0 divided 0 = ?
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To 0/0 it means you need to find a unique number x such that 0*x=0
However any number can satisfy that thus x is not a unique number
According to one of my maths professors, since division is a binary operation (namely it takes two inputs and produce an output) we don't expect it to have multiple possible values (In fact it is even worse then similar multivalued things such as solving for sin x=0 or (exp(x))^y where x,y are complex numbers (where you would end up something like exp(x*y+n*pi) as the function exp(x) in the complex plane have the same values after you walking around it in a full circle) , in that it can take ANY possible numbers in the number system in question)
Therefore 0/0 we basically cannot tell anything useful from it, hence we cannot determine anything from it alone (indeterminate).
When you get to things like calculus and compare the behavior of functions such as f(x)/g(x) as x approach a certain value a, if you get f(a)/g(a)=0/0 it means you cannot tell anything other than the two functions will separately tend to zero.
In order to tell more, you need the L'Hopital Rule which said if f(a)/g(a) = 0/0 or infinity/infinity and f'(x)/g'(x) as x tend to a exist and is a finite value L, then similarly f(x)/g(x) as x tend to a will tend to L.
In layman terms, it compares the rate of f(x) relative to g(x). If f(x) change quicker than g(x) as x approaches a (I.e f'(x) > g'(x) as x approaches a) it means the final value of f(x)/g(x) as x approaches a (BUT NOT =a) is dictated by f(x). So if f(x) approaches zero near a then f(x)/g(x) will also approach zero near a.
So what about f(a)/g(a)=0/0? , in this case since you have plugged a value (number) into a function and it split out a number, then we should treat 0/0 like a number and the argument in the 1st paragraph applies, that is, we cannot have a unique or a discrete series of answers on what 0/0 is. (Even things like modular arithmetic e.g a mod 12 system like a clock, you still have something like x=x+12 but not something like x+a=all possible numbers)
So the ultimate conclusion is that because of how numbers and division in the usual sense are defined, 0/0 gives no useful result (having an expression such that it equals to any conceivable number is not very useful in solving problem, as you always end up all possible numbers as the answer, which in reality is not the case for nearly all problems)
Footnote: There is actually a number system where 0/0 and n/0 are defined, see below
https://en.wikipedia.org/wiki/Wheel_theory
It is very bizarre in that it has two types of division and 0/0+x=0/0
Still yet to get my head around it, let alone how it is used
However any number can satisfy that thus x is not a unique number
According to one of my maths professors, since division is a binary operation (namely it takes two inputs and produce an output) we don't expect it to have multiple possible values (In fact it is even worse then similar multivalued things such as solving for sin x=0 or (exp(x))^y where x,y are complex numbers (where you would end up something like exp(x*y+n*pi) as the function exp(x) in the complex plane have the same values after you walking around it in a full circle) , in that it can take ANY possible numbers in the number system in question)
Therefore 0/0 we basically cannot tell anything useful from it, hence we cannot determine anything from it alone (indeterminate).
When you get to things like calculus and compare the behavior of functions such as f(x)/g(x) as x approach a certain value a, if you get f(a)/g(a)=0/0 it means you cannot tell anything other than the two functions will separately tend to zero.
In order to tell more, you need the L'Hopital Rule which said if f(a)/g(a) = 0/0 or infinity/infinity and f'(x)/g'(x) as x tend to a exist and is a finite value L, then similarly f(x)/g(x) as x tend to a will tend to L.
In layman terms, it compares the rate of f(x) relative to g(x). If f(x) change quicker than g(x) as x approaches a (I.e f'(x) > g'(x) as x approaches a) it means the final value of f(x)/g(x) as x approaches a (BUT NOT =a) is dictated by f(x). So if f(x) approaches zero near a then f(x)/g(x) will also approach zero near a.
So what about f(a)/g(a)=0/0? , in this case since you have plugged a value (number) into a function and it split out a number, then we should treat 0/0 like a number and the argument in the 1st paragraph applies, that is, we cannot have a unique or a discrete series of answers on what 0/0 is. (Even things like modular arithmetic e.g a mod 12 system like a clock, you still have something like x=x+12 but not something like x+a=all possible numbers)
So the ultimate conclusion is that because of how numbers and division in the usual sense are defined, 0/0 gives no useful result (having an expression such that it equals to any conceivable number is not very useful in solving problem, as you always end up all possible numbers as the answer, which in reality is not the case for nearly all problems)
Footnote: There is actually a number system where 0/0 and n/0 are defined, see below
https://en.wikipedia.org/wiki/Wheel_theory
It is very bizarre in that it has two types of division and 0/0+x=0/0
Still yet to get my head around it, let alone how it is used
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Look into abstract algebra. You can create your own system where division by zero is allowed. Sadly, you will quickly discover that it's a trivial system and there is nothing to look at.
Namely, given you have 0+x=x,1*x=x,0=/=1,-x,1/x=x^(-1),x+y=y+x, x*(y+z)=xy+xz and q*0=1
You will always end up
0=1=2=...
u always insist to know for creations and not for real
smthg divided by itself, means itself objective knowledge so the result is one, self
zero is not smthg, so its own knowlege open up infinite different ones, which actually is the positive truth result, what is always, so a plus but also free
smthg divided by itself, means itself objective knowledge so the result is one, self
zero is not smthg, so its own knowlege open up infinite different ones, which actually is the positive truth result, what is always, so a plus but also free
Problem is
Infinity is just a term saying something increase without bound (-infinity for the other direction) thus they are not treated as a number in the usual sense
In number systems such as complex projective plane (Riemann sphere) you can have n/0=infintiy by gluing the complex plane into a sphere at the point labelled infinity, but 0/0 is still indeterminate in such a system
why dont u need to speak like a jerk especially when responding to someone that speak simply, u r not infinite nor truth, u cant speak as if u r professionnal in numbers and how things work, u speak only of urself in respect that u r deforiming smthg else value
so u blow wat i said by saying wat???? that 0 divided by zero is not infinite but indertermined???? so u know wat is infinite and wat is inderminate to defend ???
infinite is the truth the positive truth that could b reversed like u do, by puting nothing over positive knowledge
infinite is the always that exist the most clearly in nothing shape, absolute is a translation of zero for positive plus of infinite truth
that is how truth existence is called freedom u can never know nor determin, which is then in existence of infinite result
0 divided by zero, confirm the point, the nothing that if u were honest or true could b u, that know nothing freedom in truth, so the 0 become free by realizing freedom, so indetermin, and the objective zero become relative to freedom factor reasons
while wat is not absolute so what is relative is not free, and when only truth exist then relative is not real, so negative less then zero
so u blow wat i said by saying wat???? that 0 divided by zero is not infinite but indertermined???? so u know wat is infinite and wat is inderminate to defend ???
infinite is the truth the positive truth that could b reversed like u do, by puting nothing over positive knowledge
infinite is the always that exist the most clearly in nothing shape, absolute is a translation of zero for positive plus of infinite truth
that is how truth existence is called freedom u can never know nor determin, which is then in existence of infinite result
0 divided by zero, confirm the point, the nothing that if u were honest or true could b u, that know nothing freedom in truth, so the 0 become free by realizing freedom, so indetermin, and the objective zero become relative to freedom factor reasons
while wat is not absolute so what is relative is not free, and when only truth exist then relative is not real, so negative less then zero
I don't quite understand, first 0/0 is infinity because x(0)=0 x can be anything.
Then you go on about its a useless equation? If 0/0= infinity, I don't think it is a useless equation. Reason why I ask is math answers questions about life as we know it. 0/0= the Big Bang? 0/0= god 0/0= infinity could define a lot, just needs to have a purpose. 1+1= 2 seems really useless too, but $1+$1 =$2 is really useful. 1 male + 1 female = a couple... So nothing (0)/ nothing(0) still equals something. The fact that it isn't 0 is very useful. The fact that it is infinity could change life as we know it.
I looked into that wheel of algebra, what I got from it was that if you had a basic multiplication chart( and/or a division chart) 0/0 will equal 1,2,3... And so on.
If $$x = \frac{y}{z}$$ then $$x z = y$$. But if $$ z = 0$$ then $$0 = x z = y$$ so $$y = 0$$ and we can learn nothing about x.
This is both why division by zero in general is undefined, since there is no number x such that $$0 x$$ is anything other than zero, and $$\frac{0}{0}$$ is too poorly defined to be a number.
Now interesting things happen with $$\lim_{x\to a} \frac{f(x)}{g(x)}$$ when $$\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$$ but manipulation of limits is not manipulation of numbers.
This is both why division by zero in general is undefined, since there is no number x such that $$0 x$$ is anything other than zero, and $$\frac{0}{0}$$ is too poorly defined to be a number.
Now interesting things happen with $$\lim_{x\to a} \frac{f(x)}{g(x)}$$ when $$\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$$ but manipulation of limits is not manipulation of numbers.
More specifically, it is non-functional.
One domain element maps to any range element. This is not a function, hence it is undefined.
Function, as understood in mathematics, is a procedure, a rule, assigning to any object a from the domain of the function a unique object b, the value of the function at a. A function, therefore, represents a special type of relation, a relation where every object a from the domain is related to precisely one object in the range, namely, to the value of the function at a.
http://plato.stanford.edu/entries/set-theory/primer.html
This is similar to a contradiction. A contradiction maps A to B and ~B.
If $$M_z$$ is the map where $$M_z \; : \; x \mapsto z x$$ then chinglu is correct, it is functional. And if $$z \neq 0$$ then $$M_z$$ is one-to-one, so its inverse map is also functional. Thus $$ z \neq 0 \quad \longrightarrow \quad M_z( \, M_z^{\tiny -1} ( x ) \, ) \; = \; M_z^{\tiny -1}( \, M_z ( x ) \, ) \; = \;x$$.
And chinglu is also right that the inverse map of $$M_0$$ cannot be a function. $$M_0 \; : \; x \mapsto 0$$ has no function as its inverse mapping except in very boring number systems which have only one number in them. (Like addition and multiplication of integers modulo 1.)
But if you had two distinct numbers, x and y, then you can prove that the hypothesis that F is an inverse function $$M_0$$ leads to contradiction:
$$ \forall x F( M_0( x ) ) = x \left( 0 \neq 1 \longrightarrow 0 = 1 \right )$$
Proof:
$$ M_0( 0 ) = 0 \\ M_0 ( 1 ) = 0 \\ M_0( 0 ) = M_0 ( 1 ) \\ 0 = F( M_0( 0 ) ) = F( M_0 ( 1 ) ) = 1$$.
In this way the hypothesis the F was an inverse function for $$M_0$$ was shown to be incompatible with the axiom that $$0 \neq 1$$. The same bogus assumption is hidden in various fake proofs that 0 = 1.
http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml
And chinglu is also right that the inverse map of $$M_0$$ cannot be a function. $$M_0 \; : \; x \mapsto 0$$ has no function as its inverse mapping except in very boring number systems which have only one number in them. (Like addition and multiplication of integers modulo 1.)
But if you had two distinct numbers, x and y, then you can prove that the hypothesis that F is an inverse function $$M_0$$ leads to contradiction:
$$ \forall x F( M_0( x ) ) = x \left( 0 \neq 1 \longrightarrow 0 = 1 \right )$$
Proof:
$$ M_0( 0 ) = 0 \\ M_0 ( 1 ) = 0 \\ M_0( 0 ) = M_0 ( 1 ) \\ 0 = F( M_0( 0 ) ) = F( M_0 ( 1 ) ) = 1$$.
In this way the hypothesis the F was an inverse function for $$M_0$$ was shown to be incompatible with the axiom that $$0 \neq 1$$. The same bogus assumption is hidden in various fake proofs that 0 = 1.
http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml
Zero divided by zero equals one. Computers cannot perform such a calculation because binary is currently insufficient. I am working on this problem (as are you) using the previously posted program that can discover any equation: x=(((a+b)*c)-d)/e.
Write4U
Valued Senior Member
Could the equation be simplified to: x/x = 1 (where x may be an infinite set of numbers). I am no programmer, but is that difficult for a computer to calculate?
Theoretically: If you divide any number by itself the result is always 1, no?
It is easy for computers to calculate with specific numbers, not so much with infinite sets.
Not always. Dividing 0 by itself does not result in 1.
Write4U
Valued Senior Member
Sorry, for not being clear about infinite sets. I meant an infinite variety of specific numbers (and included zero as a real number). The question then becomes if logically x ever can be 0 or if it is just a meaningless application?
Nonsense. One person had this idea, wrote a paper, then wrote the Wiki article about it ... a violation of Wikipedia rules. See the Talk page accompanying the Wiki page.
https://en.wikipedia.org/wiki/Talk:Wheel_theory
Nobody else takes this seriously. Just wanted to mention this in case anyone reads this thread and thinks there's more than one person in the world who takes this theory seriously.
0 is an even number.
is 0/0 still even?
is 0/0 still even?
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