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!

Do you think so?

I think people have been accepting that primary teachers can't be expert in everything, and they came back the next day and sorted it out which is good too.

The Romans for all their road building didn't even have zero (that's right isn't it?), because the concept of a sign representing nothing was so mind blowing. It's not surprising people struggle with dividing by it.

Completely wrong....as is the sweet analogy. The equation number of sweets divide by number of children tells you how many sweets each child has. In mathematical terms the dimensions of the solution are sweets / children

If you share 12 sweets between 0 children, then you are saying how many sweets does each child have?

As there aren't any children, then there is simply no answer to this question! The fact that you still have 12 sweets is irelavent.

My head hurts....

PS on my computer calculator the answer is 'not a number'.

That is true....although it's more accurate to say they had no sense of number base. We count in tens. Arab / indian mathematics was far more advanced and they could do the sort of maths that we could. The word "Algebra" actually comes from Arabic (sounds like it too) and Europeans discovered their system of maths and adopted it at some point.

That just means undefined, infinity, NaN.

Wow, that kicked off! Fascinating to see everybody's thought processes and conclusions. Glad it wasn't just me finding it both baffling and intriguing!

If this is the standard of teaching maths, that's worrying, to put mildly!
This "test" is so fundamentally wrong that if that happened to mu child I would take the trouble to make enough fuss to have whoever devised it sacked without thinking twice.

I genuinely don't see the teacher as evil and/or stupidity incarnate. I really respect how she handled it, then owned up - and that she didn't just say "Teacher is always right" and slap the kids down.

I’d bet good money that the test setter didn’t actually think that 12/0=12. The “0” will have been the error - it should have been a “1”.

These were supplementary questions for the top-set kids who are particularly able at maths - aged 10 & 11 - so I don't think they would have meant to ask 'what is 12/1?'.

I'm also in two minds as to whether it was an unfair question, too, as long as it's accepted that the answer is infinity/undefined and that the kids learn a quirk of maths as the penny drops (if they haven't come across it before) - any more than giving children multiplication questions with 'X x 0' to underline that multiplying anything by zero equals zero.

This is the only reply that made sense to me. Thank you for the explanation!! I am also of the same age demographic as those above and was definitely taught that the original number remained if multiplied or divided by zero. My school wasn’t renowned for being amazing, though. Maths was always tricky to me because my brain definitely goes for words rather than numbers. The other maths-related explanations don't help me because my logic isn’t mathematical. But this explanation does!

Thank you, I'm glad they're helpful.

As we've seen, the problem with practical illustrations is that they are easy to misunderstand. But even learning about the misunderstanding helps you to get the theory.

It also helps to think of 0 not as a number, but as a concept (which it is, really). Nothing, nobody. Not there. Infinity is also a concept rather than a number.

Teacher-made-mistake horror! Lots of people make mistakes at work. The important way is the way it is handaled.

They are absolutely right!

It is not easy....I an not an expert mathematician, but have worked in technical science all my life. You DC is almost certainly well ahead of me.

What's your definition of "particularly competent at maths"? Because there are a few of us on this thread who managed to achieve GCSE or O-Level maths and yet didn't know that 12/0 doesn't equal 12.

My phone says NOPE to the whole thing! 🤣

A lot of primary teachers aren't good at maths. It's fine in the younger stages, but gets harder to work around towards Y5 and Y6 when the kids are doing imaginary numbers, more complex multiples and stuff like number bases.

I think it’s WHY the fact that you still have 12 sweets is irrelevant is where people get confused, because it doesn’t translate to real life.

I think it can be incredibly difficult for people who ‘get’ maths to understand what the issue is for people who don’t. It seems so obvious and evident to people who understand it that it is hard for them to see that basic concepts have passed others by.

I know I never understood a single thing in maths ever, never learned times tables/long division/fractions/algebra etc. None of it… but spent hours with people explaining x=y and me saying buy why does it?! And them struggling to explain it.

Surely in the case of zero in division, it represents 'nothing' as such?

(Unlike negative numbers, where zero is represented on a sliding scale where it can go before or beyond zero on a number line.)

But zero in a division question is logically the same as asking "What would happen if I divided something by nothing at all? The logical answer would be that nothing would happen to the original number, because the dividing number was not enough to change the original number in any way/didn't exist in a dividing function. The act of dividing has to be more than simply applying a word (divided) to make a change.

wow this a long thread :) 12 / 0 is indeterminate or not defines. there are multiple ways to explain it. simply put:

Assume 12 / 0 = 12; then 12 = 0 x 12 which makes no sense

Also, imagine you are trying to split 12 into nothing parts. Is that possible? How would it work? you cannot actually do it and the answer is mathematically 'indeterminate'.

The other way to think of it is: as you try and divide 12 into smaller and smaller parts so imagine dividing 12 by 0.0000000000000000000001 and so on, the answer 'tends to' or 'goes towards' infinity.

It is hard to grasp that 0 in maths is more of a reference point and it doesn't really exist as a quantity. Same for infinity. So how can you divide by something which is not an actual 'object'.

The thing is, you could probably get an A* at GCSE and not know that 12/0 is undefined, because it doesn't come up.

However mathematically minded people, if they gave it a thought, would know from 'instinct' the answer can't be zero because as the dividing number gets smaller the answer gets larger and larger, so it just wouldn't 'feel right'.

A lot of people have no 'feel' for numbers (whether down to not being taught in a way that gives it, or aptitude, or interest), so answers don't go through the 'feels right' test.

A lot of older people were taught methods but not underlying concepts, so do things by rote rather than understanding, which messes things up for esoteric questions.

It's wrong of course (and it also implies that 12/12 = 0) but I can see someone saying "well you're not dividing it by anything are you...", which is essentially what you're saying.

Yes, I'm the same. I learned enough to get a B at GCSE but it was just learning processes/ tricks, eg. to solve this equation do this, then that, then this, then that and that gives the answer. I have never actually understood anything!

Well in that case you had shocking excuses for maths teachers, because this is not the sort of thing which mathematicians have changed the rules about.

The logic shown above shows what's going on: you need to move away from apples to an example that allows for dividing by numbers less than one.
12/1 = 12
12/0.1=120
12/0.01=1,200
The closer you get to zero, the larger the answer becomes, and it tends towards infinity.

It's more infinity than undefined, as it's an infinite amount of divisions.

it was a centrally-set test
Error on the mark sheet. Not picked up until the marks were totted up and looked at on a whole class level. It happens.

Surely if you have 12 sweets , an infinite number of children could come and take 0 away.

This is what I was trying to say, but expressed better.

Interestingly I just asked my A level top set A grade wife who flew through school without issue, and she said 12.

Then I asked my autistic home educated SEN child who needs to be taught very slowly and understand the theory/history/context of every single thing before he will take it in (as in, he couldn’t begin to understand maths until he understood why numbers are the shape they are and have the names they have) and he said 0. And understood why ‘undefined’ was a better answer.