notion of division.

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An Even Simpler Explanation: Why We Don't Allow Division by Zero

This will make sense to somebody (e.g. the hypothetical second grader) who understands division (fractions) but nothing more sophisticated.

  • Consider a fraction 0/x, where zero is divided by any number. The answer is always zero.

  • Consider a fraction x/x, where a number is divided by itself. The answer is always one.

  • Now consider 0/0. Is t answer zero by the first rule, or one by the second rule?

  • Since this causes us to have two conflicting answers, we disallow division by zero.

  • When you're older and have a bit more math, you'll understand one of the other explanations given here.

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If the second-grader can understand a little algebra, then I think this would do the trick:

For any $x,y,a$

$x=a/y$,

$x*y=a$

right?

Now you can show that if given two nonzero numbers, you can find the missing number.

But what if $y=0$?

Then it would be:

$x=a/0$

$x*0=a$

But any number multiplied by zero is zero right? Yet $a$ is nonzero. So $x$ is not a number!

In fact, $x$ keeps running away (it keeps increasing). This way, zero can never "catch" $x$, and turn $a$ to zero.

Matematically, this makes little sense. But this is how I would explain this to a second-grader.

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