School Maths: 12 divided by zero = 12?!

[WeBuiltThisBuffetOnSausageRoll]

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!

"Anything divided... by 0 will be a 0"... yet "12 apples will always be there".

Where's ya 12 apples gone then? Made them vanish?

How come? It's a pure logic. You can't have zero groups if you have some items grouped together. Then it's one group.
The 0 - or yeah, error, undefined, to be more precise is simply pure logic. And I have never been into STEM subjects. I get that some people can find it difficult. Just please stop calling it "practical' or "real life" shit. It's not.

0 is a difficult concept. As PPs have said, some ancient cultures literally didn't recognise it and many people still see it as a number rather than a concept (once you see it as a concept rather than a number, like infinity, stuff like this makes more sense). And as other PPs have said, a lot of people were taught badly.

There's no point being impatient or dismissive about it. As we see here, most people are keen to get it and indeed do get it once it's explained in the right way.

Sorry, had to keep my eyes from rolling back in my head, hence the late reply.
So what is your answer, @Sunshineparasol ?

You go first! Do explain your contradiction.

Yes.
My parents had very basic maths, their schooling being interrupted by war.
I went to a very mediocre school where most learning was from worksheets or via the medium of shouting. Occasionally teachers threw board rubbers at pupils.

"Anything divided... by 0 will be a 0"... yet "12 apples will always be there".
Where's ya 12 apples gone then? Grin Made them vanish?

They're still there in the question - but it's the answer that we're interested in!

weirdly on this, I think it was the concept of the apples. I was thinking but it is 12

But then I went onto but you cant divide by zero and then I figured out it was 5am and I hadnt slept so to come back to it today.

I have just asked DS and he went its 0 why are you asking me math questions

Yes, and in the concept of 0, there is no group. Not even you. You cant have 12 items in no group, in no place.

You're still thinking of it as you holding the items (either 12 and no equation, or 12/1). But it's division by 0. You're not there either. Or you are, but the equation isn't about you holding the items, it's about you dividing them among no people, into no groups, nowhere. It's not "how many items do you have?". It's "divide them all among nobody".

Which you can't do. Or you could stand there forever not doing it. Or people could come by forever not taking your items.

But the answer isn't 12. It's not "how many things are there?".

I wonder if the people claiming this also think that 12 x 0 = 12. Because you're 'not doing anything to it' then either.

I was trying to explain it on the level it was explained to us when we were kids and it was difficult to explain it on a very abstract level. I get your explanation completely.

So do you also think that 12 multiplied by 0 is 12?!

The mind boggles.

Here's another way of looking at it.

You are my maid, and I am Lady Muck. I give you 12 dresses and tell you to go and hang them up in the wardrobes.

But when you go upstairs, there are no wardrobes. You can't hang them up in any wardrobes.

You come back and tell me you couldn't hang my 12 dresses up but you have them with you. I say, "I didn't tell you to hold them, I told you to hang them up in the wardrobes. Don't come back until you've done that."

There are NO WARDROBES. You CANNOT hang them up in the wardrobes as I insist you do.

If you try, it'll take an infinite amount of time. The task is not to hold the dresses, it's to hang them in wardrobes when there are no wardrobes.

It is not 12. It is impossible, or infinity, or indeterminate since there are no wardrobes.

If you're willing to work with remainders then '0 remainder 12' could be a valid answer, which would correspond to the "I still have 12 apples left" logic.

That's what I meant earlier saying that the mum with the Jaffa cakes is undefined. She's not there in any meaningful sense.

The contradiction, as you call it , arises from the fact that you're thinking of what you are left with. But that was never the question. If you think like a primary child in terms of how many you put into each (non existent) group then you could come up with 0. If you take it a bit higher and think how many groups/children/baskets you could have, then the answers is infinity, because no one is taking any apples away.

In this case no - you're seeking 12 lots of 0. So you'd still have 0

I genuinely struggle with maths and threads like this feel as though they are attacking people's intelligence - which I suppose they are. If you have never had a good teacher with the patience to explain things, and never had to apply these principles then it isn't as easy as others find it.

ReneBumsWombats - you have an excellent way of making abstract concepts easily accessible in real-world scenarios!

Do you genuinely think you've spotted a mistake here that all mathematicians have missed?

I'd agree...'Division' is basically 'sharing' ..So if one has Twelve apples, and there is no one to divide them with...one still has Twelve apples.
But I'm no mathematician.

Seems a bit of a dotty question: ''Divide 12 by zero''?

ReneBumsWombats - thankyou, this is a far better explanation which I can understand.

@GimmeBiscuits

but do you understand that division and multiplication are the same thing?

Reading through this had made me think of another way of putting it (apologies of someone else has said this)
If you have zero of something then you have nothing
Therefore 0=nothing
You cannot divide by 0 as there is "nothing" to divide with/by

I remember being taught at primary that anything divided by or multiplied by 0 = 0.

The fact you were taught something at school doesn't mean it's correct. I was taught lots of things at school that were incorrect, too (I'm late fities). So were my children (early twenties).

Any number multiplied by 0 gives zero.

Numbers can't be divided by 0. As others have explained previously, dividing a number by 0.1 is the same as multiplying by 10; dividing by 0.01 is the same as multiplying by 100; dividing by 0.001 is the same as multiplying by 1,000; etc. So, as the denominator (the number being divided by) approaches 0, the quotient (the number given by dividing the numerator by the denominator) gets larger. There isn't a limit to this, because there's always a smaller number you could divide by. So the quotient will approach infinity but never reach it. Dividing by 0 is impossible.

More simply, as 12 ÷ 1 = 12, 12 ÷ 0.1 = 120, 12 ÷ 0.01 = 1,200, etc., it would be nonsensical to suggest that 12 ÷ 0 would be 12 (or 0).

We've had this discussion at home before:

DS is 10 and likes maths - he says infinity.
DP is grown up with a physics degree - he says infinity
DD is 17 doing maths A'Level - she says infinity

I scraped GCSE maths - I say 12

Take your pick!!