# 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:** 13\ **Page:** 4
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 10:57am UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/61 "2016-09-14T10:57:07Z")
In case it isn’t clear, I of course _do_ want to call -1 a number. But every reason to consider -1 a number is also a reason to consider ∞ a number, and conversely, every objection raised to ∞ as a number could just as well be raised to -1 as a number. > [@eschereal](#): > > No, -1 _is_ a number, from a very useful group of numbers. So is ∞. In fact, -1 is part of many different useful numerical systems, and so is ∞. Most notably, ∞ is part of both the affinely and projectively extended number systems, as well as the transfinite element of the cardinal 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. You can, for instance, have an infinite slope because slope is the ratio of rise to run. Without infinities, slope 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. Right. Or at least, most of the time, this is indeed how we use the term “area” (though sometimes, it is not. Cf. discussions of integration as related to area, where area on one side of an axis is countered with opposite sign from area on the other), and similarly for “length” and so on. > [@](#): > > 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. Well, that would be precisely the context where we would be inclining ourselve sto count area as 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). I don’t understand the point of this statement. What’s wrong with considering a line just a particular degenerate case of a rectangle? That’s common enough in mathematics. * * * Here is an example of the sort of thing people will insist you must say when dealing with ∞: "You said here that the integral from 0 to ∞ of 2[sup]-x[/sup] dx is equal to the difference in evaluating the antiderivative -2[sup]-x[/sup]/ln(2) at x = ∞ and at x = 0, and then plugged in -2[sup]-∞[/sup]/ln(2) - -2[sup]0[/sup]/ln(2) = (1 - 1/2[sup]∞[/sup])/ln(2) = (1 - 1/∞)/ln(2) = (1 - 0)/ln(2) = 1/ln(2). But this is not correct! ∞ is not a number, so this reasoning is erroneous! You must never think of ∞ as a number, only as a limit. Instead, the correct answer is this: The integral from 0 to ∞ of 2[sup]-x[/sup] dx really means the limit of the integral 0 to r of 2[sup]-x[/sup] dx as r grows unboundedly large. This integral is equal to the difference in evaluating the antiderivative -2[sup]-x[/sup]/ln(2) at x = r and at x = 0, which comes out to -2[sup]-r[/sup]/ln(2) - -2[sup]0[/sup]/ln(2) = (1 - 1/2[sup]r[/sup])/ln(2). In the limit as r grows unboundedly large, 2[sup]r[/sup] also grows unboundedly large, and thus 1/2[sup]r[/sup] approaches a limit of zero, and so this overall approaches a limit of (1 - 0)/ln(2) = 1/ln(2). Thus, we see that the actual answer is… 1/ln(2)." Here is the sort of thing the ancients would do in solving a problem like x^2 + 10x + 5 = 6x + 10 “We may first subtract 6x from both sides to get x^2 + 4x + 5 = 10. We may then subtract 5 from both sides to get x^2 + 4x = 5. At this point, we can no longer subtract any further from both sides. Now, recall we have a number of different quadratic formulae, depending on on which side of the = sign the terms of different degrees land. In this case, we will have to use the quadratic formula for the constant term being on one side and the other two terms being on the other side. The general solution to this particular kind of quadratic equation, x^2 + bx = c, is given by x = (sqrt(b^2 + 4c) - b)/2, so we will get that the unique actual solution is x = 1. Had we moved the bx to the other side, we would land instead in a different case of quadratic, with solution given by x = (sqrt(b^2 + 4c) + b)/2, with instead the solution x = 5, and so we might say in some sense there is a phantom ‘opposite-solution’ of x = 5 to our problem, though this does not correspond to any full-on actual solution.” I’m not making this up as merely an illustrative story; this is actually how al-Khwarizmi discussed quadratic equations in the monumentally influential “Liber Algebræ…” from which we get the term “algebra” (and “algorithm”, for that matter). Six different cases of quadratic equations handled separately, based on which sides of the equation the quantities involved arranged themselves on when left positive, directly considering only positive solutions. Negatives weren’t construed as numbers; they arose simply as comparisons, a relation between the two positive sides of an equation, either capable of being larger than the other, but this not corresponding to two signs of number which may always be brought to the one side. Eventually, however, we saw the value in reifying all the rules of reasoning about arbitrarily oriented differences between quantities on different sides into rules of arithmetic for signed quantities. It was the exact same reasoning as we ever carried out circuitously, but now viewed and described less awkwardly. All that blather about differences and sides of equations and so on WAS the rules of arithmetic for signed quantities, not some reason to reject signed quantities from numberhood. And similarly, going back to the ∞ example above: all the blather about limits and and asymptotes and so on IS the rules of arithmetic for ∞s, not some reason to reject ∞ from numberhood. The student who originally phrased their reasoning not explicitly saying “limit” all over the place didn’t make any mistakes; they have internalized such limit-based reasoning into their calculations with the number ∞, just as the student who uses the arithmetic of signed quantities isn’t not in so doing making any mistakes, but has simply internalized appropriate rules into such calculations that previously would be spelt out more cumbersomely. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 11:03am UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/62 "2016-09-14T11:03:12Z")
> [@Indistinguishable](#): > > isn’t not This double negation should have just been a single negation. ---
**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 14, 2016, 12:59pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/63 "2016-09-14T12:59:55Z")
> [@](#): > > Had we moved the bx to the other side, we would land instead in a different case of quadratic, with solution given by x = (sqrt(b^2 + 4c) + b)/2, with instead the solution x = 5, and so we might say in some sense there is a phantom ‘opposite-solution’ of x = 5 to our problem, though this does not correspond to any full-on actual solution." That 5 should of course be -5, in both instances. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 2:24pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/64 "2016-09-14T14:24:45Z")
No, I intended 5 in both instances. “Had we moved the bx to the other side”; thus x^2 = 4x + 5. And “opposite-solutions” to quadratic equations not being actual solutions, but rather just being solutions after such a perturbation. This all being the best I can do to express what is going on if I do not allow myself direct talk of negatives. ---
**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 14, 2016, 4:26pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/65 "2016-09-14T16:26:07Z")
How would al-Khwarizmi have spoken of a quadratic which does have two real positive solutions? If the only formulae available to him only had a single unique solution each, then he would only find one of them, but he would surely acknowledge that the other was also a valid solution. Or would such polynomials fall into a case for which he had two formulae? ---
**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\ **Post date:** [September 14, 2016, 5:05pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/66 "2016-09-14T17:05:08Z")
I’d vote for zero. Going either way, it is either no infinities at all, or an infinity of null quantities ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 7:38pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/67 "2016-09-14T19:38:06Z")
> [@Chronos](#): > > How would al-Khwarizmi have spoken of a quadratic which does have two real positive solutions? If the only formulae available to him only had a single unique solution each, then he would only find one of them, but he would surely acknowledge that the other was also a valid solution. Or would such polynomials fall into a case for which he had two formulae? Yes, there’s only one case where this happens, x^2 + c = bx, and here al-Khwarizmi does indeed note that there are two solutions (which he gives in words as (b/2) ± sqrt((b/2)^2 - c); for familiarity, I wrote everything above in the now more conventional framing of the quadratic formulae as all over a denominator of 2, though this presentational choice is mostly arbitrary). ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [September 14, 2016, 8:18pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/68 "2016-09-14T20:18:32Z")
> [@Indistinguishable](#): > > In case it isn’t clear, I of course _do_ want to call -1 a number. But every reason to consider -1 a number is also a reason to consider ∞ a number, and conversely, every objection raised to ∞ as a number could just as well be raised to -1 as a number. . . . Here’s one: by the Peano Axioms, every number has a successor. (n + 1.) Negative one has a successor: zero. But infinity does not have a successor. (ETA: or it is its own successor; either way, it isn’t a Peano number.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 8:30pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/69 "2016-09-14T20:30:57Z")
Of course infinity isn’t the sort of number the Peano axioms describe; they’re explicitly meant to describe only the natural numbers (or, if you like, we may call these the “Peano numbers”)! Who said anything about the Peano axioms? By the Peano axioms, -1 isn’t allowed either, mind you, as the Peano axioms tells us any property which holds of zero and is closed under successor holds of everything (which the Peano axioms apply to), which means the Peano axioms describe a world in which everything is \>= 0. Also not permitted by the Peano axioms: 1/2; within the framework of the Peano axioms, there is nothing between 0 and its successor 1. The Peano axioms are explicitly meant to describe only finite counting numbers (0, 1, 2, 3, 4, …), and nothing more. But we do not generally restrict the term “number” to this context, and have not for a very, very long time. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2016, 8:38pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/70 "2016-09-14T20:38:01Z")
(Peano himself actually didn’t include 0 in the scope of his Peano axioms, I should say, identifying his starting point with 1 instead. But now we typically use the term “Peano axioms” in such a way as that we consider the starting point to be 0.) Also: > [@Indistinguishable](#): > > The Peano axioms are explicitly meant to describe only finite counting numbers (0, 1, 2, 3, 4, …), and nothing more. Actually, I should say, the Peano axioms were more directly about finite _ordinal_ numbers than the finite _cardinal_ numbers that “counting” may be taken to refer to, though as it happens, these two concepts naturally coincide. ---
**Author:** ![raventhief](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/raventhief/32/4_2.png) [@raventhief](https://boards.straightdope.com/u/raventhief)\ **Post date:** [September 14, 2016, 10:09pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/71 "2016-09-14T22:09:17Z")
> [@dtilque](#): > > +++divide by cucumber error. please reinstall universe and reboot+++ > > How many readers will get that reference? _smacks Hex with cucumber_ ---
**Author:** ![dtilque](https://avatars.discourse-cdn.com/v4/letter/d/d6d6ee/32.png) [@dtilque](https://boards.straightdope.com/u/dtilque)\ **Post date:** [September 15, 2016, 3:06am UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/72 "2016-09-15T03:06:14Z")
> [@raventhief](#): > > _smacks Hex with cucumber_ +++out of cheese error+++ ---
**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 15, 2016, 2:47pm UTC](https://boards.straightdope.com/t/zero-times-infinity/765846/73 "2016-09-15T14:47:00Z")
> [@Indistinguishable](#): > > (Peano himself actually didn’t include 0 in the scope of his Peano axioms, I should say, identifying his starting point with 1 instead. But now we typically use the term “Peano axioms” in such a way as that we consider the starting point to be 0.) > > Also: > > Actually, I should say, the Peano axioms were more directly about finite _ordinal_ numbers than the finite _cardinal_ numbers that “counting” may be taken to refer to, though as it happens, these two concepts naturally coincide. Sing us a song, you’re the Peano Man. [Previous page](https://boards.straightdope.com/t/zero-times-infinity/765846.md?page=3)