The  asserts that our system of Arabic numerals is a geometric design where the number of corners corresponds to the number represented:    My question is: Is our Arabic number system based on a geometric design counting corners?
124256
Why are the ten digits in base 10 noted $0, 1, 2 ,3 ,4 ,5 ,6 ,7 ,8 ,9$ ? I heard it has to do with angles, but since I can't see how $9$ has $9$ angles, this may not be the historical reason ?
95245
Although I have seen many similar questions, none has been useful.   The question is quite simple: How do I reproduce the highlighted numerals in the image below?    Note that the form of the numerals addressed in questions as  is different from this one.  Thank you in advance!
41204
The Dihedral group makes sense, "Di" means two, and "hedral" means.. shape I think (I've just realised how much of what I think words mean are guesses based on experience) like a "polygon" is a 2d shape, a "polyhedron" I've always thought is just "any shape" (example 3 dimensions). So "Dihedral" means 2 dimensions, which it is. (Octagon and cube have different ... reflection+rotation groups (I hope))   The symmetric group has nothing to do with symmetries of shapes, it instead is the group of all bijections from one set to another. I suppose this means symmetry as in "abc->acb" has the "symmetric" version "$abc\to bac$".  Now the alternating group... I have no idea.   Now I discovered these in "Algebra" and "Abstract Algebra" but I've peeked at the contents page of the Combinatorics book I am reading and the symmetric is discussed there, this makes a lot of sense really, combinatorics seems to be the study of finite functions and seems to be easier than functions on the reals and stuff (which felt odd as I was sketching $x^3$ before I was considering "discreet" functions, but if anything they're easier)  So I guess that they were named because Combinatorics got there first. How? What (general case) question was being faced that lead to the solution being called "The symmetric group" (or even set, I doubt combinatorics would care that it is a group).  Sometimes how and why are similar, like turning points, "how did we decide to call $\frac{dy}{dx}=0$ a turning point" and "why" are identical. However there may be many "whys" with these groups as they apply in many areas, there can be only one how. Where the term first cropped up would be a good starting point, but I've yet to find anything.  I'll answer this myself if my searching yields anything.
42253
The question of the usability of rounded corners versus square corners has already be answered here.  My question revolves more about the consequence of having them intermingled.  I've heard some people espouse the idea that once you go down a rounded corners path, your design should really stick to that approach rather then, for example, having a square box inside a rounded one.  Is this simply aesthetic or is there some usability concern to back it up (e.g. cognitive switch from a softer less threatening to something more jarring)?  I would be very interested in any research that attempts to answer this question.
77118
I will use binary.  I claim to have a bijection $f \colon \mathcal{N} \to \left[ 0, 1 \right)$ where $\mathcal{N}$ is the set of natural numbers $\left\{ 0, 1, 10, 11, 100, \dotsc \right\}$ as follows:  $$ \begin{array}{l|l} x \in \mathcal{N} &amp; f(x) \in \left[ 0, 1 \right)\\\hline 0 &amp; 0.0 \\ 1 &amp; 0.1 \\ 10 &amp; 0.01 \\ 11 &amp; 0.11 \\ 100 &amp; 0.001 \\ 101 &amp; 0.011 \\ 110 &amp; 0.101 \\ 111 &amp; 0.111 \\ 1000 &amp; 0.0001 \\ \hspace{.6em}\vdots &amp; \hspace{.9em}\vdots \end{array} $$  It's basically the .  I claim to have everything on this list.  Diagonalization?  Go ahead!  Constructing a number that differs from $f(i + 1)$ in its $i$-th fractional digit yields $0.11111\!\ldots = 0.\overline{1}$ which is $1$, and that should not be on the list anyway!  Why does this not work? I think it has something to do with integers supposedly not having infinite digits, while reals supposedly do.
109858
I have read the excellent answers to  - my question is specifically about usage.   Is there a best practice for deciding between the variants geometric and geometrical in writing? A  shows that geometrical was more popular until about 1965, when they switched. What grammatical rules, if any, would be relevant here?
67409
Is there a word that can describe a geometric figure such as a circle or ellipse that is not centered at origin (0, 0) in cartesian space? I'd like to find an agreed upon mathematical word, but if it doesn't exist then a substitute "non-mathematical" word would do as well.
120624
Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be?    Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, would all numbers be equal?  The way I have thought Base 1 might be represented is tally marks, $0_{10}$ would be represented by nothing. So, $5$ in Base 1 would be represented by $00000$? Or we could define Base 1 to use: $\{$|$\}$ and $5$ would be |||||?
72785
I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that discusses this. Maybe it is too trivial, but I am not able to figure it out myself.   So I would like to ask you for help. Where can I find a proof for this?  Also, I am interested in the mathematical background behind plane tiling (especially periodic), If you can suggest papers on the topic it would really help.
9902
After learning about the binary number system (only 2 symbols, i.e. 0 and 1), I just thought why did we adopt the decimal number system (10 symbols) after all?  I mean if you go to see, it's rather inefficient when compared to the octal (8 symbols) and the hexadecimal (16 symbols)?
82778
Three numbers form a geometric sequence. If 5 is added to the second term, then the resulting numbers will constitute an arithmetic sequence. If 22.5 is added to the third number, these numbers will again form a geometric sequence. Find the original numbers.
61026
Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties.
79557
I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).  I say this is practically the case, so in every calculation you just assume that "$a + 0i$" is equal to the real number "$a$" and you always get the right results (this might not even be true, but I am no expert and as far as I can see this is the case). But then again, I believe that just cutting away "$+ 0i$" makes this number is not the same kind of number anymore, it's changed its structure and is not totally equivalent anymore. This is where he says I am absolutely wrong.  So, is it mathematically (strictly speaking!) perfectly fine to say that every complex number with an imaginary part of $0$ is just a real number? Or does this change the mathematical structure so much that it cannot be mathematically-valid but just something that happens to work (which is what I believe)?  I must say that I believe a complex number is one entity, whilst he believes that it is just a conglomerate of other entitites. So, as one said here, "$(a + bi) - bi = a$", I believe is barely a good argument. But then again: is it correct to view $a + bi$ as a single entity (like "$5$" or "$2$", just consisting of multiple symbols)? I'm (as a non-mathematician) not sure there either.
56182
Can somebody explain to me graphically or intuitively why this is true? Because if we think in terms of the base of the triangle formed inside a unit circle it seems like it should not be negative.
54511
"The introduction of numbers as coordinates is an act of violence." - Hermann Weyl.  A lot of people like this quote, apparently. They also seem to associate it to the manifold context in the obvious way: they interpret the quote as saying that focusing on coordinate charts is insightless, or at least dry. Although I aggree with this, I feel like the way the quote is phrased is not inclined to that interpretation. If not by the fact that Weyl is connected to geometry (and that this quote is in Bredon), I would interpret this phrase as saying that "Representing numbers by a given basis is insightless and can lead to mistakes, like thinking that divisibility is dependent of the number basis you are using." I think the words numbers and coordinates are strangely placed.  Therefore, my question is: What was the precise context on which this quote arose and, if possible, can we pinpoint exactly what did Weyl mean?
17494
What do the following numbers represent?
79687
This is a wild one: can anyone write a code that converts a number into the corresponding word? e.g.:  4 -&gt; 'four' 31 -&gt; 'thirty-one'   and so on. Obviously, it is not enough to build an array in which you can store the numbers and the word associated with it and then just print instead of the number the stored word. What complicates matters more, are numbers like 11 ('eleven'). I intend to use it for numbering equations. So, anybody up to it?