Wikipedia:Reference desk/Archives/Science/2023 April 25

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April 25

Infinity and zero

A number divided by zero, and a number divided by infinity.

Some Maths teacher are saying undefined, some maths teachers are saying infinity as answer. 2402:3A80:1C46:7462:508:3708:7AE5:926D (talk) 06:29, 25 April 2023 (UTC)

It depends on the context. We have an article on division by zero, which goes into some detail.

The short answer is that, for the ordinary sort of numbers you encounter in school, you can't divide by zero, and infinity is not one of those ordinary sort of numbers at all.

For other sorts of numbers and other notions of division, the answer may be different. See if the linked article helps.

For future reference, we have a mathematics reference desk. --Trovatore (talk) 06:33, 25 April 2023 (UTC)

Infinity is not a number. ←Baseball Bugs ^{What's up, Doc?} carrots→ 11:18, 25 April 2023 (UTC)

As Trovatore noted, it depends on what system you're working in. In the real numbers, infinity is not a number, but in the extended reals, infinity and negative infinity are both numbers. CodeTalker (talk) 15:04, 25 April 2023 (UTC)

What is infinity minus 1? ←Baseball Bugs ^{What's up, Doc?} carrots→ 16:45, 25 April 2023 (UTC)

In the extended reals, infinity minus one is still infinity. If you want to know more, I suggest reading the article CodeTalker linked. --Trovatore (talk) 17:05, 25 April 2023 (UTC)

As with anything mathematical (or really, and human created endeavour) whether something like "treating infinity as a number" or "treating division by zero as infinity" or whatever boils down to 1) is it useful 2) does it break the system we're working in. If the answer to 1) is "yes" and the answer to 2) is "no" then by all means, use it that way. There's all kinds of workarounds to these problems, such as limits, that allow us to do things like treat the limit of division by progressively smaller numbers as approaching infinity, thus for some meanings of "is equal to" we can treat something like 1/0 = ∞ as a meaningful concept. There are other contexts where 1/0 = (undefined) may make more sense. Similarly, we may treat infinity as a number (or at least, as a point on the number line somewhere out there too far to reach) in certain contexts where it is useful to do so. For example, This visual proof of the Basel problem uses the notion of a circle of infinite radius, so that there is some point, infinitely far away, at which the number line would "wrap around" and come back again. Using limits in a rigorous way one can say this without introducing any particular contradictions. --Jayron32 17:22, 25 April 2023 (UTC)

I recall in high school math, decades ago, where the teacher said you could invent something he called "R*", in which you have an object called "infinity". That's what the so-called "extended reals" are doing. ←Baseball Bugs ^{What's up, Doc?} carrots→ 17:45, 25 April 2023 (UTC)

Yes, I believe R* (or perhaps R^*?) is one of the standard names for the extended reals, though our article does not report it (the article gives ${\overline {\mathbb {R} }}$). You have to be a little careful because some of this nomenclature gets overloaded. For example I think (this is from memory) R* can also mean the multiplicative group of the nonzero real numbers. --Trovatore (talk) 18:04, 25 April 2023 (UTC)

R^* might use projective infinity (a single point at infinity, rather than separate positive and negative infinity). That is also usually how it is done when extending the complex plane to include infinity. 2601:648:8200:990:0:0:0:BC0F (talk) 19:18, 25 April 2023 (UTC)

Note in passing that, in the extended reals, you still can't divide by zero, for sort of a silly-seeming reason: You have $+\infty$ and $-\infty$, and they're different. So which one should ${\frac {1}{0}}$ equal? Because if ${\frac {1}{0}}=+\infty$, then you should have ${\frac {1}{-0}}=-\infty$. But $0=-0$, so that's a problem.

You can fix it in several ways.

• You can make $+\infty$ and $-\infty$ the same, so that the reals "circle around" at the point at infinity. That gets you what Wikipedia calls the projectively extended real line; I'm not sure this name is really standard
• You can do the same thing on the complex numbers, getting the Riemann sphere
• Or you can cut the Gordian knot by getting rid of the negative numbers, and look at the structure $[0,\infty ]$ of nonnegative reals plus a point at infinity. I don't know a name for this structure, but it's implicitly used a fair amount, especially in measure theory

In any of those structures, you can divide anything by zero except zero itself (${\frac {0}{0}}$ is still undefined), and get infinity (in particular ${\frac {1}{0}}=\infty$). --Trovatore (talk) 18:32, 25 April 2023 (UTC)

Thanks for adding formalism to some of my points above. I'm more of a recreational mathematics person, and while I understand the broader concepts, I lack the training to have the formal language to express it. --Jayron32 18:52, 25 April 2023 (UTC)

There are many infinities, of which the smallest is aleph-null. MinorProphet (talk) 21:36, 27 April 2023 (UTC)

While there is a sense in which that is true, in this context it is more likely to confuse than enlighten. Aleph null (like all aleph numbers) is a cardinal number, which means it is a measure of the cardinality (size) of a set. The infinities being discussed in this thread are points on the extended real number line, and have little relation to aleph numbers. In fact, our article on aleph null, which you linked to, says The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line. CodeTalker (talk) 01:55, 28 April 2023 (UTC)

Zerowater filter

A friend in an area with horrible tap water just got one of these. It purports to remove almost all of the dissolved solids from the water. Can anyone say how it manages to do that? I thought it was impossible to remove solutes by filtration, and you needed to distill the water or separate it with reverse osmosis. Is it possible to desalinate water with filters? Thanks. 2601:648:8200:990:0:0:0:BC0F (talk) 19:21, 25 April 2023 (UTC)

See ion-exchange resin which can remove metal ions, but may be not so great at removing chloride. Graeme Bartlett (talk) 22:12, 25 April 2023 (UTC)

What I would like is a good way to get rid of chloramine, which may indeed be a little safer than free chlorine from a public-health perspective, but has a very offensive taste. Bulky activated-carbon filters do work for a while but need to be replaced and seem wasteful. My workaround is to add a small dash of ascorbic acid powder, which I suppose reduces the chloramine to ammonium chloride (?), but it has its own taste and it's hard to tell whether it's really removing the chloramine or just covering it up. --Trovatore (talk) 23:12, 25 April 2023 (UTC)

Articles: water filter, point of use water filter, total dissolved solids. No, filtration can remove at least some contaminants. Here are listings from NSF International of filters certified by them for reduction of various contaminants.

Here's the problem with salt water: there's salt, and a lot of it. Total dissolved solids § Water classification: Drinking water generally has a TDS below 500 ppm. Higher TDS Fresh Water is drinkable but taste may be objectionable. Salt water by comparison is defined as having TDS of at least 10,000 ppm, orders of magnitude higher. Consumer filters are designed to filter fresh water; they will fail pretty quickly trying to handle salt water. I mean, in a sense you can think of a reverse osmosis membrane as just being an elaborate filter, that requires energy to force the water through it. Indeed, they even get "clogged" over time and require maintenance procedures to clean out buildup. RO and distillation are simply the most feasible ways to desalinate large quantities of water. A problem being the hypersaline brine produced by the solutes left behind, and here illustrating how much more "stuff" there is in salt water. Your typical consumer filter just traps contaminants within the filter itself; eventually it can't hold any more and then it needs replacement. --47.155.41.201 (talk) 23:38, 25 April 2023 (UTC)

Zerowater's website lists test results and notes that the filters have a limited capacity.[1] DMacks (talk) 19:49, 30 April 2023 (UTC)