Usually. There do exist nonassociative rings, the most famous being Lie algebras.
Indeed, but I have recently found a new view on this topic, a proof even:
0=0^n where n is any number not eqaul to zero.
- 0/0 = (0^n)/(0^n) = 0^(n-n) = 0^0 = 1
Either this is revolutionary, or there is a mistake somewhere in this proof. Opinions anyone?
You are correct: either it’s revolutionary or there’s a mistake. More to the point, there’s a mistake. First of all, you’re reasoning about the invertibility of 0 by using negative powers of 0, which is a no-no. First get 0^{-1}, then give negative powers.
Second, you’re assuming that 0^0 = 1, while in reality it’s just as undefined as 0/0 is.
As an addendum, never use the word “new” until you have as solid a proof of that as you should have had for the result itself. This gaffe’s hundreds of years old at the least.
And this goes for the rest of you out there who think you have some special insight that legions of professional mathematicians or physicists have missed. The absolute worst thing you can do is to overstate your case. If you have some original nugget in there, it’ll be ignored because it’s surrounded by junk and you come off as psychoceramic.
1 divided by 0 is is larger or equal to than 0 divided by 0
True or false?
I will take a stab here.
1 / 0 is infinity
0 / 0 is undefined
Is 1 / 0 greater than 0 /0 ?
No, the relationship is undefined.
For goodness sake! undefined is undefined! - you can’t have one ‘undefined’ bigger or smaller than another ‘undefined’ - undefined means they don’t exist to be compared.
6/3 = How many apples will you have in each box if you share six apples equally amongst three boxes? - answer = 2
3/6 = How many apples will you have in each box if you share three apples equally amongst six boxes? - answer = one half
1/0 = How many apples will you have in each box if you share one apple equally amongst no boxes? - answer = 'What boxes?'
0/0 = How many apples will you have in each box if you share no apples equally amongst no boxes? - answer = 'What apples? What boxes?'
I would not ask “What apples?” there, since:
0/5 = How many apples will you have in each box if you share no apples equally amongst five boxes? - answer = 0.
You can divide zero by any number (except zero), and you will always get zero. It’s just that you can divide by zero (because, as you say in yor example, you don’t have any boxes).
You’re quite right.
It is?
For what it’s worth, not everyone agrees.
OK, it is not. I read your link and it says it is undefined as well. Division is defined in terms of multiplication. 0 x infinity equals 0, so 1 / 0 cannot be infinity. I retract, though my conclusion still holds, I guess.
What do you mean by “0 x infinity”? (There are actually two parts to that question: What do you mean by infinity?, and How do you define multiplication?)
If we are talking about the real numbers or the complex numbers as usually defined, then infinity is not a number, so “0 x infinity” is not defined.
Hmmph. I retract a second time, and will endeavour never again to paraphrase a mathematician or to post again in a mathematics thread. I will quote the hopefully correct explanation from ultrafilter’s link:
::slinks off nursing wounded pride::
I’ve seen those arguments and they’re still patches. The “0^x = 0 is less important than x^0 = 1” bit is especially galling. I’m perfectly willing to concede that 1 is often a more useful patch, but it’s still a patch.
Oh, please don’t just give up. Mathematics is rigorous – intellectual hardball, if you will. There are right and wrong answers, and within a culture socially driven by soft and squishy academic disciplines with no right or wrong answers this can be off-putting at first. Anyone is wrong a whole lot more often than they’re right when they start trying. Those who get scared off by the prospect of being told “no, you’re wrong” for the first time never get anywhere, though. Don’t you think it took Evil Knievel a few crashes before he finally got jumping a motorcycle down pat?
Please, post in mathematics threads. Be wrong. Be gloriously, horribly wrong.
And then be prepared to be corrected.
Back to the original topic, why is 0/0 undefined?
If 0/0 was equal to 1 then correct 6(0/0)=3(0/0), 6=3 is wrong, BUT
if 0/0 was equal to 0 then 6(0/0)=3(0/0), 0=0 would be correct, or?
If someone could just break it down into a simple example or equation or something that would clear this all up. Perhaps Cecil is willing to take this up, or not since not everyone thinks about these kinds of things.
a/b is the unique number c such that a = bc. There’s no possibility of uniqueness when a = b = 0, so 0/0 is undefined.
What’s so hard to understand about this?
Ah, but what do you get if you add 0.999… to 0/0?
Heh. Now you are talking my language, bud. 