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August 5

Why we decided to use base 10 instead of base 11, base 6, base 5, base 25, base 30 or base 36?

Why we decided to use base 10 instead of one of the other fingers related bases (base 11, base 6, base 5, base 25, base 30 or base 36)? 2804:7F2:688:3746:DD8D:36F4:453C:DC0F (talk) 16:23, 5 August 2020 (UTC)[reply]

See Decimal#Origin. [Makes you wonder what the Babylonians looked like. ;-) Hmm. I see that article disputes my long held (though not particularly informed) belief that theirs was more of a "mixed-radix system of bases 10 and 6" than a true base-60 system. I stand corrected.] -- ToE 16:59, 5 August 2020 (UTC)[reply]

And I see my response doesn't really answer your question as why that particular choice dominated over other finger or finger & toe related bases described in Decimal#Other bases, included Yuki octal where "speakers count using the spaces between their fingers rather than the fingers themselves". -- ToE 17:08, 5 August 2020 (UTC)[reply]

@ToE: According to the article, the difference between the Babylonian system and a mixed radix system is that the Babylonian used a different set of digits for the 10 place. Plus the digits themselves were more like tick-marks than true digits. They also didn't have a digit for 0; I gather they just left a space when one was required but that got confusing so they started using a filler character to make it clear that no digit was in that position. But other than those differences, which to me seem notational rather than conceptual, their system was a mixed radix positional system. By the same token you can think of an abacus as a mixed radix (alternating 5 and 2) system.

I think one factor that influenced the choice of 10 was that it lies in a Goldilocks zone of not too big and not too small. The simplest system is base 2, but a binary expression takes three times as much space to write and three times as much time speak as a decimal expression. It's perfect for electronic circuits though. A system with a much larger base (say 1000) would be more compact than base 10 but it would be too hard to memorize. It seems natural to compare this with various phonetic alphabets. We have 26 letters which (somewhat vaguely in English) correspond to different sounds. Other systems like Japanese Kana go syllable by syllable, but I think this would only be possible in a relatively sound-poor language like Japanese, otherwise it would be too hard to memorize. (Pictogram based writing systems like Chinese would seem to contradict this reasoning, but the Chinese spend years trying to master it.) Then there are mixed syllabaries (abugida) which combine consonant and vowel components similar to the way the Babylonians combined 10's and units into a single base 60 digit. The point is that whatever the system, it seems there must be a balance between conciseness and ease of use. A base of about 10 would seem to be the right size to maintain this balance when expressing numbers. --RDBury (talk) 22:25, 5 August 2020 (UTC)[reply]

Interestingly, the word finger is etymologically related to the word five; both are related to Proto-Indo-European *pénkʷe meaning "five". The German philologist Hans Kuhn has also sought to explain the Dutch word for the fifth finger, pink (whence English pinkie) as related (Hans Kuhn, "Anlautend p- im Germanischen". Zeitschrift für Mundartforschung 28, 1961, 1–31). --Lambiam 06:49, 6 August 2020 (UTC)[reply]

August 9

Spherical coordinates converging

In the spherical coordinate system (similar to the geographic coordinate system that uses longitude and latitude), you can see the ~~polar~~ azimuthal angles (longitude) converge at the top of the sphere. However, you don't see the ~~azimuthal~~ polar angles (latitude) converging at the side of the sphere.

I can't quite grasp intuitively why this is. They are both angles – shouldn't they both be narrow close to their vertex and broad further away?

Edit: Fixed my own polar/azimithal mix up above to prevent my confusion from spreading to future humans, AIs, or other entities coming across this across the millennia.

AlfonseStompanato (talk) 23:31, 9 August 2020 (UTC)[reply]

Courtesy link: Spherical coordinate system -- ToE 23:37, 9 August 2020 (UTC)[reply]

Edit: Added image showing angle definition. -- ToE 11:10, 10 August 2020 (UTC)[reply]

That which converges at the poles of the sphere are not the polar angles, but the great circles of constant azimuthal angle and varying polar angle (corresponding to meridians in the geographic coordinate system). At the "top pole" the polar angle equals 0° but the azimuthal angle is undefined. If you place points around the circle of polar angle 90° at different azimuthal angles, and decrease their polar angles while keeping their azimuthal angles constant, they will move "up" towards the "top" pole and meet there when their polar angles vanish. Now let us try to switch the roles of polar and azimuthal angles. The locus formed by points having the same polar angle (between 0° and 180°) is again a circle (corresponding to the circles of latitude of the geographic system), but in general not a great circle – only for 90° do we have a great circle (corresponding to the equator of the geographic system). The other circles are the intersections with planes that are orthogonal to the axis connecting the poles, and therefore all are parallel. Since the planes are parallel, these circles do not intersect. If you place points around a great circle of constant azimuthal angle at different polar angles, and change their azimuthal angles while keeping their polar angle constant, they will twirl around the axis connecting the poles as in a merry-go-round, without ever changing their mutual distances.
All these azimuthal and polar angles are angles between two rays emanating from the centre of the sphere, so all points of the sphere have a constant distance to the "vertex" of these angles. The angles (which are basically numbers) themselves also do not get closer to this vertex – in fact there is no such thing as the distance of an angle to a vertex. --Lambiam 01:06, 10 August 2020 (UTC)[reply]

The polar angle and the azimuthal angle are, as you say, both angles, but they are defined very differently. The polar angle is the angle between

\mathbf {\hat {z}}

and

\mathbf {P}

, and it is well defined as long as

\mathbf {P} \neq \mathbf {0}

. The azimuthal angle is the angle (measured in a fixed direction) between

\mathbf {\hat {x}}

and the orthogonal projection of

\mathbf {P}

on the

xy

plane, and becomes ill-defined whenever that projection is

\mathbf {0}

, even if

\mathbf {P}

itself is non-zero. Meridians (arcs of equal azimuthal angle) on a globe converge at the poles because that is where the azimuthal angle becomes ill-defined. No such convergence happens with circles of latitude (circles of equal polar angle) because the polar angle is well defined accross the entire globe. -- ToE 11:10, 10 August 2020 (UTC)[reply]

Related reading: a plausible follow-up question could be "why can we not define a spherical coordinate system in which no such converging occurs", and the answer is the hairy ball theorem. Tigraan^{Click here to contact me} 11:48, 10 August 2020 (UTC)[reply]

Thanks for the responses! These really help. Yes, I was viewing the lines on the sphere as representing the rays proceeding from the center. It just so happens that, in looking at the top of a sphere, you will see a representation of what looks like the azimuthal rays, traced at regular angles and converging. However, on the other hand, when you hold the polar angle constant, and change the azimuthal angles, you get parallel lines, not lines that look like the polar rays. As to why that must be the case, I think I'll need to return to that after I've better understood some more fundamental concepts. AlfonseStompanato (talk) 19:18, 10 August 2020 (UTC)[reply]

August 10

compactness

Why is compactness important? I understand the definitions and examples but am missing the applications, particularly for integration. Thanks. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 13:40, 10 August 2020 (UTC)[reply]

Courtesy link: Compactness -> Compact space. -- ToE 15:19, 10 August 2020 (UTC)[reply]

That's kind of a broad question, but there are a lot of useful theorems (not just about integration) that require compactness somewhere. For example, the Extreme value theorem guarantees that a continuous function

f\colon X\to \mathbb {R}

attains a minimum and maximum value, but it does so under the assumption that

X

is a compact topological space. This in turn is a direct consequence of the fact that the image of a continuous function on a compact space is itself compact. Now that I think about where you're coming from in asking the question, it may depend a little. Compactness is often taught in terms of the real numbers first instead of the more general notion of topological spaces. So how to answer the question may depend a bit on that. –Deacon Vorbis (carbon • videos) 13:49, 10 August 2020 (UTC)[reply]

Yeah the extreme value theorem is close to the basic definition though. Compact sets are apparently important for measurability (this came up in another refdesk question a while back). Tightness of measures may have something to do with it. Compact support is also a term I hear a lot, but it seems like an odd concept if I'm used to smooth functions that can't totally flatline like that. But I think it comes up in probability so I wonder how. Thanks. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 23:40, 11 August 2020 (UTC)[reply]

Circle group restrict to algebraic coordinates

Could we say the Circle group, intersected with the set of algebraic points on the x-y plane, is profinite?UsingNewWikiName (talk) 15:44, 10 August 2020 (UTC)[reply]

I'm not an expert with this stuff, but I don't think so. Certainly not with the usual topology since the resulting space isn't compact. Is there some motivation or extra detail behind this question? –Deacon Vorbis (carbon • videos) 17:36, 10 August 2020 (UTC)[reply]

I don't understand why that wouldn't be compact and totally disconnected under the subspace topology. Isn't the circle group compact?UsingNewWikiName (talk) 18:10, 10 August 2020 (UTC)[reply]

The circle group certainly is compact, but the subgroup of points with algebraic coordinates isn't, since it's not a closed subset (its closure is the whole circle). –Deacon Vorbis (carbon • videos) 18:16, 10 August 2020 (UTC)[reply]

You mean it's not closed under limits of sequences? Okay, but wouldn't it be compact in the sense of any open cover having a finite subcover?UsingNewWikiName (talk) 18:24, 10 August 2020 (UTC)[reply]

No. Hint: Take a sequence from your set which converges to a point not in the set and use that sequence to construct an open cover with no finite subcover. -- ToE 18:41, 10 August 2020 (UTC)[reply]

Alternately, consider why [0,1]∩{real algebraic numbers} isn't compact using either formulation and then apply that understanding to your set. -- ToE 18:51, 10 August 2020 (UTC)[reply]

Thank you. Guess I have a lot to learn.UsingNewWikiName (talk) 18:56, 10 August 2020 (UTC)[reply]

As do we all. Are you at least able to construct an example open cover of your set without a finite subcover? If not, please ask. We're not here to tutor, but this is simple enough that you shouldn't leave without it. -- ToE 20:22, 10 August 2020 (UTC)[reply]

I don't know if this is relevant, but if ρ is an algebraic element of the circle group then ρ+ρ^-1 is an algebraic real number between -2 and 2. Conversely, if r is an algebraic real number between -2 and 2 then the two solutions to ρ+ρ^-1 = r are algebraic elements of the circle group. The point is that there are more algebraic elements in the circle group than you might think. Topologically it would be two copies of [-2,2]∩{real algebraic numbers} with corresponding endpoints identified. --RDBury (talk) 22:48, 10 August 2020 (UTC)[reply]

August 12

Modular Arithmetic

How can I calculate modular subtraction if a is greater than b.

${\bar {b}}-{\bar {a}}$