I am puzzled by zero

[royalson]

Hello there. Thank you for the opportunity to ask a math question. I hope this one doesn't come across as naiive, because on the surface level it seems rather childish, but for me, I think there is more to it than the prima facie answer that one would resort to almost without thinking.

In short: Why does 0 X 0 = 0 and not instead be undefined?

Now to build my case.

I have grown up accepting that any number divided by zero is undefined, and I can understand the reasoning behind it. However, there is an equation that I have never understood or agreed on the answer: 0 X 0 = 0.

It's taken for granted that anything multiplied by zero is zero. However, why is it that 0 X 0 yields 0? Why should the answer not be undefined?

Here is my rationale:
I would say that "multiplied by" could be synonymous with "lots of" or "instances of", so I write the following two sentences:

If there are no instances of nothing, that means there is nothing.
If there are no instances of nothing, that means there is something but how much is not defined.

Logically, which sentence makes more sense? Does not the second?
We are always told that negating a negative, nullifies that negative, we are told that "I ain't done nothin" is bad grammar because it would really mean that the person has done something, quite opposite to the intention of the speaker.

Now I'm not trying to conflate the English language with mathematical logic, but we ought to be able to be consistent in our use of language. Otherwise, what does multiplied by really mean?

Which leads me to a follow up question - Is it universally accepted that 0 X 0 = 0 more as an accepted convention or is it a mathematical paradox?

Finally, are there any famous mathematicians who have brought up this issue and have actually sided with the case that I provide that the answer should rather be undefined than 0?

Thank you for your time.

 

[romsek]

well let's see... 0 first comes up as the additive identity of some group.
what this means is that for every element of the group $x$ it's true that $x+0=x$

If we multiply both sides of this equation by some constant we get
$k(x+0)=kx$
$kx + k\cdot 0 = kx$
$k \cdot 0 = 0$

This is true for all $k$

So it must be that the additive identity multiplied by any number returns the additive identity, namely 0.
So it naturally follows that $0 \cdot 0 = 0$

 

[Debsta]

If there are no instances of nothing, that means there is nothing.
If there are no instances of nothing, that means there is something but how much is not defined.

Logically, which sentence makes more sense? Does not the second?
We are always told that negating a negative, nullifies that negative, we are told that "I ain't done nothin" is bad grammar because it would really mean that the person has done something, quite opposite to the intention of the speaker.

I can sought of see what you are getting at, but let me point out a major flaw in your argument above.

I agree that negating a negative, nullifies the negative and your examples is a classic one.

But, your statement "if there are no instances of nothing, that means there is something but how much is not defined"

can be rewriitten as " if there are 0 instances of nothing, that means there is something but how much is not defined".

This second statement does NOT include two negatives. In this case, "no" means 0 which is not negative.

 

[royalson]

well let's see... 0 first comes up as the additive identity of some group.
what this means is that for every element of the group $x$ it's true that $x+0=x$

If we multiply both sides of this equation by some constant we get
$k(x+0)=kx$
$kx + k\cdot 0 = kx$
$k \cdot 0 = 0$

This is true for all $k$

So it must be that the additive identity multiplied by any number returns the additive identity, namely 0.
So it naturally follows that $0 \cdot 0 = 0$

Thank you Romsek, for your response. I guess where I'm coming from is that it seems that 0 x 0 = 0 is presupposed to be true, hence "this is true for all $k$.

But what if 0 is an exception?
For example, we would agree that 0 is an exception when it comes to a division equation. where a / b = a defined number except when b = 0.

but to say there are zero zeros, it would mean that there is not even one instance of zero, or that for this situation there is no zero occurring, but that would not be true if the answer is zero.

I don't mean to sound silly, it's just I guess I have a different way of looking at it, which means I'm wrong I know. 99.999999999% of the world's population agrees with 0 x 0 = 0 but is it because this is an accepted convention?

I tried to use the sentences as analogies and I will readdress them with the other respondant, but allow me to use a logic example...

In binary logic, 1 represents true, or in physics "on"

suppose we have a wire connecting between a battery and a lamp with no switch.

I could say, there is no 0 (no open switch) which would mean the lamp is on because there is nothing breaking the circuit.

If I asked you, in the number 5, how many lots of 0 are there? You'd say 0.

If I said to you, in the number 0, how many 0's are there? To say 0 would be illogical, you'd have to say "we don't know" it could be 1 lot of 0, or 6517 lots of 0 , it's not defined.

Maybe it's just my silly pea sized brain, but I feel that this convention of 0 x 0 = 0 is just illogical. Or perhaps it's a paradox of mathematics? I don't know. I guess everyone else finds it to be straightforward.

Not trolling or trying to be difficult,
Oliver.

 

[royalson]

I can sought of see what you are getting at, but let me point out a major flaw in your argument above.

I agree that negating a negative, nullifies the negative and your examples is a classic one.

But, your statement "if there are no instances of nothing, that means there is something but how much is not defined"

can be rewriitten as " if there are 0 instances of nothing, that means there is something but how much is not defined".

This second statement does NOT include two negatives. In this case, "no" means 0 which is not negative.

 

[royalson]

Thank you Debsta , I was using the term negating more in the linguistic sense of nullifying or undoing.
For example, if I said to you that there are zero zero's in the number 635, that would be correct.there are also zero zeros in the number 2 and also in the number 5,774,829. So because there is no single answer that zero lots of zero would be bound to, hence undefined.
If I said there were zero zeros in the number 0, that would be incorrect, because the number 0 shows at at least one zero exists.

proposition a: zero instances of number X means there is not any instance of that number.
proposition b; In the number zero, there is at least one instance of 0.
proposition c: zero multiplied by zero means you have no instance of 0.
Conclusion : 0 X 0 cannot equal zero.

Martin and Jane play a lottery game and buy 5 lottery tickets each. Martin is very lucky but Jane is very unlucky.
Martin: how many $0 prizes did you end up with ?
Jane: All five how about you?
Martin: 0
Jane: Oh my gosh can we swap!???

Is my brain wired incorrectly? maybe, but this is currently how I see 0 X 0. Again, not trying to troll or anything, but this has plagued me for a very long time.

 

[Debsta]

For example, if I said to you that there are zero zero's in the number 635, that would be correct.there are also zero zeros in the number 2 and also in the number 5,774,829. So because there is no single answer that zero lots of zero would be bound to, hence undefined.
If I said there were zero zeros in the number 0, that would be incorrect, because the number 0 shows at at least one zero exists.

proposition a: zero instances of number X means there is not any instance of that number.
proposition b; In the number zero, there is at least one instance of 0.
proposition c: zero multiplied by zero means you have no instance of 0.
Conclusion : 0 X 0 cannot equal zero.

Hang on a minute. Using your example, you could also say that there are zero one's in the number 635; there are also zero one's in the number 2 and also in the number 5,774,829. So because there is no single answer that zero lots of ONE would be bound to, hence does that mean that zero lots of ONE is also undefined?? Clearly not!

 

[Archie]

But what if 0 is an exception?

It's not. Romsek is using abstract algebra that applies to any system (that follows some basic ideas that numbers do). If there is any number $\theta$ such that $x + \theta = x$ for all $x$, then $k \cdot \theta = 0$ (as long as $k \cdot x$ has an additive inverse, which it does in numers). Since $\theta$ is one of the numbers in the system, we get $\theta \cdot \theta = 0$. You could define a system with exceptions, but it's not the numbers that we normally deal with.

For example, we would agree that 0 is an exception when it comes to a division equation. where a / b = a defined number except when b = 0.

Similarly, we have a multiplicative identity $I$ such that $x \cdot I = x$ for all numbers $x$ that have a multiplicative inverse. It can be proved that (for systems like normal numbers), that the additive identity doesn't have a multiplicative inverse. This is why we are forced into saying that we can't divide by zero.