# 0/0 Anything divided by zero is undefined

Tags: 0 or 0, divided, undefined
 P: 625 Anything divided by zero is undefined (or infinity?), zero divided by anything is zero, and anything divided by itself is 1. so what is zero divided by zero? it seems to me that this must be an exception to at least to of the above rules. thanks for your help
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 0/0 is also "undefined" but many texts use the term "undetermined" to distinguish that it is a special case: 1/0 is "undefined" because saying 1/0= a would be the same as saying 1= a*0 which is not true, for any a. 0/0 is "undetermined" because saying 0/0= a would be the same as saying 0= a*0 which is true for all a. It is "undefined" but for a different reason- there is no one answer. The distinction is especially important in dealing with limits. If I need to find the limit of f(x)/g(x) (as x goes to some number, a, f(x), g(x) have limits separately at a) and I naievly take the limits of f(x) and f(y) separately, three things can happen: if the limit of g is non-zero, I just form the fraction to get the limit of the fraction. If the limit of g is zero and the limit of f is not, then the limit does not exist ("undefined"). If the limit of g and f are both zero- then I have to look more closely, the limit may not exist or it may be any number.
 P: 15,319 Your statement :'zero divided by anything is zero' is not a reliable statement, it is merely a 'rule of thumb' - applicable in most circumstances, but not all.
P: 15,319
0/0 Anything divided by zero is undefined

 Quote by HallsofIvy 0/0 is also "undefined" but many texts use the
Doh! Beat me by seconds! (But personally, I think my answer is more accurate and succinct.)
 P: 1,059 Look at the$$_ {limit x\rightarrow1} \frac{x^2-1}{x-1}$$
Emeritus
PF Gold
P: 11,155
 Quote by madness ... it seems to me that this must be an exception to at least to of the above rules.
...or the rules are wrong. In the second and third rules, the "anything"s are what's left after applying the first rule; they are not strictly "anything"s. Besides, when you talk about "anything", it is important to specify what set "anything" belongs in. Surely, "horses" and "laziness" don't belong in the set you have in mind.
P: 23
 Quote by Gokul43201 ... Surely, "horses" and "laziness" don't belong in the set you have in mind.
talking of horses, here's one way of understanding what's going on in terms of apples and oranges - if you had 10 apples and wanted to give an equal no. of them to 5 people, then the number of apples each one gets is 10 /5 =2 oranges.

Extending this, if you had 0 oranges and wanted to give an equal no. to 5 people, the the number each one would get is 0/5 = 0 oranges.

further, if you had 0 oranges to begin with, and wanted to equally distribute that to 0 people, then you cant talk of any specific answer to this problem - hence you say the number is undefined.

hope that helps.

P: 37
 Quote by rainbowings talking of horses, here's one way of understanding what's going on in terms of apples and oranges - if you had 10 apples and wanted to give an equal no. of them to 5 people, then the number of apples each one gets is 10 /5 =2 oranges. Extending this, if you had 0 oranges and wanted to give an equal no. to 5 people, the the number each one would get is 0/5 = 0 oranges. further, if you had 0 oranges to begin with, and wanted to equally distribute that to 0 people, then you cant talk of any specific answer to this problem - hence you say the number is undefined. hope that helps. adi
While this works perfectly for an illustrative example of why any definition of division by zero is "undefined", I think it is misleading.
The actual reason why division by zero is undefined is because undefined is to be taken literally as meaning "It is not defined". This is because the operation of division on the real numbers is defined only for pairs of numbers where the second of the pair is not zero. In other words, ( this is becoming a favorite statement of mine ), asking what "1/0" is, is essentially asking what $\sqrt{a \hspace{3} mouse}$ is. It's sillyness because it's something that is outside the scope ( or domain ) of the operator.
Obviously, my version isn't an intuitive one. Your view is like, "Don't stick your finger in the light socket because you could die". Mine is like, "Don't stick your finger in the light socket because I told you not to." Your's is more likely to prevent people from trying. The problem occurs when someone does. If they don't die, and tell their friends, we're in for a heap of trouble. I can still say, "Ok, so you didn't die. I still told you not to"
 P: 92 The way I like to think about it stems more from linear algebra. For a number a, define 1/a as the number such that a* (1/a) = 1. There is no number b that exists such that 0*b = 1, since 0*x=0. If b did exist, it would be 1/0, but it doesn't, and division by 0 is the same as multiplication by 1/0, but 1/0 doesn't exist, so what you've got is a heap of gibberish. All of this occurred to me in relation to non-invertible matrices. Lots of people argue about 0/0, but nobody seems to care if the inverse of a matrix doesn't exist. One can always think of a number as a 1x1 matrix, clears things up for me at least.
P: 111
 Quote by robert Ihnot Look at the$$_ {limit x\rightarrow1} \frac{x^2-1}{x-1}$$
I don't understand what does this have to do with it. Could you please explain? the answer is obviously 2.
 P: 91 and what about $$\frac{Sin(x)}{x}$$ at x=0 above is defined as 1, hence 0/0=1 by logic deduction
 Quote by AntonVrba and what about $$\frac{Sin(x)}{x}$$ at x=0 above is defined as 1, hence 0/0=1 by logic deduction
$$\frac{\sin(0)}{0}$$
$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$$