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

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?

Solution 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".

Solution 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.

Solution 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)

Solution 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

Solution 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.