Wikipedia:Reference desk/Archives/Mathematics/2010 July 13

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July 13

the value of 0/0

i have a doubt for long days. what is the value for 0/0??? upto my knowledge i have guessed that this will have three values as shown below....could you please give me the right answer????

case (i):

 0/0=0

reason: zero divided by any number is equal to zero.

case(ii):

 0/0=1

reason: any number divided by the same number is equal to zero.

case(iii):

 0/0=infinite

reason: any number divided by zero is equal to infinite.

i have asked this question to many teachers. all said that the answer is infinite.....but they can't explain why we shouldnot consider the remaining cases.....please explain me —Preceding unsigned comment added by 122.183.208.130 (talk) 14:35, 13 July 2010 (UTC)

Division by zero is undefined. So, it does not have a value. You can argue that it is any value you like. So, in a sense, division by zero produces EVERY answer possible. -- kainaw™ 14:37, 13 July 2010 (UTC)

You mean division of zero by zero, I suppose. It does not make sense to make the value of x/0 a finite real number for nonzero x.—Emil J. 14:44, 13 July 2010 (UTC)

See indeterminate form. --Tango (talk) 14:40, 13 July 2010 (UTC)

In computing dividing one floating point 0 by another will give a special value called Not a Number, which basically is saying it is indeterminate. In reasoning about values in automatic proofs it is liable to be set to Bottom type which is yet another way of saying the same thing - but then again one might mean any value rather than no value. In some circumstances a value can be defined for it for instance x/x is indeterminate when x'=0 but it has a limit value of 1 at that point.You might as well ask how many angels can dance on the head of a pin? Dmcq (talk) 14:58, 13 July 2010 (UTC)

If someone with computer graphing skills can upload a graph showing together the three equations 0/x = y, x/x = y, and x/0 = y, then we might see where they lead for 0/0 = y.—Wavelength (talk) 15:11, 13 July 2010 (UTC)

There is nothing to graph here. For $x\neq 0$ we have $0/x=0,\ x/x=1$, and in the real projective line $x/0=\infty$. You also have $(2x)/x=2$ and $x/(7x)=1/7$. All of these approach the indeterminate form 0/0, and because of this, there is no natural definition for it. -- Meni Rosenfeld (talk) 15:23, 13 July 2010 (UTC)

Division by zero is also relevant. -- Meni Rosenfeld (talk) 15:24, 13 July 2010 (UTC)

Your teacher, even allowing them to be a bit sloppy, isn't right to say "infinite". More often the answer is something like "unknown". Imagine that you're trying to calculate the speed of an object (which is travelling at a steady speed) using the equation speed = distance / time. If you check how far its travelled after 2 seconds and find it has travelled half a metre, then its speed is 0.5 / 2 = 0.25 metres per second. If you had checked after just half a second you would find it had travelled an eighth of a metre, so its speed is still 0.125 / 0.5 = 0.25 metres per second. If you check how far it has travelled after zero seconds i.e. after no time at all, then you would find it hasn't travelled at all, so your calculation would look like 0/0. But this is what you would get no matter how fast or slow the object was travelling, so really "the object travels 0 metres in 0 seconds" gives no information. (What your teacher was thinking of was x/0 for any x other than 0 e.g. if an object travelled 2 metres in 0 seconds then it must have been going infinitely fast.) Quietbritishjim (talk) 00:36, 14 July 2010 (UTC)

A (slightly) more rigorous way of looking at it is to look at the graph. The page inverse function shows how to get from an equation like y=(something involving x) to x=(something involving y): by flipping about the diagonal line shown on that graph. In our case we want to go from y = 0 x to x = y / 0. But y = 0 x is a horizontal line, so x = y / 0 becomes a vertical line. This backs up the idea that if you try to take 0 / 0 you have any (or all) values, whereas y / 0 for any other y doesn't make sense (or is infinite, if you allow that). Quietbritishjim (talk) 00:43, 14 July 2010 (UTC)

The fraction a/b is 'the' solution to the equation bx=a. If b≠0 then this equation uniquely determines x, (because (b≠0 and bx=a and by=a) → (b≠0 and bx−by=a−a) → (b≠0 and b(x−y)=0) → (x−y=0) → (x=y)). If a=b=0 then the equation bx=a is 0x=0 which is true independent of the value of x, and so the equation does not define x, and so 0/0 is undefined. If a≠0 and b=0, then the equation bx=a is 0=a, which is false independent of the value of x. Bo Jacoby (talk) 03:20, 14 July 2010 (UTC).

time to break out truth

In case the above isn't convincing to you, let's break out actual reality - you know, truth. So, what is 0 - it is the absence of a single one or fraction of whatever you're counting, but not negative - you don't owe any either. For example, 0 apples is the complete absence of a single Apple. Now, I am about to do strictly integer division - no fractions. Let's say you drive to my house, pick up 2 apples, drive home, put them in your basket at home, and repeat until you can't make your next drive, since there aren't enough apples left. How many times can you make that drive (physically - we are talking about reality here, not math). You can physically do that half as many times (INTEGER DIVISION) as I have apples. If I have 8 Apples, you can drive to my house, pick up two, drive home, and repeat, 4 times (8/2 = 4). After the fourth time, you can't do that again, there not being enough apples at my house to make the trip. So, that is how many times you can do it. Now what if instead of coming to my house and picking up 2 apples, you come to my house and pick up 0 apples, drive home, and repeat. After how many trips will you have to stop, there not being enough apples at my house for you to make your next trip? There is no answer to "after how many do you stop" because you do not stop. It's not that you stop after infinite times - you do not stop; the division is not exhaustive: you can always do one more. So, if the normal definition of integer division would be subtracting the divisor one time until you can't do that any more* (because you run out of the dividend), then you will never meet that definition: you will never "run out" of the dividend when you're taking 0 from it. So, there is no solution to how many times you are going to be able to do that, if the process is never finished. This is like asking: if every time you say something, your wife says something, and every time your wife says something, you say something, then if you are the first to speak, how many times will the last person to speak have spoken? There is no "last person to speak" by that definition, so it is a meaningless question, it fails to refer. Asking what 5/0 is, is the same as asking what the value of the last prime is. There is no last prime, and there is no end result to the division. Saying 5/0 is "infinite" is just as wrong as saying the "last prime" is "infinite". The last prime is not infinite, since there is none.84.153.208.32 (talk) 10:13, 14 July 2010 (UTC)

• obviously this is the normal definition. When I ask you how many times you can remove 2 apples from 8, your answer isn't "Oh, you have a lot of choices. You can do it 0 times, 1 time, 2 times, 3 times, or 4 times - these are the physically possible alternatives facing you when you are looking at 8 apples and are to remove 2 a certain number of times, these are the possibilities of how many times you can do that.". Yes, "linguistically" the question "how many times can you remove 2 apples from 8" has the answer "0, 1, 2, 3 or 4", but that's not how we understand it. We understand "how many times until you can't do it again." That's the definition, and by that definition when you divide by 0 you do not get "any answer you want", any more than when you divide 8 by 2 you get "any answer you want, from the choices 0, 1, 2, 3, or 4" -- the definition is to do it until you can't do it one more time, and since there is no such point in the calculation, there is no answer when dividing by 0. It's like me telling you how to calculate Mango's number: you start with 2, then keep doubling (2, 4, 8, 16, 32) until you get to a negative number: that's Mango's number. What is the value of Mango's number? (Obviuosly it is not "infinite", or "any number you want", or architecture-dependent). No, there is no Mango's number. It's the same if we say Mango's number is defined as "4/0": there is no value, the definition does not admit of a point at which you can say, "Okay, now I have Mango's number", just as you can't reach a point where you can say "Okay, now I have the largest prime." You can't say "Okay, now I have the value of 4/0".

You are correct in saying that there is no answer in the real numbers, but many mathematicians and also many non-mathematicians are happy with R, the Extended real number line, where the answer to x/0 is plus or minus infinity for non-zero x. Dbfirs 08:25, 16 July 2010 (UTC)

The extended real number line is not suitable for division by zero. For this you need the real projective line, where 1/0 is unsigned infinity. -- Meni Rosenfeld (talk) 09:07, 16 July 2010 (UTC)

Another way round is used in IEEE floating point where they have Signed zero. So you either have plus infinity equal to minus infinity or you have plus zero being different from minus zero, take your pick ;-) Dmcq (talk) 11:47, 16 July 2010 (UTC)

I considered linking to the real projective line article, but most people are happier with plus infinity being distinct from minus infinity. Dbfirs 17:06, 17 July 2010 (UTC)