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!

Yeah, that's sort of how I think of it. If you cut 12 whole pizzas in to halves, you will make 24 portions of pizza. (12 divided by 1/2 = 24)

Or if you quarter your 12 whole pizzas, you will make 48 portions. (12 divided by 1/4 = 48)

Cut in them into eighths will make 96 portions. (12 divided by 1/8 = 96)

Etc.

And when you're sick of Lady Muck insisting you spend infinity hanging designer dresses up in no wardrobes, divide them all by half and hand in your notice!

Does it?

This is Matt Parker - a great online maths communicator

Half goes into anything twice, to put it more simply. If you take half away from anything, you'll get to 0 after doing it twice.

Well, not if you halve the remaining half after doing it once, but you know what I mean.

For dividing by fractions I prefer to think of division as shorthand for repeated subtraction, rather than sharing into groups.

So 12/6 is “I have 12 apples. How many times can you take 6 apples from me?”. You can do that two times.

12 divided by a half: “I have 12 apples. How many times can you take half an apple from me?” You can do that 24 times.

This can work for dividing my zero too. I have 12 apples, how many times can you take zero apples from me? You can do that over and over forever - an infinite number of times.

If you have a number and take away half of the original number twice, then you get 0

But if you have a number and take away 0.5. you don't get 0 after 2 goes. Unless it's 1

And as you said, if you take away half of the number, then half the new number etc...well it just goes on. But doesn't hit 0

‘Dividing by zero’ isn’t the same as ‘don’t do any dividing’. ‘Don’t do any dividing’ would just be 12 = 12. You’ve done nothing to the 12. Dividing by zero means ‘take your 12 apples, split them equally between zero groups, and tell me how many are in each group’. You can’t do it, it’s impossible. The answer isn’t 12, there are not 12 apples in each of the non-existant groups.

Of course in the physical world, having been unable to divide your apples into zero groups, you still have your 12 apples, but the sum doesn’t care about that. The sum wasn’t asking ‘how many apples do you have left, after attempting, and not being able, to divide by zero?’ It ONLY wanted to know how many apples were in each of the zero groups.

Or put another way the sum 12 divided by 4 isn’t ‘I have 12 apples, I split them into 4 equal groups, what am I left with?’ (Which would be zero, and seems to be how some people are reading the 12 divided by 0 question). It’s ‘I have 12 apples, I split them into 4 equal groups, how many are in each group?’

Speaking of halves and doubles, I love this maths question.

There is a lake. For some bizarre reason, it doubles in size every day until it's full. It is full after 48 days. On what day was it half full?

Originally the 12 apples are in a group of 1. Or 1 group of 12 apples. You can make them into 12 groups of 1

Now make them into a group of 0. Or into groups of 0

I can imagine asking a class of 6 year olds to do that and see what they decide. Preferably in their own groups.

There is a lake. For some bizarre reason, it doubles in size every day until it's full. It is full after 48 days. On what day was it half full?

This is a very old one.

That reminds me of..

When I was 6-years-old, my sister was half my age. She is now 15-years-old. How old am I?

In transreal arithmetic, zero divided by zero is defined to be the number 0/0.

In real arithmetic, that pupils learn in school, if is a definition that a number times zero is zero.

If you have nothing, and you have 12 groups of nothing, or nothing add nothing repeatedly, you still have nothing.

I can do the bat and ball as well

It has been possible to divide by zero since 1957 when Patrick Suppes, a logician who formalised quantum physics, was trying to find a logical way to block division by zero. He ran into lots of trouble and then discovered that division by zero is possible using the relationship x/0 = 0. This developed into the Suppes-Ono algebra. This approach to division by zero is logically sound and algebraically sound but it cannot support calculus so it is of very limited use. By contrast, transreal arithmetic is logically sound, algebraically sound, and does support calculus. It has practical applications, for example it makes all computer calculations more reliable and makes some of them more accurate. Despite these advantages, schools do not teach how to divide by zero.

People from all over the world send me links to news about division by zero and other aspects of transmathematics. Yesterday someone sent me a link to this Mumsnet thread. I don’t usually comment on other people’s discussions but this time the OP seemed unusually aware of the issues surrounding division by zero and had a good reason to want to know how to do it - to help build the confidence of DS.

OP - if you show your son my 12/0 = ∞ answer and he understands it, he will be absolutely certain he was right to challenge the teacher. If he doesn’t understand it, at least that experience will teach him something.

He ran into lots of trouble and then discovered that division by zero is possible using the relationship x/0 = 0

I'm not a mathematician so I really hesitate to get involved in this one and I'm very ready to defer to the more advanced people.

But this sounds a lot to me like "it is possible to divide by 0 if you base it on a false claim about dividing by 0".

In the mathematics taught in school, division by zero is undefined. Learn what is needed to pass the A-Level exams and, if interested, also learn how to divide by zero. There is a lot more mathematics than is taught in schools.

Some schools have away days where pupils learn university level mathematics. It’s a rich world out there that does not fit within the confines of the school room!

Maybe he could come and visit you and stay at the infinite hotel?
If 12 / 0 = infinity, then infinity times 0 = 12?

How would you explain that to a child?

All options are on the table.

Real arithmetic, which is taught in schools, says zero divided by zero is undefined.

Suppes-Ono algebra says any number divided by zero is zero.

Transreal arithmetic says any positive number divided by zero is positive infinity, any negative number divided by zero is negative infinity, and zero divided by zero is nullity.

There are areas of mathematics that define 00 = 1 but in real arithmetic, which is taught in schools, 00 is undefined. In transreal arithmetic 0^0 = 0/0, where 0/0 is the transreal number nullity.

What do you get if you multiply positive infinity by negative infinity?

Or if you add positive and negative infinity together?

Can you put positive infinity into groups of negative infinity?

I appreciate that there's much more to maths than is taught in schools (which is as it should be, not everyone is an advanced maths whiz and there are other things that they need to learn too) and that my understanding is not very advanced, which is why I'm not enjoying wading into this one.

But I can't see how you could legitimately prove anything when starting on the premise that x/0 = 0. You might as well be basing it on the premise that 1 + 1 = 0. It's simply not true. There's no way of proving that x/0 = 0 because it doesn't.