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!

Such demonstrations are spoofs, not proofs. They are mathematical jokes that make a point. The point of the 1 = 2 spoofs is that in real arithmetic, zero does not have a multiplicative inverse. You are not supposed to believe that 1 = 2.

There are also a lot of spoofs that division by zero is impossible. You are not supposed to believe these. They exist only to demonstrate errors in mathematical thinking but some people take them seriously!

Mathematics is an abstract game. Mathematicians are free to make any assumptions and then work out the consequences.

I notice you've not answered the infinity calculation questions.

Those are the consequences of your assumptions

This is my last answer for today.

In transreal arithmetic, positive infinity times negative infinity is negative infinity: (1/0) x (-1/0) = (-1 x 1) / (0 x 0) = -1/0.

Adding positive and negative infinity together gives nullity: (1/0) + (-1/0) = (1 - 1) / (0 + 0) = 0/0.

There are minus one groups of positive infinity in negative infinity.

Bye for now!

-1 groups

I can just imagine that.

Is it a big group?

Is this what eating shrooms feels like?

See because I am a 'wordy' and not 'mathy' person, 12 makes perfect sense to me.

I read 0 = nothing = the division cannot take place therefore the sum total is the same as it originally was. Exactly the same as 12 plus 0 is still 12, 12- 0 still 12, etc.

Putting it into a real world scenario 'Bob has 12 apples. He decides to divide them equally between all the people who come to his party. However nobody shows up. How many apples does Bob have? 12/0 = Bob still has 12!

Given your DS is 10/11 that's the sort of level I assume the q would be at rather than introducing pure maths/degree level alegbra at that stage!

Surely in all subjects you learn the basic/right 99% of the time answer first, and only at a higher level get into the 'aha but technically....' element, whether that's maths, languages, history...etc.

And how do you divide into the -1?

I'm aware I might be making a complete tit of myself here but that's fine if I learn something. I realise we're beyond practical correlates now, but if I understand this correctly, we're not just dividing sweets by no children, we're dividing by negative children, which is even less. We're hanging dresses in fewer than no wardrobes.

We could put an infinite number of dresses into an infinite number of no wardrobes? Dresses forever into nothing, forever?

I've followed the discussion so far (thank you, everyone) but I've lost it now. Can't follow this reasoning at all.

Putting it into a real world scenario 'Bob has 12 apples. He decides to divide them equally between all the people who come to his party. However nobody shows up. How many apples does Bob have? 12/0 = Bob still has 12!

That's the answer to the question "how many apples does Bob have?" It's not the answer to "how many apples does each guest have in the dividing?"

As there are no guests, there is no answer. Indeterminate. You can't can't say they'd eat have 2 if 6 turned up, because that's not the question; that puts guests into the equation, even if Bob's only working in theory. The point is, with no guests, there is no answer to how many apples each guest gets. In terms of the equation, they don't exist.

Not dividing, or dividing by 1, is not the same as dividing by 0.

Technically, he has already divided the 12 apples into 1 group.

So he has 1 group of 12.

If you have 12 apples in a group of 1 and then ask a child to divide them into 0 groups. I think they might look at you with bewilderment.

As before...0 is difficult. I think I've misunderstood it in the -1 groups, positive infinity into negative infinity thing, so I'm hoping someone can explain it.

Hang on, I'm an idiot. You can divide and multiply with negatives, of course you can.

I'm clearly missing something on James' reasoning, but I'm pretty sure he isn't actually dividing by 0.

I think what James is doing is like when they thought it was annoying not having an answer to "what is the square root of -1" so called it i and then saw where that led.

It's being annoyed that there isn't an answer to 0/0 so giving it a made-up answer and then messing around with what results from that.

James has clearly decided for his own reasons that "trans" maths would be popular on MN ;)

All of mathematics rests on some base assumptions which everyone agrees on (like x = x). These are called axioms. Originally maths started from people doing "sweets/pizza maths" based on the real world, but mathematicians are playful & nerdy and sooner or later they started asking themselves: "hey, what if we made a different set of base assumptions - can we work out what would be the consequences? What would that new system look like?"

James is describing a system where they thought, hey, what if we invented a new thing called "nullity" and defined it as 0/0. How would maths rules change if that thing existed? ginger.readthedocs.io/en/latest/help/transmaths.html

You might remember imaginary numbers, which came from saying let's define something called "i" which is the square root of -1. Or you could think of it as "what if the number line... actually also went vertically?!" The "rules" that emerge as a consequence of that are actually really useful in some fields like electrical engineering.

New question for the group since we are all agreed on 12/0.

What about 12/-0...? (negative zero)

Hang on a moment.

Are you saying zero is positive?

I guess under trans maths, it can be positive or negative. Or neither

Negative zero is a computer science concept.

Out of interest, I just did a Google to see what is out there re requirements in curriculum at primary to explain that diving by zero is impossible. Didn't find anything, but did find this resource which is supposed to be SATs practice wrongly explaining that if you see a question asking you to divide by zero, it always equals zero. Caution to all using ready made resources!

https://myminimaths.co.uk/arithmetic-16-practice-question-4/

I will now try to do my bit and try to contact the website to point this out.

Surely 12 (or any number) divided by nothing leaves the original 12 (or whatever number)!!

No. That's simply not dividing, or dividing by one.

What would 'dividing 12 by 1" look like?

You would have 1 group of 12. Which is what you already have when you have "12".

So when you have 12 objects, they are already divided into 1 group (i.e. divided by 1)

en.m.wikipedia.org/wiki/James_A._D._W._Anderson

To be polite.....not a mainstream idea!

Now that it's on mumsnet, it's real! (or transreal, which is even more valid I think)

I'm beginning to think my flippant comment earlier wasn't so wide of the mark.

No, you are not left with the original number. If you have 3 apples that is still 1 group of 3 apples. For there to be zero groups of 3 apples, the apples would have to cease to exist. As long as there are 3 apples in existence they are a (or 1) group of apples.