School Maths: 12 divided by zero = 12?!

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My DS came home yesterday quite sad and frustrated because he other classmates had lost marks in a school maths (his best subject) test for not getting the 'correct' answer that '12 divided by zero = 12'.

His reaction, upon coming home, was to look up the expected result of dividing by zero on several reputable maths sites and, as expected, none of them gave the answer that 'X divided by zero = X'. They all backed up his (and my) reckoning that the only possible correct answers could be 'undefined/impossible', 'infinity' or (possibly, at a real outside semantic push) 'zero'.

Thankfully, the teacher raised it the following day (I don't know if she had looked into it herself - it was a centrally-set test - after seeing the pattern of usually-able children unexpectedly all getting it 'wrong') and re-instated all the lost marks; but I'm still baffled as to how anybody could arrive at that answer in the first place, and it's bugging me!

Suppose the sum had been the simpler '12/2=6', the reckoning process would mean that you could have 12 apples and remove 2 apples 6 times, thus ending up with zero apples afterwards (as a valid 'checksum'); equally with '12/12=1', you could remove 12 apples once and again end up with the sum-validating 0 apples; but if 12/0=12 were true, you could thus remove 0 apples 12 times and be left with 0 apples - but of course, you wouldn't have 0 apples left after that: you would have 12 apples left; indeed, just as you would have if you removed 0 apples a thousand billion times!

Was this just a brain-fart by a tired maths test-setter - and one that wasn't immediately obviously wrong to a maths teacher yearning for half-term, who initially insisted that it was right when it was queried - or is there some kind of maths/philosophy train of reasoning that any boffins out there know of by which you could legitimately justify/argue that 'X/0=X' can indeed be correct, the same as 'X-0=X' naturally would be?! It doesn't really matter, obviously, but it's still irritating me a bit!

How few, I mesnt

I’m in my forties and was definitely taught that dividing by zero = infinity/undefined. But my primary maths education was really good and we were doing algebra by year 6 (C of E primary school with old fashioned teachers!)

I’ve just asked DD (10) who said infinity, but also said she’s recently been arguing with “the mathsy boys” at school who refused to believe her that this was true 😂

It is actually quite hard to explain why you cant divide by zero. I hasten to add...I am not an expert mathematician, and have not read the whole thread. If people know better, then I am perfectly happy to be corrected. Having said that, I think it's a classic case of proof by contradiction.

If you allow division by zero, then you can set out a series of seemingly logical algebraic sets that can prove (for example) that 1 = 2. There are many YouTube videos of this sort of thing....many claiming to prove that the whole foundation of maths is wrong.

As it is very clear that 1 does not equal 2, then there must be some step in the logic that is false. The false step is that there is no solution to the equation y = x/0 because division be zero is impossible.

Anybody teaching maths at any level should know this.

I'm in my forties and was taught that anything divided by zero is zero. I was never taught why though so thank you to those who have explained.

My DC who is a maths genius - I mean a maths genius - says it is not infinity but indefinite.

Algebraic steps...

But in the equation, you're not really there holding the sweets as a group to be included in the division. You're just a vessel for the equation, and the equation is to divide them by nobody.

That's brilliant Rene I love your explanations!

I wasn't bad at Maths in school (did higher level for my Leaving Cert (A levels in Ireland) but that was because I just worked at it, my mum was a Maths teacher but I never 'got' it either.

It's a great idea to put into practical illustrations as you have done!

@CrescentMoons ,

I think your son is splitting hairs, regardless of his ‘genius’. Any non zero number divided by zero is infinite. Of course, as a poster said above, infinity is a nebulous concept, but I think more useful as undefined.

No. If 12/1 = 12 then 12/0 can’t also be 12.

I'm in my 50s and certainly was not taught the answer was 12! If has always been infinity.

I don't think it's unreasonable for the teacher not to have read the whole paper. Even if manually marking, I would only look at the answers and mark it. Once you have everyone's marked you can do a gap analysis to see what type of question the children are getting incorrect and then you can do more work on that. That would be why he/she noticed that the 'bright' group of children all had that question wrong.

I'm coming to you lot when DS starts doing maths concepts at school that are beyond me. Which is basically everything after the times tables.

Infinity isn't a number.

If you think about representing division as fractions

1/2 1/3 1/4 1/5 1/6 .....

The number on the bottom is what you are dividing by, the fraction is the answer to the division. 1 divided by 2 is 1/2

As the number on the bottom gets bigger, the fraction gets smaller (1/6 is smaller than 1/2, 1/10000 is smaller than 1/10, 1 divided by a million is smaller than 1 divided by 2)

So as the number on the bottom gets bigger and bigger and bigger, the fraction gets smaller and smaller and smaller.

We say 'as the number on the bottom tends to infinity, the fraction tends to zero.'

But that doesn't mean that 1/infinity is zero. It's undefined.

A = B
A^2 = AB
A2 - B2 = AB - B^2
(A - B)(A+B) = B(A - B)

Divide by (A - B) .... important step!

A + B = B

A = 2B start point is A = B, so replace B with A

A = 2A ... and now divide by A

1 = 2.

The problem with the "proof" is that if A = B, then A - B must be zero.

As 1 does not = 2, dividing by zero does not produce a result. It is not a mathematical operation!

BadNomad "Don't focus on the sweets. Focus on the other number. The question is about the other number.

There are 12 sweets on a table. 4 children come along and divide the sweets between themselves. How many sweets does each child have? 12/4=3

So, there are 12 sweets on a table. No children come along. How many sweets does each child have? None. Because there are no children.

The question is never "how many sweets are there?"
Thank you, this is the clearest explanation of them all. Somehow Passed 'O' level maths but have never been any good at it...my brain is just not practical enough!
My primary school maths teacher was absolutely terrifying (I swear she was a real life Mrs Trenchball and you simply never asked how or why, you learned by rote and gawd help you if you couldn't....many an hour spent on the naughty seat in the corner with the dunces hat on because I couldn't learn by rote!) unfortunately it meant maths has always brought me out in a sweat.

Actually, what this thread has shown is that maths is really hard to teach. I agree the teacher handled it well.

Third line should be A2 - B2 = AB - B^2

This thread explains (to me) exactly why I was never any good at maths!! Mathematical thinking doesn't seem to follow real life.
If I have 12 of something and I divide it by 0, it means I haven't shared those objects with anyone so I still have 12 of them. Simples.

Infinity, NaN

I was coming here to say the same. We were taught 0= nothing, therefore if you divide by nothing, you do nothing, so the original number is unchanged.

This is beginning to feel like another teacher bashing thread.

I think the teacher sounds great. She didn’t dismiss the problem and do nothing. She thought about it overnight, came back to it and tried to work it out with the able pupils. That in itself shows children what learning is all about. Mistakes happen and that’s how we learn. Even Year 6 teachers.

dividing by zero does not produce a result. It is not a mathematical operation!

This explains it best to me as a non-mathematician – thanks @Sceptic1234. I discovered this relatively recently (this has never come up!) when playing with DS's new calculator – when x/0 = error. I would explain it to myself it by saying that if you divide 12 apples by no people then you just don't do any dividing. (And they rot, I expect, when no-one comes to claim any apples.)

My first instinct is do the apples exist or not? This is a maths class though, not philosophy! It's the apples that weren't needed in the question. We all know 12 x 0 is 0 but asking kids to divide objects, I.e apples is a stupid trick question. Good on the kids for noticing this and questioning it. Clearly no one else did!

Thank god, I thought I was going mad. Mid 50s too

Are people seriously claiming they're being taught basic maths incorrectly?

You don't have to be a qualified maths teacher, or even particularly competent at maths to know that 12/0 does not equal 12.