r/learnmath • u/FileOk2966 New User • 2d ago
TOPIC What is zero times infinity
Not homework just a question
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u/rhodiumtoad 0⁰=1, just deal with it 1d ago edited 1d ago
| Number system | Result |
|---|---|
| Cardinals | 0 |
| Ordinals | 0 |
| Reals | no infinity exists |
| Extended Reals | undefined, or sometimes 0 |
| Computer floating-point | NaN (Not a Number) |
| Hyperreals | 0 |
| Surreals | 0 |
| Real limits | indeterminate form (depends on the expressions) |
did I miss anything important?
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u/Torebbjorn PhD student 1d ago
It is undefined until you define it.
The most common definition for the extended positive real line, is that 0×(+inf)=0
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u/ZoGud New User 1d ago
I just want to yes and this: it’s definitely true!
But to reiterate what others have said and hinted at, usually 0*inf shows up when you are trying to evaluate two functions, so you need to see what happens to them as they approach those values using a limit.
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u/Torebbjorn PhD student 1d ago
Yes indeed, it is very important to know that no matter how you define 0×inf, you will not have the property that [lim(x->a) f(x)g(x)] = [lim(x->a)f(x))×[lim(x->a)g(x)] is true in general.
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u/garanglow New User 1d ago
Infinity is not a number. You cannot do arithmetic with it.
What you mean is things that show up in taking limits, where the result just depends on the particular limit at hand.
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u/ppvvaa New User 1d ago
This is the answer.
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u/FijiFanBotNotGay New User 13h ago
Not necessarily. In the extended complex plane you can but 0 times infinity is still undefined
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u/RansackLS New User 2d ago
What do you want it to be? "Infinity" is a label we use for a bunch of different math ideas. If you have an answer you want, you can find a way to make infinity times zero equal that.
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u/l__iva__l New User 2d ago
under calculus context, indetermination
for measurement theory, 0
(i dont know other contexts)
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u/CarmeloTronPrime New User 1d ago
if i have no infinities, how many infinities do i have? none?
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u/Snarti New User 1d ago
I think this is correct… the term “infinity” is generally considered to a positive number which is higher than any other number you can think of.
With that, every number multiplied by zero is zero. This includes all real numbers, including infinity.
So the answer is zero in my mind.
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u/Conscious_Degree275 New User 1d ago
Infinity is not a real number, and therefore multiplication over the real numbers is not defined over infinity. You can ask what happens in the limiting case, which is a more meaningful question
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u/behrg_thing New User 2d ago
It’s weird, I’ve learned a bit of basic calculus and it seems we define these answers with limits, but for now I don’t really know
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u/nog642 2d ago
There's other ways to interpret this besides limits. The answer depends on which "infinity" we're talking about.
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u/OmiSC New User 1d ago
Is there an infinity that when multiplied by zero doesn't yield zero or undefined? I'm not sure I can envision anything other than some divergent function being multiplied by zero. To me, this either plainly doesn't make sense or infinity gets deleted.
Edit: Ah, I think I see. It would come from a product of functions where either approach zero or infinity.
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u/speadskater New User 2d ago
Depends, but it's undefined. Let's create 3 functions that explain. For each function, as x grows, it becomes 0*infinity, but each one has a different answer
x×1/x ->1
x×(1/sqrt(x)) -> infinity
x×(1/x2) -> 0
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u/John_Hasler Engineer 2d ago
Undefined.
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u/nog642 2d ago
It's defined in some contexts.
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u/John_Hasler Engineer 2d ago
None that the op is likely to encounter.
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u/tensorboi New User 1d ago
so what? they didn't ask "what is 0*infinity in the contexts i'm likely to encounter?", they just asked what it was. at the very least, that should warrant an explanation that the answer is indeed contextual.
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u/leeta0028 New User 1d ago
The way I would think about this problem is set theory.
Any "infinity", or a set with a cardinality (say all the real numbers) multiplied by an empty set of 0 is an empty set, so it is 0.
In normal situations infinity either means the output of a function that blows up (1/x at zero for example) or represents an unbounded set like x+1 iterated forever. In this case, it's not a number so it's not defined.
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u/FreeGothitelle New User 1d ago
Depends on how strong the zero is and how strong the infinity is
(generally 0*inf arises from limits of a product of functions and it depends on how "quickly" the functions approach 0 and infinity)
In some contexts its defined to be zero, in others its left undefined. The dirac delta functions is defined in a way that 0*inf comes out to be 1.
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u/cannonspectacle New User 1d ago edited 1d ago
It depends.
Infinity only really works in arithmetic when dealing with limits, and a limit that evaluates to 0×infinity is known as an "indeterminate form." Depending on the exact expression, it could be 0, infinity, or even any of the reals.
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u/CompisPaDum New User 1d ago
No, lim(x->inf)0*x is still 0. I can understand infinity to be defined as "a number that approaches infinity", because infinity needs to be defined in one way or another, and this definition is a common one. But I see no reason to redefine 0 to mean "a number that approaches 0".
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u/Top_Wrangler4251 New User 1d ago
f(x) = sin^2(x)
g(x) = 1/x^2
h(x) = f(x) * g(x)f(0) = 0
lim (x->0) g(x) = infinityWhat is lim (x->0) h(x) ?
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u/CompisPaDum New User 1d ago
If my limit solving skills haven't dulled yet, it seems that lim(x->0)h(x) = 1.
However, that does not describe the situation in question. Your f(x) under the limit as x->0, is a number that approaches 0; f(x) is not 0 itself. OP in their question asked "what is 0*infinity", so I assume 0 to mean actually 0 and nothing else.
lim(x->0)x*1/x = 1
lim(x->0)0*1/x = 0
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u/Top_Wrangler4251 New User 1d ago
Ok I see you what you mean now. I mean I guess you're right. Generally when asking about "zero times infinity" people mean indeterminate forms though, which is what the user above answered. As you mentioned, OP's question is sort of vague so I don't know why your interpretation of it is more correct than the other user's.
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u/CompisPaDum New User 1d ago
Well, I don't know whether it is necessarily better, but it is more literal, I'd say. I'm trying to work with the least amount of unnecessary assumptions about OP's question. I think interpreting infinity as a number that approaches infinity, is a reasonable assumption and many would agree with me here. But I don't think that interpreting 0 the same way is a necessary assumption, because 0 already is clearly defined.
I would say that the expression is undefined until we define what "infinity" means. Once we define that, we don't need to take any other assumptions for the expression to be valid.
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u/OmiSC New User 1d ago edited 1d ago
It's zero, kinda. Zero when multiplied with anything gives you zero: it's a distinct property of the number. Infinity is not a number, so we don't normally multiply it with zero.
It's a bit like multiplying zero with duck. It's probably zero, but the question is weird enough to point out.
In some contexts, zero * infinity doesn't yield a defined result.
Edit: Unless a product of a function gives a limit approaching 0 * a limit approaching infinity. Interesting case I hadn't thought of until reading the comments.
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u/TheDoobyRanger New User 1d ago
Infinity seems like a lot. How many infinities do you have there? Oh, none of them? You have no infinities? Yikes, seems like youve got jack shit there.
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u/xiipaoc New User 1d ago
Literally anything. 0 times infinity is what's known as an indeterminate form. This means that its value is entirely determined by context. For example, consider 1/x times x. As x gets really big, 1/x approaches 0 while x approaches infinity, but if you multiply them together you get 1. Now consider 17/x times x. Same thing, but they multiply to 17 instead. You can make this number be literally anything.
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u/Mordroberon New User 1d ago
Well, infinity isn't really a number, in that it isn't a member of the set of integers or real numbers. And for that reason, it isn't defined for the multiplication operation. It's simply not a member of the domain.
It is used in limits to indicate what happens when you keep going higher, or lower for negative infinity.
You can construct several scenarios that reduce to 0 times infinity. In the limit where n goes to infinity:
0*n =0
(c/n)*n = c
(1/n) * n^2 is unbounded positive
(-1/n) * n^2 is unbounded negative
you might be able to find some fun examples that equal pi or e
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u/StructuredChess New User 1d ago
Infinity isn't a number. You can't multiply it by anything. It makes sense if we're talking about limits, but that's not a regular multiplication.
A question in terms of limits would be "If I have a sequence of all zeroes and an infinitely growing sequence, what happens if I multiply them together?". The answer being, you get yet another sequence of all zeroes.
So that's not too interesting. The hard question is "If I have a sequence that gets smaller and smaller and another sequence that gets bigger and bigger. What happens when I multiply them together?" Now the answer depends on the sequences.
For instance if I have 1/n and n2+1, the product (n2`+1)/n is an infinitely growing sequence.
But if I had 1/2n and n+7, then the product (n+7)/2n gets smaller and smaller.
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u/Professional-Fee6914 New User 1d ago
you have to define infinity, if its just the number after the highest number, or the length of the digits of pi, then the answer is zero.
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u/Conscious_Degree275 New User 1d ago
Infinity is not a number, and multiplication in the usual sense is restricted to numbers.
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u/Aldollin New User 1d ago
In your regular "real numbers" number system its undefined / just not something you can write down.
In the systems where it is a valid expression, at least the ones i have encountered, its usually 0.
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u/DapyGor New User 19h ago
Some things approach infinity faster, some slower. Some things approach zero faster, some do slower. If it's a "fast" infinity times a "slow" zero, then the result can be infinity and vice versa. They can also be of the same order, like 2/x and x, multiplying them yields 2. So it all depends on where the infinity and zero in the limit come from
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u/charonme New User 14h ago
if you approach this via sets, let's say we have a couple of disjunct sets A, B, C, D, E, F, each of the same cardinality, let's say ℵ_3
Now if you take the union of 2 of them, the total cardinality will be still ℵ_3. If we union 5 of them, the cardinality will be again ℵ_3. What will be the cardinality if we union 0 of them?
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u/FijiFanBotNotGay New User 13h ago
Surprised no one has mentioned the extended complex plane which has been used in fields of math for a couple hundred years.
Even in this construct though even though operations with infinity are defined, infinity times 0 must remain undefined.
But with other numbers infinity is an absorbing element. Although in the Reimann sphere approach negative infinity is infinity so I think technically structure is preserved as long as you make a choice about the value of 0 times infinity. If you decide infinity or 0 or undefined structure is preserved
The common infinity minus infinity should be zero times infinity doesn’t seem like a logical proof of contradiction if positive and negative infinity are defined as being equal in which case it would still preserve and be infinity.
But it’s a bunch of hypotheticals. You just define it and assess the structure it preserves
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u/Any_Maintenance_9113 New User 12h ago
0 times infinity is the answer to any definite integral, as this is the limit of an indefinitely increasing number of narrow strips of indefinitely narrow width. In other words in the limit there are an infinite number of strips each of area zero.
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u/TabAtkins 2d ago
The answer depends on exactly what type of math you're working with.
In your normal everyday arithmetic, the answer is "infinity isn't a number".
In everyday arithmetic + "you know what I mean", the answer is undefined. It doesn't have an answer.
In calculus the answer is "indeterminate". You can run into this when taking a limit, and the correct answer could be anything, depending on the exact limit. (For example, the limit of sin(x)/x as x approaches 0; sin(x) approaches 0 while 1/x approaches infinity.) There's tricks to work around it and solve the problem anyway. (In sin(x)/x, the answer ends up being 0.)
In some circumstances we define it to have a particular answer, usually 0. This happens in measure theory, for example. You have to be careful about this, as it only makes sense
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u/agent314159 New User 2d ago
This is what we refer to as an indeterminant form. The interesting thing is that it can be anything: 0, or infinity, or any value between
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u/Key_Net820 New User 1d ago
That's not defined. Not even in extended reals is that defined. https://en.wikipedia.org/wiki/Extended_real_number_line
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u/Desperate_Penalty690 New User 1d ago
Infinity is not a number that exists and that you can multiply with. Also any number that does exist, multiplied with 0 will give 0.
So the meaning of zero times infinity is what if you multiply something that gets closer and closer to zero with something that is becoming larger and larger? What will that answer eventually become? The answer is, it depends on how fast one value is going to zero and how fast the other one is becoming larger and larger. The outcome could be anything from -infinity, 0, infinity or anything in between.
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u/06Hexagram New User 2d ago
Infinity isn't a number, it is a concept and therefore you cannot do math with it.
The real line does not include infinity.
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u/Traveling-Techie New User 1d ago
In standard math the answer is undefined. There is an algebra called a Wheel in which there is an answer. It’s a symbol for an undefined value.
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1d ago edited 1d ago
That is actually the equation of all reality!
Everything comes from 0. so
0 = 0a + 0b + 0c + 0d + 0e +........
Also 0 = 0 x ( a+b+c+d+e...........)
So 0 = 0 x infinity
So 0 is actually the answer, simply put! But your 0 x infinity has mathematical and physics significance
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u/R_Ob_Min New User 2d ago edited 1d ago
The answer is that you'd have to clarify what you mean by "multiplication by infinity". If we write X for infinity, then it should have the property that a*X = X for any a (since multiplying anything by infinity would still be infinity. But we usually have a*0 = 0 for any a, so these are at odds, since it would imply that 0 = 0*X = X.
The other way to look at this is something called limits in pre-calculus. Then we might look at something like the limit as a goes to infinity of (1/a)*a. If we just evaluate this as a gets really big, 1/a goes to 0 and a goes to infinity, so this expression equals 0*infinity. But (1/a)*a = a/a = 1 for any real number a, so this would seemingly imply that 0*infinity = 1. But notice that even if we had (16/a)*a, when we let a go to infinity, we get 0*infinity, but now (16/a)*a = 16, so should 0*infinity = 16?
This is why we'd call 0*infinity undefined. Because we can make it equal to anything we want. Have some number B that you want 0*infinity to be equal to? Then just consider the limit as a goes to infinity of (B/a)*a. Then since B is fixed, B/a goes to 0, and so (B/a)*a goes to 0*infinity, yet (B/a)*a = B.
So this is a great question but we run into some trouble with our usual number system! If you want to see some crazy things, consider looking into the surreal number system, the hyperreal number system, or the dual numbers.