# Zero Times Infinity. **URL:** \ **Category:** Factual Questions\ **Created:** [September 13, 2016, 12:02pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846 "2016-09-13T12:02:54Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![dougie\_monty](https://avatars.discourse-cdn.com/v4/letter/d/439d5e/32.png) [@dougie\_monty](https://boards.straightdope.com/u/dougie_monty)\ **Post date:** [September 13, 2016, 8:46pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/41 "2016-09-13T20:46:04Z")
> [@Indistinguishable](#): > > […] **dougie\_monty** clearly did not intend to restrict themself […] I am not concerned with your poor grammar. What is that supposed to mean–the plural pronoun? Unless you want me to report this post I expect you to explain this phrasing immediately! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 8:55pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/42 "2016-09-13T20:55:42Z")
Er, I think you’re joking, but in case you’re serious: though your username is quite possibly your real name and therefore perhaps an indication of your gender, I don’t actually know it and didn’t want to presume, and thus used singular “they”, as I frequently do without any great thought put into it. ---
**Author:** ![TroutMan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/troutman/32/6721_2.png) [@TroutMan](https://boards.straightdope.com/u/TroutMan)\ **Post date:** [September 13, 2016, 9:02pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/43 "2016-09-13T21:02:55Z")
> [@Indistinguishable](#): > > All numbers are concepts. There is no reason to single out ∞ as somehow non-numbery but i and -3 and 1/2 and 0 and 7 and all the rest of them as perfectly numbery. Sure there is. If I use the numbers -3 or 0 or √5 or 6_i_, they mean exactly the same thing no matter how I use them. But “infinity” doesn’t have a single value. The number of integers and the number of real numbers are both infinite, but those are different values. Now, depending on how you define “number”, that might be a distinction without a difference. But to say that claiming infinity is not a number is the same as claiming 0 is not a number isn’t fair. It’s difficult to find definitions of number that don’t include 0; there are plenty of definitions of number that [don’t include infinity](http://mathforum.org/dr.math/faq/faq.large.numbers.html). ---
**Author:** ![dougie\_monty](https://avatars.discourse-cdn.com/v4/letter/d/439d5e/32.png) [@dougie\_monty](https://boards.straightdope.com/u/dougie_monty)\ **Post date:** [September 13, 2016, 9:19pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/44 "2016-09-13T21:19:42Z")
> [@Indistinguishable](#): > > Er, I think you’re joking, but in case you’re serious: though your username is quite possibly your real name and therefore perhaps an indication of your gender, I don’t actually know it and didn’t want to presume, and thus used singular “they”, as I frequently do without any great thought put into it. I explained in a thread about Usernames, several years ago, that mine was sort of a shortened version of the name I most often go by, Doug Montgomery. It would take quite a stretch of the imagination to construe ‘my gender’ as other than male. Do we understand each other? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 9:29pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/45 "2016-09-13T21:29:01Z")
> [@TroutMan](#): > > Sure there is. If I use the numbers -3 or 0 or √5 or 6_i_, they mean exactly the same thing no matter how I use them. But “infinity” doesn’t have a single value. The number of integers and the number of real numbers are both infinite, but those are different values. > > Now, depending on how you define “number”, that might be a distinction without a difference. But to say that claiming infinity is not a number is the same as claiming 0 is not a number isn’t fair. It’s difficult to find definitions of number that don’t include 0; there are plenty of definitions of number that [don’t include infinity](http://mathforum.org/dr.math/faq/faq.large.numbers.html). There are plenty of notions of number that don’t include 6i or sqrt(5) or -3 or 1/2 or all the rest of them, as well. Hence people calling “6i” imaginary in the first place. If numbers are meant to count sizes of sets, as they were originally intended to do, well, all of these are right out. None of these have anything to do with that, nor make any sense on such an account of what numbers are. But all this means is that there are different kinds of different numbers, for different purposes. Speaking of sizes of sets, the number of integers and the number of real numbers are only different if you are choosing to count them as different; if you’d like to work in a system of cardinalities in which all infinite cardinalities are identified (a perfectly coherent and reasonable thing to do, for many purposes), they are the same. But, again, counting sizes of sets is not the only thing numbers do, given such numbers as sqrt(5), etc. In fact, you say sqrt(5) means exactly the same thing, no matter how you use it. I disagree (or rather, I say this depends on what “means exactly the same thing” means, and only amounts to tautology; yes, sqrt(5) is always sqrt(5), and in the same sense, ∞ is always ∞). Are there not multiple square roots of 5? Not only 2.2360679775 and -2.2360679775, but also 4 and 7 in mod 11 land, and the matrix [[1, 2], [2, -1]] among many others, and so on and so on. Shall we say then that “sqrt(5)” then is not a number but a process, and only 2.2360679775 is a number proper? You can contort yourself like that if you want, but this is madness. sqrt(5) refers to a number, or rather, a multiplicity of numbers in different numerical systems. And the same is true of all the rest of them. The cardinality 6 (existing within a discrete context in which there is no such thing as -1 or 1/2), the ordinal 6, 6 as a measure of ratios of unsigned lengths, 6 as a signed integer, 6 as an entity within the “complex numbers”… all of these are in some sense different concepts under the umbrella of 3, and none of this is to deny that 6 is a number. (And at times, we may wish to draw finer distinctions that are coarsely lumped together under 6: when we say that the limiting value of (x - sin(x))/x^3 at x = 0 is 6, we are lumping together behaviors like “oscillates around”, “approaches always staying under”, “approaches and actually reaches”, etc., all into the single entity 6, which at other times we may wish to distinguish between. Again, none of this means 6 isn’t a number (or if you like, many different numbers in different contexts)). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 9:30pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/46 "2016-09-13T21:30:21Z")
> [@dougie\_monty](#): > > I explained in a thread about Usernames, several years ago, that mine was sort of a shortened version of the name I most often go by, Doug Montgomery. It would take quite a stretch of the imagination to construe ‘my gender’ as other than male. > Do we understand each other? I’m sorry for causing offense by refraining from assuming things about a stranger I did not know. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 9:35pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/47 "2016-09-13T21:35:01Z")
> [@Indistinguishable](#): > > Shall we say then that “sqrt(5)” then is not a number but a process Rather, “[not a number but a concept](http://boards.straightdope.com/sdmb/showpost.php?p=19625998&postcount=29)” > [@Indistinguishable](#): > > (x - sin(x))/x^3 The numerator and denominator here should be switched (but anyone who cares already realized that). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 9:37pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/48 "2016-09-13T21:37:57Z")
_Sorry, ignore this_ ---
**Author:** ![Hector\_St\_Clare](https://avatars.discourse-cdn.com/v4/letter/h/b5ac83/32.png) [@Hector\_St\_Clare](https://boards.straightdope.com/u/Hector_St_Clare)\ **Post date:** [September 13, 2016, 9:48pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/49 "2016-09-13T21:48:28Z")
> [@Francis\_Vaughan](#): > > I’m trying to avoid i/0 being infinity, but it occurred to me that the objection to 1/x goes away for the case where it is 0/x as we don’t have the problem of it having different signs on each side - we can collapse onto 0 from both sides. Thus we can side step the need to worry about an undefined number. > > For infinite and transfinite I would be making an argument that we are using the 1-1 correspondence definition of cardinality. For every possible pairing of numbers the answer is zero. The transfinite numbers are built from the powerset of infinity. It is a constructive process. I would say that if multiplication by zero has essentially the semantics “none of these” it doesn’t matter how you construct the transfinite number, none of them is zero. You’re approaching the concept of a limit, but you’re doing it improperly. (Which is OK: most people before Newton, even very smart ones, didn’t really know how to think about limits). Again, look up L’Hopital’s rule to understand how to properly think about ratios involving zero or infinite quantities. The issue here is that different ratios may both involve infinite quantities (and thus look, in some sense, like infinity / infinity), but they might converge to very different limits, and that’s why infinity \* zero is undefined. ---
**Author:** ![dougie\_monty](https://avatars.discourse-cdn.com/v4/letter/d/439d5e/32.png) [@dougie\_monty](https://boards.straightdope.com/u/dougie_monty)\ **Post date:** [September 13, 2016, 9:49pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/50 "2016-09-13T21:49:23Z")
> [@Indistinguishable](#): > > I’m sorry for causing offense by refraining from assuming things about a stranger I did not know. Fine. 🙂 ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [September 13, 2016, 9:51pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/51 "2016-09-13T21:51:57Z")
I am not a mathematician so whatever I type is going to be deeply flawed, but I am having a hard time figuring out how zero times infinity could be anything other than zero. The zero should negate any sum no matter how big or what it is. Nothing multiplied by something is nothing. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 10:02pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/52 "2016-09-13T22:02:12Z")
Let’s put it this way: zero times infinity definitely encompasses zero (and for the very reasons you think it does), but (in suitable contexts) also other possible values as well. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 10:07pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/53 "2016-09-13T22:07:12Z")
So many typos… In addition to the previously noted two, > [@Indistinguishable](#): > > Hence people calling “6i” imaginary in the first place. Scare quotes belong on “imaginary”, not “6i”. > [@](#): > > 2.2360679775 Should have a trailing ellipsis in each instance. > [@](#): > > all of these are in some sense different concepts under the umbrella of 3, and none of this is to deny that 6 is a number. “3” here should of course have been “6” ---
**Author:** ![eschereal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/eschereal/32/18939_2.png) [@eschereal](https://boards.straightdope.com/u/eschereal)\ **Post date:** [September 13, 2016, 10:31pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/54 "2016-09-13T22:31:20Z")
Not sure about zero, but I am certain that infinity times eight is cloverleaf. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 13, 2016, 10:50pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/55 "2016-09-13T22:50:53Z")
Well, tilted cloverleaf. To get a conventional cloverleaf, you also have to multiply it by sqrt(i). ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [September 13, 2016, 11:24pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/56 "2016-09-13T23:24:42Z")
> [@Indistinguishable](#): > > And the same is true of all the rest of them. The cardinality 6 (existing within a discrete context in which there is no such thing as -1 or 1/2), the ordinal 6, 6 as a measure of ratios of unsigned lengths, 6 as a signed integer, 6 as an entity within the “complex numbers”… all of these are in some sense different concepts under the umbrella of 3, and none of this is to deny that 6 is a number. (And at times, we may wish to draw finer distinctions that are coarsely lumped together under 6: when we say that the limiting value of (x - sin(x))/x^3 at x = 0 is 6, we are lumping together behaviors like “oscillates around”, “approaches always staying under”, “approaches and actually reaches”, etc., all into the single entity 6, which at other times we may wish to distinguish between. Again, none of this means 6 isn’t a number (or if you like, many different numbers in different contexts)). I think the problem in this thread and many like it is that you are defining “number” in mathematics and other posters are defining “number” in arithmetic. You simply aren’t speaking the same language. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [September 13, 2016, 11:37pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/57 "2016-09-13T23:37:40Z")
r = |sin 2Ω| is the equation for a cloverleaf in polar … ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 13, 2016, 11:53pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/58 "2016-09-13T23:53:55Z")
> [@Exapno\_Mapcase](#): > > I think the problem in this thread and many like it is that you are defining “number” in mathematics and other posters are defining “number” in arithmetic. You simply aren’t speaking the same language. I don’t see that arithmetic vs. wider math makes much difference here. In even grade school arithmetic, we already have a multitude of number concepts: numbers that count “How many?” (0, 1, 2, 3, ), numbers that count differences (here come negatives), numbers that count ratios (here come fractions). We see already that the basic concept of number expands and expands to cover new abstractions. And for many of these applications, incidentally, it would not be unreasonable to consider the particular abstraction of infinity as a number: If numbers mean only “How many?”, then -1 and 1/2 aren’t numbers. And ∞ is a number. (“How many primes are there?”) “No, 1/2 is a number because you can have something 1/2 a foot long”. Oh, ok. You can have one length that’s 1/2 as long as another, so 1/2 is a number. Fine. If numbers mean only ratios of lengths, then, again, -1 isn’t a number, and ∞ is. (“How many times as long as a point is a stick?”) “No, -1 is a number because I can owe you one dollar.” Well, yeah… money’s a game whose rules we’ve made up. If we wanted, you could owe me red banana dollars, and have rules for how to handle that situation. “Well, ok, -1 is a number because it’s what you get when you subtract 6 from 5”. One might say “You can’t subtract 6 from 5. Sure, it looks like you can, but properly speaking, we must say there’s no such thing as the number -1, just the concept of the difference, between something and something one higher. -1 isn’t a number; it’s a difference. Just like ∞ isn’t a number; it’s a limit”. But this is a foolish way of hamstringing our perspective and our language. In arithmetic, we already see that there are multiple kinds of numbers, useful for different purposes, and thus should already be comfortable with the idea that we could make new numerical systems encapsulating ∞ cleanly. And so we might well consider ∞ a number, in just the same way as all these others. ---
**Author:** ![eschereal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/eschereal/32/18939_2.png) [@eschereal](https://boards.straightdope.com/u/eschereal)\ **Post date:** [September 14, 2016, 1:02am UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/59 "2016-09-14T01:02:34Z")
No, -1 _is_ a number, from a very useful group of numbers. You can, for instance, have a negative velocity because velocity is the vector quantity of speed. Without negatives, vector calculations (like composing billiards shots or plotting space junk trajectories), would be a pain in the ass. But, for example, a rectangle of, say, 5 x -3, does not have an area of -15 because negative area does not make sense. Even if it is being used in some sort of math where it offsets with rectangles of positive dimension, its area is still not negative. But if you had a rectangle of 0 x ∞, you would not actually have a rectangle, so you could not express its area in any meaningful way (you would have a line). ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [September 14, 2016, 7:24am UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/60 "2016-09-14T07:24:47Z")
> [@Velocity](#): > > I am having a hard time figuring out how zero times infinity could be anything other than zero. The zero should negate any sum no matter how big or what it is. Nothing multiplied by something is nothing. Zero times anything is zero. But also, infinity times anything is very big. Something’s got to give way. [Cite.](https://www.youtube.com/watch?v=PNoEGTZzaAE&t=12) [Previous page](https://boards.straightdope.com/t/zero-times-infinity/765846.md?page=2) [Next page](https://boards.straightdope.com/t/zero-times-infinity/765846.md?page=4)