# Thread: Solution to 1/0

1. Could there be a possible solution to 1/0?   2.

3. No, 1/0 is not defined- you could attempt to say that "it's infinity", but it is not actually defined.

If you were to write: then that would essentially be it, there's no real, definable answer to it.  4. Originally Posted by Aaronfresh
Could there be a possible solution to 1/0? Nope  5. Originally Posted by Aaronfresh Could there be a possible solution to 1/0? Why you can't divide by zero:

Once only was I ever asked by a student of my maths classes, "why cant' you divide by zero?".

This was unusual surprise for it was a student in his late teens/early twenties
I think the reason for this is, you're introduced to division at an early school age.
Its natural to ask your teacher the question. "Sir why can't you divide by zero" [apologies for the sexual discrimination]
Sir, either doesn't know or he realises that his pupils are too immature to understand the reason why.

So you grow up with 'you can't divide by zero' as an absolute law (more powerful even than the law of gravity).
Its hard-wired into your brain. Hence my surprise.

But I knew I could construct a proof which would be understandable to the whole class.

So here it was: (First explain what I'm going to do and then do it)

The method of proof is 'indirect proof', also called 'proof by contradiction', or 'reductio ad absurdum' (meaning reduce it to an absurdity)
This a valid method in standard logic and is widely used in logic and mathematics.

You assume the opposite (negation) of the statement you're wishing to prove.
You then show that this assumption leads to a contradiction (or absurd situation).
It follows using standard 2-valued logic that the assumption was false, and hence the original statement was true.

1. You can divide by 0 [ assumption]
2. Let x be variable standing for a real number (x is an arbitrary real number)
3. So x / 0 = r [ the set of reals is closed under division (inc. 0 by assumption) division represented by the symbol / ]
4. So x = r 0 [simple algebraic transposition, stands for multiplication ]
5. So x = 0 [ rhs: since any real number multiplied by 0 = 0, so r 0 = 0 ]

Now here is the crux: There is a rule in predicate calculus called 'universal instantiation' which states that if you can
show that any arbitrary member (x) of a set has a property (P), then all members of x have property P.

6. So all real numbers are equal to 0.
At this point I could have stopped, with the absurdity, but I decided to push on into 'the real world':
I started counting out aloud how many people there were in the classroom. "0, 0, 0 .... there's no one here!"

"How many people outside? 0, 0, 0,.... there's no one outside!
0, 0, 0, ... there's no buildings in this college.
0, 0, 0, ... there's no towns in England.
0, 0, 0, ... there's no countries, there's no world.
0, 0, 0, ... there are no planets.
0, 0, 0, ... there are no stars, no galaxies, no universe!"

At that point I turned to the student who asked the original question and said "Is that absurd enough for you?"

[ Howls of laughter ]

Thus 'You cannot divide by zero!'

So there is no possible solution to 1/0  6. Originally Posted by Elterish  Originally Posted by Aaronfresh Could there be a possible solution to 1/0? 1. You can divide by 0 [ assumption]
2. Let x be variable standing for a real number (x is an arbitrary real number)
3. So x / 0 = r [ the set of reals is closed under division (inc. 0 by assumption) division represented by the symbol / ]
4. So x = r 0 [simple algebraic transposition, stands for multiplication ]
5. So x = 0 [ rhs: since any real number multiplied by 0 = 0, so r 0 = 0 ]

Now here is the crux: There is a rule in predicate calculus called 'universal instantiation' which states that if you can
show that any arbitrary member (x) of a set has a property (P), then all members of x have property P.

6. So all real numbers are equal to 0.
At this point I could have stopped, with the absurdity, but I decided to push on into 'the real world':
I started counting out aloud how many people there were in the classroom.
That seems a lot like this logic:

Given that "0*1=0*2", prove that 1=2.

1. 0*1 = 0*2 (as we know, this equation is true) (multiplication denoted by "*")
Reason: Given. Known to be true.

2. (0*1)/0 = (0*2)/0 (division denoted by "/")
Reason: Simple algebraic process of inverse operations to simplify the equation.

3. 1 = 2
Reason: The product of (0*1) and (0*2) simplified to 1 and 2 due to the inverse operations in step 2.

Negation:
So, we just proved 1=2. Wait, wait, wait... we just proved that 1=2. An absurdity?!
Oh never mind, it's false. Because dividing by zero is undefined in the first place. Therefore, 1 does not equal 2.
(implying that 0 you can only multiply by zero, not divide, except for 0/0)

I've seen this proof used a lot to negate division by zero, from reductio ad absurdum. (which I've had to use a lot when I used to be a Christian apologetic)

I don't find this logic convincing though... The reason we get 1=2 is because multiplication/division with zero involves indeterminate forms.

Take for example the equation " 0=0*x "
Tell me, what is x? You can't determine its value because it is indeterminate, it could be 1 or 2. Just like this equation: " 0/0 = x " (apparently you can only divide by zero unless zero itself is also the dividend)

The inverse operations process works assuming there's a defined solution for x. In indeterminate forms, x can be several things, that's why you get 1=2.

So that's why inverse operations must be used carefully in case the value is indeterminate.

Otherwise, you just cancel out certain terms and see what's left. Which is certainly not the case here So, that knocks down one wall to division by zero, in my view.  7. No 1/0 is undefine its greater than infinity.  8. Originally Posted by Nancy No 1/0 is undefine its greater than infinity.
Something that "undefined" cannot be greater, less or equal to anything, so 1/0 has no place on the number line. Neither is infinity a number, it simply denotes unbounded growth or size.  Bookmarks
 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Terms of Use Agreement