How to explain that division by $0$ yields infinity to a 2nd grader

Question

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?

Answer 1

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".

Answer 2

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.

Answer 3

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)

Answer 4

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

Answer 5

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.

Answer 6

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.

Answer 7

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.

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

Answer 8

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.

Answer 9

This might be useful to you for inspiration http://www.khanacademy.org/math/arithmetic/number-properties/v/why-dividing-by-zero-is-undefined

Answer 10

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).

Answer 11

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.

Answer 12

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.

Answer 13

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.