Cancelling zero

• B
Gold Member
Why does this?

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member 587159
I have no idea what that image is trying to convey. I can give you one message: dividing by 0 is always forbidden and makes absolutely zero sense.

symbolipoint
Gold Member
Of course, it is forbidden. But those two examples are a proof. If you have a zero, and zero goes into zero, you have exactly one zero, right? It is not "fringe" I didn't particularly care to be called fringe for asking about this. I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth. That's all that image is trying to convey.

member 587159
Of course, it is forbidden. But those two examples are a proof. If you have a zero, and zero goes into zero, you have exactly one zero, right? It is not "fringe" I didn't particularly care to be called fringe for asking about this. I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth. That's all that image is trying to convey.

What does "if you have a zero, and zero goes into zero, you have exactly one zero" mean?

phinds
Gold Member
... those two examples are a proof...
A proof of what? That if you perform a meaningless operation that you can pretend that the answer is valid?

undefined314
Gold Member
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.

phinds
Gold Member
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.
@Atlas3, you are, in military parlance, pissing up a rope.

berkeman
fresh_42
Mentor
A correct answer to this question is: junk the book.

Zero isn't part of the multiplicative group of our ring. Hence the question about division by zero is as meaningful as whether there are unicorns and to which species they belong. It is not "forbidden", it is simply not existent. Now one can ask, if we can extend the multiplicative group by the zero of the additive group, to which the answer is no, as we will run into contradictions:
$$0^{-1}\cdot 0 \stackrel{(1)}{=} 1 \text{ and } 0^{-1}\cdot 0 \stackrel{(2)}{=}0^{-1}\cdot (1+(-1))\stackrel{(3)}{=}0^{-1}\cdot 1 + (- 0^{-1}\cdot 1)\stackrel{(4)}{=}0$$
1. definition of the multiplicative inverse
2. definition of the additive neutral
3. distribution law
4. definition of the additive inverse
Hence the answer is: The question is nonsense, and any attempt to attach a meaning is necessarily in vain.

berkeman, SchroedingersLion and eys_physics
PeroK
Homework Helper
Gold Member
2020 Award
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.

I think the argument is as follows:

Because ##0 = 1 \times 0## then ##\frac{0}{0} = 1##

One problem with this argument is that we also have:

Because ##0 = 2 \times 0## then ##\frac{0}{0} = 2##

Etc.

mfb and DaveE
symbolipoint
Homework Helper
Gold Member
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.
No. That division process goes on forever, with no stopping point. Division By Zero is meaningless and is undefined.

Gold Member
Thank you for the better topic Title. I appreciate the moderation.

Mark44
Mentor
I think the argument is as follows:

Because ##0 = 1 \times 0## then ##\frac{0}{0} = 1##

One problem with this argument is that we also have:

Because ##0 = 2 \times 0## then ##\frac{0}{0} = 2##

Etc.
This is exactly why the expression ##\frac 0 0## is meaningless. We in mathematics are picky about arithmetic -- we like any division problem to have only one answer.

Gold Member
Well I don’t know the formatting markup to display such nice notations. I know it’s pretty stupid to conversation about an undefined. As far as one answer for any division problem. What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer. I’m not contesting anything. Just rattling around on a few thoughts.

fresh_42
Mentor
Well I don’t know the formatting markup to display such nice notations. I know it’s pretty stupid to conversation about an undefined. As far as one answer for any division problem. What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer. I’m not contesting anything. Just rattling around on a few thoughts.
##a \cdot 0 = 0## is due to the distributive law in a ring. It combines multiplication and addition, but is does not extend the multiplicative structure by the additive neutral! We just have ##a \cdot 0 = a \cdot (1+(-1))=a\cdot 1 + a\cdot (-1)=a + (-a) = 0##. The element ##0^{-1}## is still undefined, it does not exist.

Gold Member
This is exactly why the expression ##\frac 0 0## is meaningless. We in mathematics are picky about arithmetic -- we like any division problem to have only one answer.
These kind of questions must have bored the crap out of the posters/moderators with such strange notions. However, I did ask a simple thing. Is it useful? Not one comment to the question. I have been schooled instead. I think the title is useful if you knew where to put it. I also received replies stating the state of the nature of the question. Comments followed up by phrases of military parlance of urinating in a vertical fashion are amusing. I suppose it is hard to have an option about things that were left undefined since the 1500's. Thanks to those that took the time to distribute the proofs. It read a bit condescending but I can live with it. Thank you.

jbriggs444
Homework Helper
What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer.
It is all right for many different expressions to all yield the same result. A bit inevitable, perhaps. There are so many expressions and so few results.

It is not all right for one expression to yield many different results. We normally require each expression [with no free variables] that produces a result must produce the same result every time it is evaluated.

fresh_42
Mentor
I think you have got more answers than the question deserved, esp. as there are already dozens of such threads which can easily be found by a forums search. The short answer is: ##\dfrac{0}{0} =1## is not useful, because it will result in logical contradictions. Instead you have been explained why these contradictions will arise. I bet you would had been equally unsatisfied by a simple "No!" which is a complete answer and barely better than not one comment to the question, that wouldn't had schooled you, and didn't contain phrases of military parlance. Furthermore, this
I suppose it is hard to have an option about things that were left undefined since the 1500's.
doesn't make sense either, as there is no such option.

I consider your criticisms unfair, because - as I pointed out - seemingly there will be no answer which would have pleased you. But I will respect your wish: next time I will try to answer your questions by a simple yes, no, correct or false. It will save me a lot of time, I only wished you would had said this right from the beginning.

The question itself is as if you had asked: "What if I fly a plane under water, will it increase gas mileage?" Please ask yourself how you would have answered to such a question.

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Mark44
Mark44
Mentor
I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth.
The category is neither number theory nor discrete mathematics -- it's just plain old arithmetic. And no, it's not true that 0/0 = 1, for reasons already given.

However, I did ask a simple thing.
Yes, and it has been answered.

Is it useful?
Not at all, as has been amply pointed out in the responses.

Not one comment to the question.
This is post #18 -- you have received lots of good responses.

think you have got more answers than the question deserved
Indeed.