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My little brother (6 years old) asked me a question ("What is $0*0$?") and gave an answer to his own question which I found ridiculous so I refuted it but he still thinks he is right.

He says that the first $0$ in $0*0$ means "no". He argues that $0$ multiplied by $6$ is $0$ because there is no 6. Then he argued that $0$ multiplied by $anything$ is $0$ because there is no $anything$. Since there is "no anything", it's nothing and thus 0. I told him it doesn't help his case of $0*0 \ne 0$ 0 that he just said $0*(anything) = 0$. Then he told me to be quiet and said that the 2nd zero is more important. Zero literally means "nothing" to my brother so he said $0*0$ means there is no "nothing" which means that it is "something" so $0*0 = something$. Since $0$ is "nothing" and he concluded that $0*0 = something$, he says my argument of $0*0 = 0$ is flawed because L.H.S. = $something$ and R.H.S. = $nothing$ which aren't equal to each other. It's good that he's questioning why things work they are but I feel like it's just wordplay and I told him that he shouldn't confuse Math and English and that he is playing with words. He is pretty stubborn and still thinks he's right. For multiplication, visual representations of what is going on is helpful to young children but $0*0$ isn't exactly the best thing to visualize. What is the best way to convince him that $0*0 = 0$, at least in the arithmetic we're dealing with in this context?

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    $\begingroup$ Holy crap. What are you gonna do when he learns about powers and starts asking you about $0^0$ ? $\endgroup$
    – shalop
    Commented Mar 22, 2015 at 0:56
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    $\begingroup$ It might help, but I'm not vouching for this, to rid him of the idea that $0=\text{negation}$. $\endgroup$
    – Git Gud
    Commented Mar 22, 2015 at 0:58
  • $\begingroup$ Better suited for Mathematics Educators S.E.. $\endgroup$
    – Git Gud
    Commented Mar 22, 2015 at 1:00
  • $\begingroup$ @Shalop: I believe that would be easier to explain once he decides that $0*0 = 0$ :) Hopefully, by then he understands that $0/0$ is undefined which is what $0^0$ hinges on. $\endgroup$
    – calculusbyparts
    Commented Mar 22, 2015 at 1:02
  • $\begingroup$ What is 1*0*0? Or 0*1*0? Are they same or different to him? $\endgroup$
    – Valtteri
    Commented Mar 22, 2015 at 1:19

4 Answers 4

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Try framing multiplication $a \times b$ as the area of a rectangle with sides of length $a$ and $b$.

So a $2 \times 3$ box has area 6:

enter image description here

A $2 \times 0$ box has area 0, as does a $0 \times 3$ box.

enter image description here enter image description here

And a $0 \times 0$ box has area…?

enter image description here

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In my opinion, you should change his terminology away from "(any/no)-thing" and into any number.

A more correct sentence would be "any number times zero equals zero".

Then remind him that zero is a number.

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  • $\begingroup$ That's a good suggestion but if I said that, I fear he would ask why 0 is a number in the first place because to him, 0 represents nothing. $\endgroup$
    – calculusbyparts
    Commented Mar 22, 2015 at 1:06
  • $\begingroup$ Then I suppose that might be a good first step, to get him to realize zero is a number. Perhaps for a 6y/o, you could try explaining that any (exact) response to the question "how many apples do you have" is technically a number including "none(=zero)", "one", "two", etc... but that there are other types of numbers out there too. (hopefully that last bit won't throw him off too much but won't prejudice him against fractions, negative numbers, and irrational numbers later on) (I say exact to avoid "lots", "more than 5", etc... as numbers) $\endgroup$
    – JMoravitz
    Commented Mar 22, 2015 at 1:11
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Try this. Let $a$ be any number. Then $a\cdot 0 = a(0 + 0) = a\cdot 0 + a\cdot 0$. Subtract to get $a\cdot 0 = 0$ for any number $a$. Hence it works for $a = 0$.

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    $\begingroup$ Implying that a 6yearold will follow algebra. $\endgroup$
    – JMoravitz
    Commented Mar 22, 2015 at 1:00
  • $\begingroup$ Thank you for the answer. However, that was his part of first "argument" that $0*anything = 0$ but that $anything$ couldn't be $0$ since $0$ meant $nothing$, not $anything$. $\endgroup$
    – calculusbyparts
    Commented Mar 22, 2015 at 1:01
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Let's say I have a school. The school has three classrooms, with two kids in each room. (Clearly an underfunded school.) Hopefully you can convince him — perhaps drawing it might help? — that there must be $3\times2=6$ kids in the school.

Now, let's say that there is another school, but with no rooms and no kids in each room. (Horribly underfunded.) How many kids are in the school? $0$. So $0\times0=0$.

(A problem might arise if he would ask for a similar proof that $0\times2=0$, since that corresponds to "a school has has no rooms, and two kids in each room." How can there be two kids in each room if there are no rooms? I'm sure you'll manage.)

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  • $\begingroup$ No rooms and no kids in each room? That does really make sense. As for your last example, can't we just say that the school has two rooms, with no kids in any of the rooms? $\endgroup$
    – user207710
    Commented Mar 22, 2015 at 1:36
  • $\begingroup$ @Ahmed True, but perhaps he'll still ask what the sentence would mean, anyway. (P.S. When you said "That does really make sense," was that a complement or a typo?) $\endgroup$
    – Akiva Weinberger
    Commented Mar 22, 2015 at 1:38

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