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How to explain why division by zero is undefined.

If we try to divide 2 by 0, why isn't that just infinity? Well, infinity isn't really a number. It's just a concept and it represents a place that you can't ever really get to. Math deals with numbers, and it just can't handle this.

Here's a way to think about it. If I divide 20 by 5, I get 4 as the answer. Turning that around, it says that I would need 4 groups of size 5 to make 20. So what would 20 divided by zero mean? How many groups of size zero would I need to make 20? There is no number of groups that would ever add up to 20. Even if you could add up an infinite number of zeros, the total would still be zero. You just can't get to 20! The answer is therefore undefined.

This is by no means mathematically rigorous, but it gives one an intuitive picture of why you just can't do that.

Comments

I like that explanation! I have a middle-school math student who is often curious about things and comes up with questions about things like that. I'll have to remember that example in case he ever asks me about that sometime.

Thanks for the comment!  I've seen this question addressed once in a while, and the answer was always couched in deep mathematical terms that didn't give you anything that you could intuitively see.  I came up with this when tutoring a student about fractions.  It was a simple twist to make one particular explanation about fractions apply to division by zero.  (I saved it for more advanced students.  She didn't need to be worried about this while trying to understand fractions.)

Hi Gene. Here is another point of view: 
Let's assume, that 20 divide by "0" is some, different from zero, number "n"
20 / 0 = n
then
n * 0 = 20
but we know that n * 0 = 0

Hi Nataliya,

I've seen a lot of your work in Answers.  You're one of a select few who I really admire for your math skills.  For those of us for whom a mathematical proof clearly demonstrates a point, your method is great, but I've long looked for something that would make sense to a youngster who hasn't yet gotten to the point where a valid proof really "feels" like a proof.  I was well into college math before I realized that seeing a derivation was really a proof to me at the "gut feel" level.  I think what I came up with is something that accomplishes what I wanted. Thanks for your comment!