Wikipedia:Reference desk/Archives/Mathematics/2020 March 3

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March 3

Division by 0

Why is 0/0 undefined? i read the article saying that it is because theres nothing you can divide to get a assuming a isnt 0. what does that mean? Also, if a/0=undefined, does infinity times = something? UB Blacephalon (talk) 03:08, 3 March 2020 (UTC)

The article you mention is presumably Division by zero. You may also be interested in the article Indeterminate form, and specifically its section §Indeterminate form 0/0. Note that in the context of "the indeterminate form 0/0", we are not actually dividing zero by zero, but using it as the shorthand for an expression involving limits (a concept from Calculus). -- ToE 05:07, 3 March 2020 (UTC)

For your questions, lets restrict ourselves to the arithmetic of the Real numbers.

You ask the meaning of "a/0 has no meaning, as there is no number which, when multiplied by 0, gives a (assuming a ≠ 0)".

Perhaps using specific numbers will help. We write that 8/4=2 because 2 is the number which when multiplied by 4 gives back 8. (2*4=8)

Now suppose someone asserted that 8/0=72. (Here 8 plays the role of a in the article lede.) That would mean they claim that 72*0=8, which we know is not true because 72*0=0. You have the same problem for every value answer you come up with because there is no real number which when multiplied by zero yields 8, so we say 8/0 is undefined. The same holds for all a/0 for non-zero a; they are all undefined.

You also ask about when a=0, about 0/0. Well, suppose you said that 0/0=7 because 7*0=0. But someone else might say 0/0=13 because 13*0=0. So 0/0 has the opposite problem that 8/0 has. Where nothing satisfies 8/0, every number seems to satisfy 0/0. But if you wish to define 0/0 then you need to choose one value, and no value is better than any other, so 0/0 is left undefined as well. (In a sense, 8/0 isn't anything so it is left undefined, but 0/0 is everything, so it is also left undefined. That's a mathematically sloppy way of expressing the idea, but perhaps it helps.)

When working with limits, there are some indeterminate forms of 0/0 which will evaluate to 7 and others which will evaluate to 13, but remember that those are not pure arithmetical divisions, but are expressions from calculus.

Finally, be careful with concepts of infinity as it is used in many different mathematical contexts to mean different things, and problems arise from confusing contexts. Infinity is not a real number, so in the realm of real number arithmetic it does not make sense to talk about multiplication by infinity. In limits, infinity is used as a shorthand for growing without bound (with formal mathematical definitions), and note that Indeterminate form gives several forms which involve infinity. -- ToE 05:54, 3 March 2020 (UTC)

While I understand where you're coming from, wouldn't technically 0/0=0 because 0*0=0? Plus, why is 0^0=1? With infinity, in my case it represents every number because it IS every number one can think of but even it falls to 0 because any number divided by zero is 0. Wouldn't that count? UB Blacephalon (talk) 12:54, 3 March 2020 (UTC)

The problem is that n*0=0 for any n and there is no reason to favour one answer over any other. 0⁰ = 1 as an empty product. And infinity is certainly not "every number"; usually it rather means "growing without bound" as ToE says. Double sharp (talk) 14:14, 3 March 2020 (UTC)

Yes, growing numbers, meaning every number, or infinity, right? UB Blacephalon (talk) 16:17, 3 March 2020 (UTC)

"Growing numbers" does not mean "every number". Here is a sequence of growing numbers: 0, 0.9, 0.99, 0.999, 0.9999, 0.99999, ...  It will never even get to 1.  --Lambiam 16:43, 3 March 2020 (UTC)

More to the point, an unbounded continuous function may pass through every number (or at least every number in the direction it is unbounded), but that it not true for all functions in general. Also, "infinity" is not a nickname for "every number". -- ToE 18:36, 3 March 2020 (UTC)

To elaborate on the problem _𝄪 mentions, if technically 0/0 = 0 because 0×0 = 0, then by the same argument 0/0 = 42 because 42×0 = 0. But if 0/0 = 0 and 0/0 = 42, then 42 = 0, which is technically somewhat problematic.  --Lambiam 16:40, 3 March 2020 (UTC)

Isn't there an equation method with that? UB Blacephalon (talk) 16:42, 3 March 2020 (UTC)

I don't know what you mean by "an equation method".  --Lambiam 16:47, 3 March 2020 (UTC)

Like an equation you can use for this exact method in Algebra or something. And with your example, those are all valid numbers to use with this problem. UB Blacephalon (talk) 16:50, 3 March 2020 (UTC)

Which exact method? Though this be madness, I don't see a method in't. And of course I only use valid numbers in my examples. I don't even know what invalid numbers are.  --Lambiam 16:55, 3 March 2020 (UTC)

Maybe because you can find multiple numbers at once (y=8 and 2 on a graph). Therefore it could by any and all numbers making the possibilities infinite, right? UB Blacephalon (talk) 17:07, 3 March 2020 (UTC)

Is that an answer to my question?  --Lambiam 17:32, 3 March 2020 (UTC)

Yes. Sorry I put it in the wrong way. UB Blacephalon (talk) 17:43, 3 March 2020 (UTC)

Are you suggesting that if we define division as the inverse of multiplication such that a/b is the number x such that x·b=a, then you wish to define 0/0 as the set of all numbers x such that x·0=0, which is all numbers, and you want to call "all numbers" infinity? There are two problems here. First, for reasons of utility, we want division to be an operator which, where defined, always returns a single number, and not in certain cases returns a set. Second, infinity is not another name for "all numbers". There may be infinitely many numbers (and this is yet another kind of infinity, different from ∞ or +∞ & -∞ mentioned in Division by zero#Calculus -- see instead Cardinality of the continuum), but infinity is not the name of the set. If you are working with the reals it is ℝ. But you are onto something, in that indeterminate forms with limits which look like 0/0 can often be evaluated to determine the actual value of the limit. For a trivial example, consider f(x)=13x/x. f(0) is undefined, but the limit of f(x) as x→0 is 13. -- ToE 18:29, 3 March 2020 (UTC)

I knew I was! So everything I was taught in school about infinity was a lie? Hmm.....I'm saying that all the number we know of could easily fit into 0/0=a and a*0=0. So I thought infinity would work but apparently I was wrong. There's gotta be something,right? UB Blacephalon (talk) 18:43, 3 March 2020 (UTC)

@Blacephalon: It was not a lie, but rather a simplified approach at a specific case. Similarly what you were told about a line, a natural number or a set were all simplified, partial presentations of what geometry, number theory or set theory say about them. --CiaPan (talk) 13:13, 4 March 2020 (UTC)

So it could be both? UB Blacephalon (talk) 19:54, 4 March 2020 (UTC)

Every number is a solution to the equation 0 * X = 0, and division by 0 (including division of 0 by 0) is not defined. --JBL (talk) 11:08, 5 March 2020 (UTC)

So it's any or every number, right? UB Blacephalon (talk) 13:34, 5 March 2020 (UTC)

If by "it" you mean the string of symbols "0/0" then the answer is "no, 0/0 is not defined". (If you meant something else, I don't understand your last comment, and please clarify.) --JBL (talk) 19:53, 5 March 2020 (UTC)

I find the following approach helpful when contemplating this question. On the number line, no point relies for its value on the ratio 0/0. Similarly, the ratio 0/0 cannot be assigned to any point on the number line. (We might imagine 0/0 is equal to 1, but it is easy to show that 0/0 is not equal to 1. We might imagine 0/0 is equal to 0, but it is easy to show it isn’t equal to 0.) The conundrum is solved by saying we simply don’t assign a numerical value to 0/0.

When we find an algebraic expression which, for one value of the independent variable, the numerator and denominator are both zero, it is likely the expression defines a function with a removable discontinuity so we go in search of the limit at that value of the variable. Sometimes the limit proves to be 0, and sometimes it proves to be 1, but it is just as likely to be some value other than 0 or 1. Dolphin (t) 13:40, 5 March 2020 (UTC)

Yes what I mean is that any number could fit in to the equation (some equal 0,1,2 or more), so there has to be some way to figure this out algebraically. I thought maybe it would be infinity but I was wrong. When number line comes in to play, 0/0 would equal 0 because you're not going anywhere there. And how does an error come into play here? UB Blacephalon (talk) 20:28, 5 March 2020 (UTC)

It's not clear to me what you are now questioning, but perhaps this will help:

• The quotient 0/0 (as in the number zero divided by the number zero) is undefined. Period.
• The solution set for x·0=0 is ℝ (all real numbers), assuming that is your domain. So there are infinitely many solutions, but "infinity" is not a synonym for "everything". And infinity itself is not a solution to that equation because infinity is not a real number. Moreover, if you happen to be working with the extended real number line ℝ ∪ {−∞, +∞} or the projectively extended real line ℝ ∪ {∞} where infinity of some form is considered a number, then it is still not a solution to that equation because −∞⋅0 and +∞⋅0 in the former and ∞⋅0 in the latter are also undefined. (And they are undefined for the same reason 0/0 is. When they appear in limits they are indeterminate forms.)

Note that the calculus can be taught without any appeal to an extended real line, and that when symbols such as +∞ are used, it is in specific contexts of unbounded growth. For instance, lim_x→af(x)=+∞ is understood to mean that ∀ M ∃ δ>0 s.t. |x-a|<δ ⇒ f(x)>M. While a symbol for infinity is used in the easier to read shorthand, the definition uses only real numbers. You will learn these and be expected to recite them, hopefully not from memorization but instead from learning the language.

• If a function has the form of a quotient such that when evaluated at a particular value both the numerator and the denominator are zero, then the function is undefined for that value. Period. For example, if f(x)=(13x-13)/(x-1), then f(1) is undefined. Note that you cannot simplify f(x)=(13x-13)/(x-1)=13(x-1)/(x-1)=13 because that involves the assertion that (x-1)/(x-1)=1 which is not true for x=1. You can, however, simplify f(x)=13 where x≠1.
• But, as your intuition suggests, it may be possible to evaluate the limit of such a function at that value. For the function above it is obvious that lim_x→1f(x)=13, but that is not the same as saying that f(1)=13. f(1) is still undefined. In precalculus class you will work an interminable number seemingly trivial exercises such as that, but they are designed to teach you the fundamentals and language so that in calculus class you can investigate more interesting ones such as lim_x→0sin(x)/x.

-- ToE 14:06, 7 March 2020 (UTC)

While my brain currently hurts from reading that, you do make a lot of good points. I was thinking of the infinity answer but your lim"f"("x")=13 problem does make sense, so why isn't that the technical answer? Plus, I've always thought that 0/0=0 because 0*0=0 and it is the only number that would work, right? UB Blacephalon (talk) 11:59, 9 March 2020 (UTC)

But 13 *IS* the technical answer to that limit. It just isn't the answer to 0/0. The limit is concerned with the path taken, but the simple division knows only the end state. When you mash those keys on your calculator it can only give you one answer; it can't ask you how you arrived there. There are infinitely many functions which arrive at 0/0, each taking a different path and each having a different value in the limit. No one value is preferred over the others, so the simple division 0/0 is left undefined. -- ToE 16:30, 9 March 2020 (UTC)

With apologies to Thomas Merton. -- ToE 17:35, 9 March 2020 (UTC)

The South Sea King was Act-on-Your-Hunch. He said, "Zero divided by any number is zero. So $\lim \limits _{x\to 0}0/x=0$." The North Sea King was Act-in-a-Flash. He said, "Any number divided by itself is one. So $\lim \limits _{x\to 0}x/x=1$." The King of the place between them was No-Form. 0/0. Undefined. He said nothing. Now the South Sea King And North Sea King Used to go together often To the land of No-Form Where they were joined by the King of the projectively extended real line, Who was fond of saying, "Any number divided by zero is infinity. So $\lim \limits _{x\to 0}x/0=\infty$." No-Form treated them well. So they consulted together They thought up a good turn, A pleasant surprise, for No-Form In token of appreciation. "Quotients," they said, "have a single value For simplicity, for ease of application of the algebraic limit theorem, And so on. But No-Form Has no value. Let’s assign him one." So after that They put holes in each other's arguments, Continuously, for seven days. And when they finished the seventh day, They all lay dead. Lao Tan said: “To oversimplify is to destroy.”

Deep...........OH! You're saying because it has no official answer we can't give it a single quotient! But wouldn't it have infinite answers then because we can't give it a single answer therefore it would be ALL numbers? UB Blacephalon (talk) 19:27, 9 March 2020 (UTC)

The other way around. Because it doesn't have a single quotient (and that's because 0/0=13 is every bit as expectable and correct as 0/0=0), we can't give it an "official" answer. And 0/0=x can't have infinite answers because they would contradict each other. 0*x=0 can have infinite answers because they don't contradict each other (one reason being that x is not alone on one side of the equation). 93.138.43.92 (talk) 20:38, 9 March 2020 (UTC)

Then shouldn't the answer be 13? If not, any special rules we can make up? UB Blacephalon (talk) 16:07, 10 March 2020 (UTC)

13 is just a random number I chose. The part in parentheses should have really been "For any real numbers X and Y, 0/0=X is every bit as expectable and correct as 0/0=Y, and vice versa". You can't choose 0/0=13 because e.g. 0/0=7 makes as much sense, and if you choose both (or even all real numbers) to be correct, then you get the contradiction 0/0=7, 0/0=13 => 7=13. You have to leave it undefined to be consistent. There are no special rules or tricks here.

You can of course create some kind of mathematical framework where nearly everything about real numbers is correct but 0/0 is assigned some real number value (or infinity or zero or whatever you choose), but you'll end up with contradictions and your framework will be inconsistent, making it of little use to anyone. 93.136.6.251 (talk) 01:14, 11 March 2020 (UTC)

We seem to be going in circles here and I don't understand where the hang-up is. Does any of this help?

• Division is an operation which takes two real numbers as input and, where defined, returns one number as output. That one number must be the same value every time the operation is invoked with the same input. Its output value is dependent solely on its specific numeric inputs, and were 0/0 defined, then it would have to be defined as a single particular value and not give a different answer today than it did yesterday.

With division defined as the inverse of multiplication, it lets you undo multiplication. If you have the product x⋅y, you know the value of y, and you want to determine the value of x, then you can compute x⋅y/y=x. For example, 1⋅2=2 and 13⋅2=26, and we can divide by 2 to recover the 2/2=1 and the 26/2=13. If 0/0 were defined, then you should be able to use it to reverse both 1⋅0=0 and 13⋅0=0, so you need 0/0 to be both 1 and and to be 13, but it can only be a single fixed value if it is defined at all. If you arbitrarily chose one value, then if would fail to reverse other products, so division would no longer be the inverse of multiplication (its raison d'être). The only option is to leave it undefined.

• Every number multiplied by zero is zero. So if you take any one unknown number and multiply it by zero you get zero, and all information about that original number is as lost as if it has fallen into a black hole. There is no reversing multiplication by zero, so zero divided by zero is undefined.

• Consider this riddle: I think of my two favorite numbers and write down their value on cards in front of me. When I multiply the first number by 2 I get 14. What was my first number? Easy: You compute 14/2=7. Now I tell you that when I multiply the second number by 0 I get 0. What was my second favorite number? Being able to answer that question is equivalent to defining 0/0. Good luck doing that.

-- ToE 02:52, 11 March 2020 (UTC)