# Infinity, squared? **URL:** \ **Category:** Factual Questions\ **Created:** [September 27, 2003, 11:00pm UTC](https://boards.straightdope.com/t/infinity-squared/204219 "2003-09-27T23:00:24Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![mangeorge](https://avatars.discourse-cdn.com/v4/letter/m/d78d45/32.png) [@mangeorge](https://boards.straightdope.com/u/mangeorge)\ **Post date:** [September 27, 2003, 11:00pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/1 "2003-09-27T23:00:24Z")
I was reading an article, [multiverse](http://www.ravenblack.net/thoughts/multiverse.html), in which the author mentions infinity squared. I’ve encountered this before, and it would appear that some math geeks believe that infinity and infinity squared can indeed exist seperately. How so? ---
**Author:** ![Not\_In\_Anger](https://avatars.discourse-cdn.com/v4/letter/n/3ab097/32.png) [@Not\_In\_Anger](https://boards.straightdope.com/u/Not_In_Anger)\ **Post date:** [September 27, 2003, 11:06pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/2 "2003-09-27T23:06:46Z")
Infinity squared equals infinity. Just like zero times zero equals zero. I wish math questions were usually as user friendly as that. ---
**Author:** ![mangeorge](https://avatars.discourse-cdn.com/v4/letter/m/d78d45/32.png) [@mangeorge](https://boards.straightdope.com/u/mangeorge)\ **Post date:** [September 27, 2003, 11:14pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/3 "2003-09-27T23:14:24Z")
Well I agree, **Not In Anger**. But I’ve read where some think differently. I guess I _could_ find a cite. BTW; One times one equals one. One is closer to infinity than to zero. ---
**Author:** ![BarnOwl](https://avatars.discourse-cdn.com/v4/letter/b/e95f7d/32.png) [@BarnOwl](https://boards.straightdope.com/u/BarnOwl)\ **Post date:** [September 27, 2003, 11:22pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/4 "2003-09-27T23:22:30Z")
Boring so far. ---
**Author:** ![Apex\_Rogers](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@Apex\_Rogers](https://boards.straightdope.com/u/Apex_Rogers)\ **Post date:** [September 27, 2003, 11:30pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/5 "2003-09-27T23:30:27Z")
I believe infinity squared is a separate entity from infinity because of the fact that it can be either the result of multiplying infinity \* infinity or -infinity\*-infinity. Since infinity and negative infinity both exist separately, then infinity ^ 2 must be different. Just a guess/hazy memory of what a professor once said. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [September 27, 2003, 11:38pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/6 "2003-09-27T23:38:01Z")
As cardinal numbers, they are the same, but as ordinals, they are different. As you may know, there are different orders of infinity. For simplicity, I’ll just take omega (or aleph-naught, as some might call it), the set of natural numbers: {0, 1, 2, 3,…} omega[sup]2[/sup] is different from omega–think of it as omega x omega, or omega copies of omega. It looks like this: {0, 1, 2, 3,… omega, omega + 1, omega + 2, omega + 3,… omega+omega, omega+omega+1, omega+omega+2, omega+omega+3,… omega+omega+omega, omega+omega+omega+1,… . . } In other words, picture a “copy” of the natural numbers. Next, picture another copy of the natural numbers after _that_. And another after _that_, and another after _that_,… So that, ultimately, you have as many copies of the natural numbers, one after the other, as there were natural numbers _themselves_ in the first place. That is the ordinal known as omega[sup]2[/sup]. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [September 27, 2003, 11:43pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/7 "2003-09-27T23:43:08Z")
There exists a bijection between the Cantor set, which has outer measure zero, and the Cartesian product of countably infinite copies of the real line, each of which has outer measure infinity. This bijection, sometimes called the Peano space-filling curve, demonstrates that the two sets are in some sense the same size. Both have the cardinality of the continuum, c. Compared to this craziness, infinity^2 = infinity is considerably easier to wrap your mind around. ---
**Author:** ![mangeorge](https://avatars.discourse-cdn.com/v4/letter/m/d78d45/32.png) [@mangeorge](https://boards.straightdope.com/u/mangeorge)\ **Post date:** [September 27, 2003, 11:47pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/8 "2003-09-27T23:47:43Z")
Here’s [one](http://www.student.math.uwaterloo.ca/~jjeellio/IM1concepts.html) idea. ---
**Author:** ![mangeorge](https://avatars.discourse-cdn.com/v4/letter/m/d78d45/32.png) [@mangeorge](https://boards.straightdope.com/u/mangeorge)\ **Post date:** [September 27, 2003, 11:51pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/9 "2003-09-27T23:51:45Z")
> [@](#): > > \*Originally posted by Antiochus \* > \*\*Boring so far. \*\* So jazz it up a little. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [September 28, 2003, 2:00am UTC](https://boards.straightdope.com/t/infinity-squared/204219/10 "2003-09-28T02:00:13Z")
**amore ac studio** , I thought a space filling curve was a continuous map from the reals onto the plane (or some higher dimension). As for the OP, in addition to what I said above, I suppose it’s also possible that the writer confused infinity^2 with 2^infinity. For any cardinal number (including infinite) x, 2[sup]x[/sup] is always strictly larger than x itself. For example, omega (the natural numbers as I mentioned above). If you take 2[sup]omega[/sup] (which can be described as the collection of _all possible subsets_ of the natural numbers), that’s a strictly larger infinity than omega itself. 2[sup]omega[/sup] is also the cardinality of the reals (so the “size” of the set of reals is strictly larger than the “size” of the set of natural numbers). Unfortunately, we have _no idea_ how much larger 2[sup]omega[/sup] (size of the reals) is than omega (size of the naturals). 2[sup]omega[/sup] could be the next higher cardinal than omega, known as omega-1 (this is known as the Continuum Hypothesis), or it could be _much_ larger. In fact, it’s consistent with standard (ZFC) set theory for 2[sup]omega[/sup] to be _any_ cardinal larger than omega, so long as that cardinal has cofinality omega. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [September 28, 2003, 2:51am UTC](https://boards.straightdope.com/t/infinity-squared/204219/11 "2003-09-28T02:51:52Z")
> [@](#): > > _Originally posted by Cabbage_ > I thought a space filling curve was a continuous map from the reals onto the plane (or some higher dimension). That’s the usual definition, which is why I said the bijection from the Cantor set to R[sup]infinity[/sup] was only _sometimes_ called the space-filling curve. You’ll note that a bijection from the Cantor set to R[sup]n[/sup] can be constructed via the composition of the [Devil’s staircase](http://users.forthnet.gr/ath/jgal/math/cantor.html) (which is continuous and monotone increasing) with a bijection from [0,1] to R, followed by the usual space-filling curve for R[sup]n[/sup]. The construction of a bijection from the Cantor set to R[sup]infinity[/sup] is roughly analogous. ---
**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 28, 2003, 2:53am UTC](https://boards.straightdope.com/t/infinity-squared/204219/12 "2003-09-28T02:53:36Z")
From the [Introduction](http://www.student.math.uwaterloo.ca/~jjeellio/IMintro.html) of the page that **mangeorge** cited: > [@](#): > > ``` > For the purposes of Infinite Math, it must be expressed that infinity can be acted upon by mathematical operations, just as any finite value can be acted upon by mathematical operations. For example, six times infinity is six infinity. It cannot be assumed, as in Finite Math, that since infinity is a concept and not a reachable value, infinity is equivalent to six infinity, nor can it be assumed that infinity is equivalent to infinity plus one. Only infinity plus one is equal to infinity plus one in Infinite Mathematics. It is easily seen by this definition that there are numbers greater than infinity, since infinity plus one is farther along the number line. These are infinite numbers, and they are discussed in Chapter 3. > > ``` I’m not about to read Chapter 3 at this hour, and I don’t know whether this is just crackpottery or not, but from this paragraph it is clear that his Infinite Mathematics is distinctly not the infinite math that Cantor defined. ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [September 28, 2003, 5:26am UTC](https://boards.straightdope.com/t/infinity-squared/204219/13 "2003-09-28T05:26:27Z")
There are several ways to extend the arithmetic operations to apply to infinite numbers. **Cabbage** has already discussed Cantor’s cardinal and ordinal numbers (where infinite numbers are called transfinite). Cantor’s cardinal and ordinal numbers are discrete. John Conway’s surreal numbers are continuous. You can do just about any operation on his infinite numbers that you can do on finite numbers – take square roots, subtract them, divide by them. Any surreal number can be squared, and if the number is greater than 1, the square will be greater than the number. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [September 28, 2003, 5:47am UTC](https://boards.straightdope.com/t/infinity-squared/204219/14 "2003-09-28T05:47:29Z")
The author of the page **mangeorge** cited mentions in the first few paragraphs that it’s obvious that 1/infinity = 0. I’m going to look it over tomorrow, but I’m already a bit cautious. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [September 28, 2003, 6:46am UTC](https://boards.straightdope.com/t/infinity-squared/204219/15 "2003-09-28T06:46:37Z")
Frankly, it looks like the author of the page is either: [ul] [li]Not using the concept of infinity the way we have used it in all previous threads here, and the way I’ve always seen it used in the serious texts, or[/li][li]Completely incompetent at the math he professes to teach to others and is making it up as he goes along.[/li][/ul]It’s a pity that more of the site isn’t up yet, but when people begin to say that they’ve [developed a whole new branch of math](http://www.student.math.uwaterloo.ca/~jjeellio/int&ab.html), I begin to wonder about how many wheels they have on the ground. Quote of interest: > [@](#): > > his is an extremely new branch of mathematics that a good friend of mine, JAMES R. WALLACE, and I have recently developed. It has the significance that Calculus has had, since Newton first used it for motion problems. Infinite Math is very complex and is often very difficult to understand. This is partially due to its very recent discovery, but also due to the abstract nature of infinite concepts. Eh, very few things are as significant to math as the Calculus. _Damned_ few, in fact. Makes one wonder why I’ve not heard of it, or of these people, before. Anyway, let’s take a little example from the site: As he says (and has been quoted in this thread), normal mathematical operators work on infinity. He also states that 1/inf = 0. Therefore, let’s do a little algebra. inf \* 1/inf = 0 \* inf 1 = 0 Ah, a little inconsistency! Unless he wants to retract one or more of the above assumptions he’s made, he’s fallen into a fairly serious trap. ---
**Author:** ![Eleusis](https://avatars.discourse-cdn.com/v4/letter/e/f4b2a3/32.png) [@Eleusis](https://boards.straightdope.com/u/Eleusis)\ **Post date:** [September 28, 2003, 6:52am UTC](https://boards.straightdope.com/t/infinity-squared/204219/16 "2003-09-28T06:52:48Z")
I’m not a mathematician but it seems to me that anything other than zero _times_ infinity = infinity. p.s. I only entered this thread because the board lied and told me it had zero posts. I wanted to be the first with a witty remark after a ham(p)ster eaten OP. ☹ ---
**Author:** ![Sqwert](https://avatars.discourse-cdn.com/v4/letter/s/a9adbd/32.png) [@Sqwert](https://boards.straightdope.com/u/Sqwert)\ **Post date:** [September 28, 2003, 8:00am UTC](https://boards.straightdope.com/t/infinity-squared/204219/17 "2003-09-28T08:00:53Z")
> [@](#): > > Therefore, let’s do a little algebra. > > inf \* 1/inf = **0 \* inf** > 1 = **0** Derleth, **zero \* infinity** is not necessarily zero. It is usually undefined (or indeterminate). That is precisely how integral calculus works. Integral calculus shows instances where you can define it. Or more to the point how you can calculate it. Take for example determining the area under a curve. Calculus takes the area and breaks it up into an infinite number of parts, each with zero thickness (and therefore 0 area). Totaling up all the areas you get : 0 + 0 + 0 + 0 + 0 + … (an infinite number of times or [0 \* infinity] ) = the area under the curve. Depending on the curve, that area can be zero, +/- infinity, or any number in between. a visual representation: [http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html](http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html) ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [September 28, 2003, 10:00am UTC](https://boards.straightdope.com/t/infinity-squared/204219/18 "2003-09-28T10:00:05Z")
Those infinite mathemtatic people look like cranks (or jokers, or overenthusiastic high-schoolers) to me. Their method is just defining a symbol oo equal to 1/0, and, if I read the site correctly, assuming that it can then obey all the normal arithmetic rules, which paints them into a corner immediately: 1/oo=0=(1/oo)/2=(1/2oo) rearranging gets 2oo=oo, which they explicitely say it doesn’t (alternatively, they say you can cancel oo, so 2=1) There _are_ number systems that have an infinity obeying normal rules of arithmetic - see [hyperreal numbers](http://www.wikipedia.org/wiki/Hyperreal_numbers) and [surreal numbers](http://www.wikipedia.org/wiki/Surreal_number) - but they have to have infinitessimals (numbers greater than zero, but less than any positive real number) as well. The site gropes at the edges of this by saying they use “a positive and a negative zero.” It’s even possible to use infinitessimals in calculus instead of limits, though it was ages before anyone did it rigorously, despite everyone thinking that way 🙂 ---
**Author:** ![Mbossa](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@Mbossa](https://boards.straightdope.com/u/Mbossa)\ **Post date:** [September 28, 2003, 11:06am UTC](https://boards.straightdope.com/t/infinity-squared/204219/19 "2003-09-28T11:06:05Z")
> [@](#): > > \*Originally posted by Eleusis \* > **I’m not a mathematician but it seems to me that anything other than zero _times_ infinity = infinity.** I’m not a mathematician either (I’m dabbling in a bit of complex analysis at university, but that’s just for fun). But from what I understand, the symbol “infinity” is never actually defined in such a way that makes, for example, “2\*inf = inf” any more meaningful than “k) + :mad: -& = 🆒/🆒 )(((()(” (at least not in the more ‘traditional’ branches of mathematics). However, “infinity” can be _part_ of another ‘symbol’. In this case, an appropriate ‘symbol’ would be the limit. So we could say something like “lim(x-\>inf) 2x = inf”, which expresses the same idea as “2\*inf = inf”, except that it’s nice, meaningful, and well-defined (so long as we make certain obvious assumptions, like x being a real number). As an added bonus, it’s also true! Try not to think of “lim(x-\>inf) 2x = inf” as an equation, despite the equals sign. In particular, don’t try anything silly like rewriting it as “(lim(x-\>inf) 2x)/inf = 0”, or we’ll be back where we started. Another way of writing “lim(x-\>inf) 2x = inf” is “As x-\>inf, 2x-\>inf”, which doesn’t have an equals sign. This way is probably nicer, but it won’t impress the girls/guys/whatever. The formal definition (and proof) involves more epsilons and deltas than is appropriate for my first post to this message board :). But basically it says that “as a variable x increases without bound, 2x also increases without bound.” Note the absence of any mention of equality. A great thing about using this limit notation is that it doesn’t foul up the rules of arithmetic any more than necessary. If infinity was a symbol we could plug into equations willy-nilly, we couln’t even be sure that obvious statements like “x = x” are true anymore!! Granted, mathematics is already quite dirty with all the problems that zero causes, but why make things worse than they already are? So yeah, basically you’re right. Another way of saying what you want to say is that as a variable x increases without bound, x multiplied by any non-zero number also increases without bound. Well, I guess that’s not quite right, because if that non-zero number is negative, it _decreases_ without bound. So, in a nutshell, lim(x-\>inf) nx = inf, where n\>0 lim(x-\>inf) nx = -inf, where n\<0 These can be proven, but just not by me right now. (now I’d better brace myself for my post to be ripped apart by people who actually know what they’re talking about) ---
**Author:** ![moriah](https://avatars.discourse-cdn.com/v4/letter/m/b2d939/32.png) [@moriah](https://boards.straightdope.com/u/moriah)\ **Post date:** [September 28, 2003, 3:04pm UTC](https://boards.straightdope.com/t/infinity-squared/204219/20 "2003-09-28T15:04:57Z")
What’s the infinite root of infinity? Or infinity to the infinite power? [Next page](https://boards.straightdope.com/t/infinity-squared/204219.md?page=2)