Wikipedia:Reference desk/Mathematics: Difference between revisions
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<noinclude>{{Wikipedia:Reference desk/header|WP:RD/MA}} |
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[[Category:Non-talk pages that are automatically signed]] |
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[[Category:Pages automatically checked for incorrect links]] |
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[[Category:Wikipedia resources for researchers]] |
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[[Category:Wikipedia help forums]] |
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[[Category:Wikipedia reference desk|Mathematics]] |
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[[Category:Wikipedia help pages with dated sections]]</noinclude> |
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= December 21 = |
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== Calculate distance between centers of circles given area of intersections == |
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I need help checking my work. I'm working on a representation of multiple regression effect sizes using a Venn Diagram for educational policy research. I see that [[Lens (geometry)|the area of a geometric lens]] has a closed form solution. Given a [[Venn diagram]] made from circles <math>A</math>, <math>B</math>, and <math>C</math> with centers <math>X</math>, <math>Y</math>, and <math>Z</math>, that <math>r=R=\sqrt{1/\pi}</math>, and <math>A_{A\cap B}=0.23</math>, then <math>d_{XY}\approx 0.429</math>. Can someone validate this? The Venn diagram is represented at [https://i.stack.imgur.com/XyK4N.png this link here].[[User: schyler|<span style="color:#458B00;">Schyler</span>]] ''([[User talk: schyler|<span style="color:#00688B;">exquirito veritatem bonumque</span>]])'' 04:07, 21 December 2022 (UTC) |
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:Circle <math>C</math> has no role. I cannot replicate these numbers. When each circle passes through the centre of the other circle, so the circle centres are <math>r\approx 0.56419</math> apart, a rhombus of two equilateral triangles fits within the lens. The area of this rhombus is <math>\tfrac{\sqrt 3}{2\pi}\approx 0.27566,</math> so when <math>d_{XY}=0.429<r</math>, <math>A_{A\cap B}>0.27566,</math> contradicting <math>A_{A\cap B}\approx 0.23\,.</math> My calculations give me that distance <math>d_{XY}=0.429</math> gets you <math>A_{A\cap B}\approx 0.52785\,.</math> Conversely, to get lens area <math>A_{A\cap B}=0.23\,,</math> I find we need <math>d_{XY}\approx 0.73938\,.</math> --[[User talk:Lambiam#top|Lambiam]] 07:38, 21 December 2022 (UTC) |
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:Yes, circle <math>C</math> has no role. Let <math>\theta</math> be the angle at X (or Y) between the lines to the junctions of ABD and of CEFG. The lens has area <math>A = R^2\left(\theta - \sin \theta \right)</math>, where <math>R=\sqrt{1/\pi}</math>, and the diagram says <math>A_{A\cap B}</math> (D+G) has area 0.03, so <math>\theta - \sin \theta = 0.03 \pi</math>. That gives <math>\theta \approx 0.8235</math> (in radians). The length from X (or Y) to the midpoint of XY is <math>R \cos(\theta / 2)</math>, and twice this gives <math>d_{XY}\approx 1.0341</math>. My trig is rusty and I may have made some errors, but I think the principle and the order of magnitude are right. If <math>A_{A\cap B}</math> were 0.23, we'd get <math>\theta \approx 1.1377</math> and <math>d_{XY}\approx 0.9507</math>. [[User:Certes|Certes]] ([[User talk:Certes|talk]]) 13:31, 21 December 2022 (UTC) |
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::For <math>\theta=0.8235,</math> we have <math>(\theta-\sin \theta)/\pi\approx 0.02864,</math> not quite <math>0.03</math> but at least in the ballpark. But in the second case, you missed a factor <math>\pi</math>: for <math>\theta=1.1377,</math> we have <math>\theta-\sin \theta\approx 0.23</math> while <math>(\theta-\sin \theta)/\pi\approx 0.07322.</math> --[[User talk:Lambiam#top|Lambiam]] 15:52, 21 December 2022 (UTC) |
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::Here are the calculations for lens area <math>0.23,</math> step by step: |
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:::<math>\begin{array}{cccc} |
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\theta &=& 1.71254 \\ |
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\sin \theta &=& \sin 1.71254 &\approx& 0.98997 \\ |
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\theta-\sin \theta &\approx& 1.71254-0.98997 &\approx& 0.72257 \\ |
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(\theta-\sin \theta)/\pi &\approx& 0.72257/3.14159 &\approx& 0.23000 \\ |
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\cos(\theta/2) &\approx& \cos 0.85627 &\approx& 0.65526 \\ |
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2R\cos(\theta/2) &\approx& 2 \times 0.56419 \times 0.65526&\approx& 0.73938 \\ |
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\end{array}</math> |
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::--[[User talk:Lambiam#top|Lambiam]] 16:03, 21 December 2022 (UTC) |
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:::Oops. Thanks, that looks more credible. [[User:Certes|Certes]] ([[User talk:Certes|talk]]) 18:40, 21 December 2022 (UTC) |
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Okay, well this is interesting. Yes, in the example, circle C has no role here. Someone said "When each circle passes through the centre of the other circle," but I do not think that is probable. Here were my steps: |
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Circles <math>A</math>, <math>B</math>, and <math>C</math> have centers <math>X</math>, <math>Y</math>, and <math>Z</math> and radii <math>r_A=r_B=r_C=\sqrt{1/\pi}</math>. The centers form <math>\triangle XYZ</math> and intersect such that <math>Area_{A\cap B}=0.03</math>, <math>Area_{B\cap C}=0.11</math>, <math>Area_{A\cap C}=0.23</math> |
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<math>d_1=\overline{\rm XY};d_2=\overline{\rm XZ};d_3=\overline{\rm YZ}</math> |
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<math>Area_{A\cap C}=r^2_A\cos^{-1}(\frac{d^2_1+r^2_A-r_C^2}{2d_1r_A})+r_C^2\cos^{-1}(\frac{d^2_1+r_C^2-r^2_A}{2d_1r_C})-2\Delta</math> |
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<math>\Delta=\frac{1}{4}\sqrt{(-d_1+r_A+r_B)(d_1-r_A+r_B)(d_1+r_A-r_B)(d_1+r_A+r_B)}</math> |
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Simplifying: |
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<math>\cancel{0.23=2(\frac{1}\pi \cos^{-1}(\frac{d^2_1}{2d_1\sqrt{\frac{1}{\pi}}}))-\frac{1}{2}\sqrt{(-d_1+2\frac{1}{\pi})(d_1+2\frac{1}{\pi})}}</math> |
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<small>oh I see this was my problem, I think... I distributed the 2 onto <math>\frac{1}{4}\Delta</math></small> |
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<math>\cancel{0.23=2(\frac{1}\pi \cos^{-1}(\frac{d^2_1}{2d_1\sqrt{\frac{1}{\pi}}}))-\frac{1}{4}\sqrt{(-d_1+2\frac{1}{\pi})(d_1+2\frac{1}{\pi})}}</math> |
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Update: <small>well, here i am checking my own work again. i found another error. i deleted some values of d within <math>\Delta</math>. I should know better than to ask for help right away... i can do this... but the doubt is strong in this one</small> |
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<math>\cancel{0.23=2(\frac{1}{\pi} \cos^{-1}(\frac{d^2_1}{2d_1\sqrt{\frac{1}{\pi}}}))-\frac{1}{4}\sqrt{(-d_1+2\frac{1}{\pi})(d_1+2\frac{1}{\pi})}}</math> |
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<small>no that's wrong too, <math>r=\sqrt{\frac{1}{\pi}}</math>... i got it mixed up in <math>\Delta</math> again...</small> |
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:You can shift two equal-sized circles such that each passes through the other's centre; I used this special case merely to easily establish bounds that were violated by a purported solution. |
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:The function <math>f(\theta)=(\theta-\sin \theta)/\pi</math> is [[Transcendental function|transcendental]], and one cannot hope to solve the equation <math>f(\theta)=x</math> by a combination of algebraic and trigonometric manipulations for rational values of <math>x,</math> except when <math>x</math> is an integer, in which case the equation is solved by <math>\theta=x\pi.</math> --[[User talk:Lambiam#top|Lambiam]] 09:01, 22 December 2022 (UTC) |
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= December 24 = |
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== [[New Math|The New Mathematics]] == |
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Greetings, keepers of the eternal flame! |
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I normally give this desk a miss since I am a mathematical nincompoop. But I learned recently through our article that ''Time'' placed New Math on a list of the 100 worst ideas of the 20th century. But did it produce anyone who became a full-time mathematician? I was one of the guinea-pigs subjected to this cruel and inhumane treatment in the UK from around 1968 aged 8. My views are somewhat coloured because I missed the entire first term of lessons, and I was thrown in the deep end with no explanation whatever. My 'mental arithmetic', times tables etc. were previously fairly good. [[WP:OPINION|As an aside]], did anyone else here suffer this unusual punishment? Did it work for you? At least I got to understand binary, which came in useful when teaching IT in later years. |
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:Obviously, some of the people who underwent this treatment became full-time mathematicians; it is not like there was a ten-year gap in the age distribution of academic staff at the maths departments of US universities at the turn of the century. But can one one say it "produced" them? Did they manage to develop their mathematical prowess even in spite of an ill-advised hyperabstract approach, or did it perhaps come natural to their already (for their age group) advanced abstract insights? It may be impossible to produce other than anecdotal evidence. Possibly the highest honour for one's mathematical achievements is to be a recipient of the prestigious [[Fields Medal]], which was given in 1998 to American mathematician [[Curtis T. McMullen]], the doctoral adviser of the late [[Maryam Mirzakhani]], the first woman to be given the Fields Medal. McMullen was born in 1958 and therefore received his primary maths education during a period in which the abstract approach was in vogue. I have no specific information, though, about how he was schooled and whether there were perhaps stimulating extra-curricular circumstances. --[[User talk:Lambiam#top|Lambiam]] 17:39, 24 December 2022 (UTC) |
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:I'm the right age to have gotten some of the 'new math', though I'm not sure how much filtered through to the rural grade school I attended. It may have been a bad idea, but I can see how it might have seemed like a good one at the time. "The important thing, " in the words of Tom Lehrer, "is to understand what you're doing rather than to get the right answer." Put another way, it seemed like a good idea to put less emphasis on rote memorization and blindly following recipes and more emphasis on learning the concepts involved. It would be like becoming a better baker by learning what yeast and gluten do rather than just memorizing different recipes for bread. If you put it that way it sounds like the new math would even get kids to stop hating math (as much). Perhaps the new math did help at make some people more open to abstract concepts and help them in more advanced studies. On the other hand if your interest in math ends with being able to follow your bank statement then the new math was probably a waste of your time. I think what made the idea truly bad is that it was implemented despite being based on what were only theories on the best ways of learning. So instead of saying "It sounds interesting, let's try some of the ideas on a few of the brightest students first to if it's an improvement," it was "It sounds great, let's completely revamp the curriculum and roll it out to the entire school district." Such a radical adoption of untested methods is rarely a good idea. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 20:20, 24 December 2022 (UTC) |
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::An additional problem was that many teachers did not really understand mathematics, and this new maths even less. They taught it the only way they knew, which was just the same way as before: as a recipe to be followed mindlessly, step by step. --[[User talk:Lambiam#top|Lambiam]] 21:09, 24 December 2022 (UTC) |
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::[https://www.youtube.com/watch?v=UIKGV2cTgqA New Math] by Tom Lehrer - worth a listen 😀 I've always liked maths so I never had to learn the stuff in class as I'd worked most of it out long before. Except for geometry where there was stuff new to me. [[User:NadVolum|NadVolum]] ([[User talk:NadVolum|talk]]) 00:45, 25 December 2022 (UTC) |
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:::Also worth watching is a longer film (36 minutes), [https://www.youtube.com/watch?v=lvEcFJANVQo ''What happened to the 'new math'?''], that provides a historical perspective. One of the things I learned was the detrimental role played by the profit-driven logic of the US market for elementary-school textbooks. --[[User talk:Lambiam#top|Lambiam]] 06:17, 25 December 2022 (UTC) |
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::::Thanks for the link. I didn't know that new math was considered successful in high schools before they tried to expand it to elementary schools. I've read "Why Johnny Can't Add" but I don't remember it as being so much a critique of new math specifically but more of the way it failed to address more systemic problems in education. One example I remember is a teacher who miscopied an example from her math for teachers class, and went on to faithfully pass on the now useless example her students. It's been several decades since I read it though, so my impressions aren't too reliable. Another educational movement which I more experience with is the so-called [[Moore method]], which has some similarities with new math only it's geared toward advanced students. The up-side to my mind was that I got lots of practice proving theorems in point set topology without having to look them up. The down side was that it took an entire semester to cover the material you would have gotten in the first two weeks of a course taught in a more traditional way. Everything's a trade-off I guess. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 12:54, 25 December 2022 (UTC) |
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:Next step, post-fact maths [https://www.youtube.com/watch?v=Zh3Yz3PiXZw] :-) [[User:NadVolum|NadVolum]] ([[User talk:NadVolum|talk]]) 13:49, 26 December 2022 (UTC) |
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::Wow, thanks everyone for your interesting and insightful comments, plus the links. I feel a lot happier now about my inability to 'get' maths. Cheers all, [[User:MinorProphet|MinorProphet]] ([[User talk:MinorProphet|talk]]) 09:20, 28 December 2022 (UTC) |
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= December 26 = |
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== Deducing formulae from data == |
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Are there books that tell you how to deduce or guess the formula for an un-understood process for which you have measured data? Can anyone recommend a text? |
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In junior high school we did things like plot pairs of points on graph paper, such as 1, 1.21; 2, 2.20; 3, 3.23, etc and realise that within measurement uncertainty it is presumably a straight line, formula Y = a + bX where a is 0.21 and b is 1. Big deal, real life is not that simple. We also plotted sets of points that looked just like the formula is square law, Y = aX^2. Again, big deal. There are of course other simple obvious things like exponential growth/decay. But real processes are often not that simple. |
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In high school and 4 years of university engineering degree, we never looked at this topic again. |
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There are a multitude of curve fitting textbooks on least squares regression, fitting generalised formula such as Y = k1 + k2X^2 + k3X^3 .... etc etc. These are not what I need. You can always fit a curve with such methods, if you throw enough terms in, or decide to accept arbitary error. But it tells you nothing about how the process that produced the data actually works, or help you extract the real curve from the "noise" and measurement uncertainty. [[User:Dionne Court|Dionne Court]] ([[User talk:Dionne Court|talk]]) 05:49, 26 December 2022 (UTC) |
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:I'm afraid there is no general approach to finding a relatively simple formula that fits observed data within the measurement error. It may be instructive to read the section {{section link|Planck%27s law#History}} and see the long process leading up to Planck's 1900 construction of an empirical formula, which he called an "improvement" of [[Wien approximation|Wien's equation]]. One can see the combination of heuristic theoretical approaches (Wien's thermodynamic arguments, Planck's resonant electric oscillators) with high-quality experimental data at work, and yet it appears that in the end success emerged as the result of patch work, splicing Wien's and [[Rayleigh–Jeans law#Historical development|Raleigh's formula]]s together without theoretical underpinning. One may hope that in the not-too-far future an AI can tirelessly try and fit parametrized models in order from simple to more complex, using some appropriate measure of complexity. Compare the (pre-computed) [[Inverse symbolic calculator]], which will [https://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=1.644934&lookup_type=simple quickly] inform me that my calculated numerical value 1.644934 may be [[Basel problem#zeta_2|<math>\zeta(2)</math>]], which, where I'm coming from, is simpler than <math>\tfrac{1867}{1135}.</math> --[[User talk:Lambiam#top|Lambiam]] 09:21, 26 December 2022 (UTC) |
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::I haven't bothered with them but I believe there are various programs o the web for guessing a formula from some data. They put together various formula and give results ordered by a measure of simplicity and how well they fit the data. I guess they might do Planck's law. If the possibilities that are programmed in don't include the right answer at least they'll probably give you a good approximation for physical processes - even if they can't give the prime numbers for instance. [[User:NadVolum|NadVolum]] ([[User talk:NadVolum|talk]]) 14:08, 26 December 2022 (UTC) |
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:::A considerable time ago there was some hubbub about a program that discovered laws of physics,<sup>[https://www.wired.com/2009/04/newtonai/]</sup> but after a brief flurry of interest I never heard of it again. --[[User talk:Lambiam#top|Lambiam]] 20:48, 26 December 2022 (UTC) |
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:::The term for this appears to be [[Symbolic regression]]. Our article lists a few apps. Some are free. A comparison is lacking; without examining each individually, it is not clear which ones can handle multivariate models. --[[User talk:Lambiam#top|Lambiam]] 12:52, 28 December 2022 (UTC) |
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= December 28 = |
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== Why are Reeve tetrahedra skewed? == |
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[[File:reeve_tetrahedrons.svg|thumb|upright=1.25|Reeve tetrahedra showing that Pick's theorem does not apply in over two dimensions]] |
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Would anyone know why John Reeve picked (1, 1, ''r'') as the top vertex of the [[Reeve tetrahedron]]? (0, 0, ''r'') gives the same result yet is much easier to visualise and draw. Thanks, '''[[User:cmglee|cmɢʟee]]'''⎆[[User_Talk:cmglee|τaʟκ]] 16:21, 28 December 2022 (UTC) |
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:Choose <math>r=2</math>, so the four vertices are lattice points, having positions <math>(0,0,0),</math> <math>(1,0,0),</math> <math>(0,1,0)</math> and <math>(0,0,2).</math> Then, next to these four, also the lattice point <math>(0,0,1)</math> lies on the surface of the tetrahedron. --[[User talk:Lambiam#top|Lambiam]] 19:57, 28 December 2022 (UTC) |
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::To clarify a bit for who aren't familiar with them, the point of the Reeve tetrahedra is to show there is no three dimensional version of [[Pick's theorem]]. If the fourth vertex is (0, 0, r) then the number of lattice points on the surface vary with r, and that invalidates the example. Reeve tetrahedra have the same number of interior lattice points (0), face lattice points (0) and edge lattice points (0), the number of vertices is the same (4), and vertices are lattice points, but the volumes are different. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 00:01, 29 December 2022 (UTC) |
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= December 29 = |
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== Mathematical constants that are not real numbers == |
== Mathematical constants that are not real numbers == |
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:The basic [[quaternions]] are commonly denoted by <math>\mathbf i,</math> <math>\mathbf j</math> and <math>\mathbf k.</math> --[[User talk:Lambiam#top|Lambiam]] 00:08, 30 December 2022 (UTC) |
:The basic [[quaternions]] are commonly denoted by <math>\mathbf i,</math> <math>\mathbf j</math> and <math>\mathbf k.</math> --[[User talk:Lambiam#top|Lambiam]] 00:08, 30 December 2022 (UTC) |
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:Our article [[Greek letters used in mathematics, science, and engineering]] states that <math>\omega</math> denotes a complex cube [[root of unity]], a claim also found in the ''The Concise Oxford Dictionary of Mathematics''.<sup>[https://books.google.com/books?id=c69GBAAAQBAJ&pg=PA333&dq=%22cube+root+of+unity%22&hl=en]</sup> I think, however, that <math>\omega</math> more generally denotes a [[Root of unity#Group of nth roots of unity|primitive <math>n</math>th root of unity]], not a specific constant. --[[User talk:Lambiam#top|Lambiam]] 00:46, 30 December 2022 (UTC) |
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Revision as of 00:47, 30 December 2022
Mathematical constants that are not real numbers
Where integers count as a subset of the reals. By mathematical constants I mean numbers with names or symbols connected to them, like π, e, the Hardy-Ramanujan number 1729, and so on. Among constants that are not real-valued, I can think of the imaginary unit i, and some transfinite ordinals and cardinals like ω or . Are there any more, particularly complex-valued ones? I keep expecting a few numbers like to have "names", but none come to mind. The constants should be numbers, which I suppose is a loose term in this context, but let's not count matrices, graphs, etc. Thanks. 2601:648:8200:990:0:0:0:B9C2 (talk) 21:44, 29 December 2022 (UTC)
- There is a positive infinitesimal surreal number denoted by --Lambiam 00:04, 30 December 2022 (UTC)
- The basic quaternions are commonly denoted by and --Lambiam 00:08, 30 December 2022 (UTC)
- Our article Greek letters used in mathematics, science, and engineering states that denotes a complex cube root of unity, a claim also found in the The Concise Oxford Dictionary of Mathematics.[1] I think, however, that more generally denotes a primitive th root of unity, not a specific constant. --Lambiam 00:46, 30 December 2022 (UTC)
