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!

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!)

The same 46 - thing is we had calculators in 80s in school and dividing by zero gave an error.

I had C of E primary and DH basic state primary and we were both taught algebra at primary - yet other parents at last school insist this couldn't have happened.

I read some comment under a why California school are failing you tube video - and everything seems to be going wrong but someone saying they were a teacher was arguing algebra shouldn't be taught before 16 as brains were able to grasp concepts before that - both DS and I sat our GCSE early at 15 and did very well and algebra was one of many concepts being tested.

I think Maths is the main subject where success is entirely dependant upon the teacher.

We had real concerns about maths teaching at their first primary school - so looked online and found mathsfactor - which I am so grateful for as they have rock solid basics.

At secondary in top sets they've tended to have good teachers - though DD1 did have a non maths teacher for half the year in Y7 and then one who thought being top set they could push through really fast - but on-line sites and good subsequent teachers and she was fine.

You are confusing the total number of somethings with the number of somethings in each group.

I have 12 apples....I divide them into 3 groups, each of 4 apples.

I then forget how many apples I have. So I count the number of groups - 4 (call the number of groups g)

I then count the number of apples in each group - 3 (call this n)

I can now use these numbers to work out the total number of apples.

Total number = g x n
Total number = 3 x 4 = 12

Suppose I kept them as one group...same applies

Total number = 1 x 12.

Mathematically whe you divide you create a new number which tells you how many apples are in a group.

You simple cannot divide a number of apples into zero groups....the statement has no meaning.

You can keep them as one group, or you can divide them into any number of groups (although you may have to start chopping them up) but it is impossible to divide thrm amongst zero groups, the statement just has no meaning.

A bit more esoteric.....but if you could divide by zero then

0/0 = 1 (because any number divided by itself is zero)

.....and in answer to a comment above, I learned basic algebra during my last year at primary school (age 10 / 11) in the UK during the late 60s. That was perfectly standard. We used to do "problems" which were basically written descriptions of situations, that would have a question at the end.

Something like "Mrs Smith buys two pounds of apples and four pounds of bananas for a total of £1.90. Apples cost 45 p per pound, how much does a pound of bananas cost?" (Except it would be in pre decimal currency).

The trick was to rewrite into simple algebra....in this case:

2a + 4b = 1.90; what is the value of b if a = 0.45?

This was absolutely standard stuff for last year at primary school and we definitely taught to use simple algebra in this way. I hated problems.... school maths book just dozens and dozens of them.

If you have 12 apples and you don't separate them into individual piles, you have put them into 1 group. You still have 12 apples. 12 / 1 = 12

If you have 12 apples and you put them into zero groups, the question doesn't make sense. You have to put them into at least 1 group. So the answer is undefined. 12 / 0 = undefined

It's why your logic doesn't work ("zero is nothing, so if you do anything with zero, the original number remains unchanged").

You understand instinctively that zero groups of X = zero, not X.

0 groups of 12 apples = 0 apples.
12 groups of 0 apples = 0 apples.

It is not the case that 'doing something with 0' means the answer to the calculation is the same as the original number

So the reasoning doesn't work.

.....and in answer to a comment above, I learned basic algebra during my last year at primary school (age 10 / 11) in the UK during the late 60s. That was perfectly standard. We used to do "problems" which were basically written descriptions of situations, that would have a question at the end.

Se that what I thought totally normal to get taught algebra last year or so of primary but I was surrounded by parents who insisted it couldn't have happened and algebra was hard .

Interestingly some of DC friends come over and enjoy playing dragon box - which is basically algebra gamified- and because they didn't realise they enjoyed it and later on when it's obvious they were surprised they could do it.

Physicality is of no consequence in maths as a taught principle so the fact they are talking about apples etc is actually irrelevant

I was always taught with division that it is about how many times the number you are dividing by fits into the number you are dividing!

0 is unusual as it indicates empty/nothing if you google is it a number there are huge explanations.

Maybe the question should be if you don't divide 12 by anything what do you get...

Those of you who can do it, for whom the concept is straightforward, and who were well taught- well done, that's nice for you.

Those of us who were incorrectly taught, didn't do any concepts in maths, just rote mechanical stuff, and haven't needed to know since are not thick.

Honest!

I actually relearnt alot of maths via my daughter and her homework from school including online homework - I also went and found out there is loads of stuff online, I knew my own maths was poor but not because of ability but teaching.

Physicality is of no consequence in maths as a taught principle so the fact they are talking about apples etc is actually irrelevant

That's easy to say when you understand abstract concepts. Some of us need these things to mean something functional for us to be able to understand them. Hence why apples and boxes works.

@EerieSilence - it’s easy to say ‘I wish people would cop on’ if you understand maths. My brain just doesn’t compute maths, the subject is largely alien and foreign to me and explanations tend to go over my head. However I have found some of the explanations on here really helpful so I guess if I’d have had the right teacher I might have achieved a better understanding of it.

No, that's the wrong question. It gives you 12 because you aren't dividing. It's not an equation, it's just the number 12. And you are being told to divide! Just by nothing.

Hang the dresses up in no wardrobes and tell me how many dresses are in each one. It's not 12 and it's not even 0 because that would mean there's a wardrobe with no dresses in it. In fact there are no wardrobes at all, so it's impossible, infinite or indeterminate, depending on how you look at it.

Of course you're not thick. The concept is difficult. If it weren't, it would always be taught properly and everyone would get it immediately.

But lots of maths doesn't have a physical correlate

The imaginary number i is defined as the square root of -1. That's not a number with any physical existence. It's only abstract.

I think that's the biggest single leap you need to make when learning maths. And in the end it's a leap of faith, there is no such thing as the square root of -1 apples.

@ReneBumsWombats thats my point. "Don't divide" gets you the 12 answer

That's all true, but even understanding the problems with practical applications helps you to understand the pure maths theory. Practical applications are the best place to start. The sweets/children and dresses/wardrobes ideas have enabled several people on the thread to understand the concept and also why they stop working at a certain point.

They've helped people to understand the concept of 0 too.

Yes. Which is why so many people struggle with maths.

I understand some algebra because I can apples and oranges it. Fractions get turned into pizza. I can understand trig in so far as I can visualise circles and can therefore visualise angles. But sine, cos and tan can fuck off. And take surds with them.

I think I was unclear in my original post. I was told "zero is nothing..." but looking at multiplying by 0 that seems incorrect. However, I have never been asked to divide anything by zero before.

I got very confused some years ago trying to work out something statistical - I have no idea what it was or why but after several hours of confusion I asked my lecturer (I went back to education late in life) and he said that the mathematical rule was in this case a set number (Either 1 or 0). Don't ask what as I honestly have no idea.

Yeah, no criticism of your metaphors intended.

I was responding to the other poster, because I think that leap from 'maths which represents the real world' to 'maths which only exists in maths world' is a leap that some people don't/won't/can't ever make.

And I think the confusion over the main question on this thread comes from the fact that people are trying to fit a question from the second category into the first. There comes a point where you have to recognise that there is some maths that you just can't visualise or represent on a number line, and you either take that on faith and carry on with that, or you stop there. I think a lot of people stop there forever, and that is why this question is doing so many heads in.

Sorry if this has been said earlier:

m x n = k if and only if k/n = m
So k/0 = m if and only if 0 x m = k.
But as long k is non-zero, there is no m such that 0 x m = k.
Because zero times anything is zero.
So k/0 is undefined.

There are respectable ways of treating infinities mathematically. You can use infinity symbolically -- for instance, you can talk of sequences approaching infinity to mean that the sequences get larger and larger and are not bounded.

You can describe a set as being infinite -- the set of all natural numbers; the set of all irrational numbers. You can also introduce meaningful ordering between infinite sets, and meaningfully compare the size of different infinite sets (shocker: some infinities are bigger than others.) But without tampering with the familiar laws of arithmetic, I think by far the best way to think about k/0 is as undefined.

By definition, multiplication is the opposite of division. So if a number (A) is defined by the equation....

A = B / n

Then the number B can be defined by the equation....

B = A x n

Let us assume that division by zero is possible. This will allow us to define a number (A) using the equation

A = B / 0

Where B is a number greater than zero.

Since multiplication is the opposite of division, B must be defined by the equation...

B = A x 0

However, any number multiplied by zero is zero, and B is greater than zero by definition. So this equation cannot be correct since a number greater than zero cannot be equal to zero.

Assuming that that division by zero is possible thus leads to an impossible conclusion. Division by zero must therefore be impossible.

Mathematicians call this proof by contradiction.

Virtually the same thing said in previous post...

You cannot divide things into groups of zero.
Apples, sweets, pizzas, or dresses.

12/3 means divide the 12 into groups of 3, and you get 4 groups.

12/1 means divide into groups of 1. So 12 groups.

12/0 means divide into groups of zero. It cannot be done. So the answer is undefined.

I like turtles.