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!

Teachers introduce the idea of division by saying things like the following.

There are twelve apples, and three children. The apples are shared equally. How many apples does each child get?

This justifies the mathematical operation of division in the case that the all the numbers involved are counting numbers. This is a justification, not a mathematical definition. The justification breaks down for other kinds of numbers. For example, there are twelve apples shared equally into pi piles. How many apples are in each pile? The correct answer involves mashing the apples and making transcendentally large piles of apples. This is more than a primary school pupil can understand. At some point, one has to switch to using mathematical definitions and doing abstract mathematics. Dealing with negative numbers is one such case.

If you had a calculator that used transreal arithmetic, any calculation involving a decimal point would be twice as accurate, even if you calculations used only real numbers. So you don’t have to understand transreal numbers to get a benefit. So long as someone does understand, they can make your life better.

(The decimal point thing is a bit of a simplification but its close enough to true for the purposes of this discussion.)

Yes, I really am dividing by zero.

You do understand how "trans" maths goes down on MN?

I like your post. It is mathematically well informed but I didn’t approach Mumsnet, I came in response to an invitation.

-0 = 0 so whatever you think 12/0 is, 12/(-0) is the same thing as 12/0.

Similarly, whatever you think 0/0 is, -0/0 is the same thing as 0/0.

Yes, there is a computer science concept of -0, but it is inconsistent because -0 is equal to 0 but 1/(-0) is not equal to 1/0 so -0 is not equal to 0. Here -0 being equal and not equal to 0 is the contradiction. This sort of mess can be fixed by using transreal arithmetic as a foundation for floating-point arithmetic.

Is zero an even number?

What does halving zero look like?

www.bbc.co.uk/news/magazine-20559052

Apparently it caused issues when cars with numbers that had an even last digit could come into the centre on certain days and odd could come in on other days.

French mathematicians regard zero as positive. Everyone else regards zero as neither positive nor negative. This makes talking to French mathematicians a little bit tricky.

This answer, x/0 = x, is what Suppes-Ono algebra gives but that algebra is inconsistent with trigonometry and calculus so it is setting up pupils for failure.

A reasonable approach is for school teachers to say that division by zero is not defined in the mathematics that is taught in schools but it is defined and taught in university level mathematics.

In Brazil, some undergraduate school mathematics teachers and both undergraduate and postgraduate mathematics and physics students are taught transreal arithmetic and transmathematics. Brazil is positioning itself to leapfrog the native English speaking countries - all of which believe that division by zero is impossible. Wow! Are you guys in for a shock!

Right, I am out of here. There is an individual in Beijing who wants to learn how to divide by zero and I know just enough Mandarin to teach this. If you have serious questions, you can track me down.

Enjoy your Sunday,

Bye!

Do you think that question about zero being even was for you?

You do realise this is a public forum with comments for all to see and respond to?

en.m.wikinews.org/wiki/British_computer_scientist%27s_new_%22nullity%22_idea_provokes_reaction_from_mathematicians

To put it mildly this stuff is not mainstream......

But they are not zero groups, they are in one group

Mumsnet: where everyone's a size 8, their DH is 6'2" and earns six figures, they're stuffed at 8pm after a massive salad at 1, they look 20 years younger...and can divide by 0.

Although that last one was a man.

Suppose a = b
Then multiply by a gives a2 = ab^
Subtract b2 gives a2 - b2 = ab - b2
Factorising gives (a+b)(a-b) = b(a-b)
Divide by a-b gives a+b = b
So if a=1, then as a=b, b= 1, so substituting gives 1+1=1, ie 2=1.

There is no puzzle. All this makes apparently sense, until you spot that this line

Divide by a-b gives a+b = b makes no sense

it compares 2 non-existing results, that can not be at all.

As a-b=0 , "Divide by a-b" means a forbidden or impossible operation called "division by zero" that CAN NOT produce any result, meaning

(a+b)(a-b) divided by a-b = (a+b)(a-b) divided by zero = no result / impossible operation

b(a-b) divided by a-b = b(a-b) divided by zero = no result / impossible operation

so in a+b = b you are comparing in fact two "non-results" - two non-existing results of two impossible operations.

If you gave any arbitrary value to a (and automatically the same value to b, as a=b) and used a computer to do these two calculations, it would simply grind to a halt at the first division by zero and throw an error message. There would be no next step. After "Divide by a-b" there can't be anything - as there is no result to work on further.

All you have proven is that you have to be careful not to be led into doing a "division by zero" when zero is expressed in a hidden or not so obvious form such as a-b where in fact a=b.

@DaSilvaP Quite. Issue well explained.

I posted something up thread....v simple so will post it again....

Multiplication is the opposite of division.

So if a number A is defined as:

A = B/n

Then the number B can be defined as:

B = A x n

If we assume that it is possible to divide by zero, then we can define a number A as:

A = B / 0 , where B is a number other than zero.

Since multiplication is the opposite of division, the number B can be defined as:

B = A x 0

Since any number multiplied by zero is zero, this equation must be false since we know B is not zero.

The "proof" that 1 = 2 is just an elaboration of this. Mathematicians call this proof by contradiction. If we assume division by zero is possible, then this assumption leads us to an impossible conclusion. This effectively proves that division by zero is not possible.

However.....this proof is based upon the assumption that multiplication is the opposite of division. This is one of the Axioms of maths.

As far as I can see, the transreal maths described above assumes that the second assumption is false, and that division by zero is possible. This requires the creation of an alternative form of maths. Link to wiki news above shows discussion of this..... to put it is not a theory that is widely accepted!

I'm sure you'll be thrilled to be in a bus driven by a professional driver who can't make the difference between the clutch and the brakes. After all, it's just a silly mistake, nothing to worry about?

After re-reading this thread it looks like the teacher was maybe using a test prepared by someone else, someone preparing teaching material for hundreds of schools.

If it turns out to be the case that those "teaching the teachers" can't get the basics right ... it would be even worse.

- it was a centrally-set test - so it was that bad, the teacher at least corrected it.