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Is there any reason to “anti-simplify” this expression?

I was tutoring a precalculus student, and the question at hand was asking to find the angle between two vectors, given the formula $$\cos \theta = \dfrac {\mathbb{u} \cdot \mathbb{v}}{\|\mathbb{u}\|\...
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37 views

Why use the letter “k” in the function transformation formula $f(x - h) + k$?

This is strictly a historical "why is it the letter k rather than say v for vertical" question -- is it the initial letter of something from a specific language? Is it arbitrary? While we're at it, ...
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Unclear passages in the paper “On a New Class of Theorems in Elimination Between Quadratic Functions” by J. J. Sylvester

I'm writing an essay about the origin of some mathematical terms in the work of J. J. Sylvester. He first used the word matrix in his paper Aditions to the Articles "On a New Class of Theorems" and "...
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48 views

How was statistics formulated [closed]

I'm sorry for the naive question, but I've only been taught statistics at a high school level (extremely basic) and it always remained a mystery as to how certain concepts in statistics existed (...
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75 views

Great mathematical fusions in math history

Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together ...
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20 views

Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
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2answers
126 views

Gauss's thesis: Theory of equations or hypergeometric functions, or both?

I had read in multiple places, and always believed, that the first good proof of the fundamental theory of algebra (that a polynomial of degree $n$ over $\Bbb{C}$ has $n$ roots, with suitable ...
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53 views

The use of binary numeral system for theoretical results

I wonder how mathematics would be changed if we were been using binary system in calculations instead of decimal .. Could theory of mathematics would change a little ? Are there known ...
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19 views

Original statement of Wiener's $1/f$ theorem

I'm studying Wiener's 1/f theorem, and I got curious about which was its original statement.I've been looking online but found nothing. I want to know if Wiener also proved the $n-$dimensional ...
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1answer
77 views

19th history mathematics puzzle [closed]

Consider the following fictitious conversation between 19th century mathematicians: Teddy: This new field of calculus seems valid, but in some ways it’s sketchy. It needs to have a sound foundation ...
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36 views

Games based on sheaves (or sheaf gluing?)

my question is quite simple (and I hope it is NOT stupid). It is well-known that many successful games have clear mathematical underpinnings (see for example http://web.mit.edu/sp.268/www/rubik.pdf ...
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88 views

Question regarding Fermat's last theorem for n = 4

I am reading through a proof of Fermat's last theorem for $n=4$ and I see this statement... "Since $x^4+y^4=z^4$ has a nontrivial solution, then $x^4-y^4=z^2$ also has a solution." I have tried to ...
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135 views

Did Cantor both prove and disprove the Continuum hypothesis?

I have been listening to the podcasts of A Brief History of Mathematics on BBC Radio 4. In the episode on Georg Cantor the narrator, Prof. Marcus du Sautoy, says that one day Cantor proved that there ...
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46 views

Could Euclid have proven Dedekind's definition of real number multiplication?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
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1answer
77 views

What operations have represented 2^2? [closed]

I knew = 2.2 and 2.2= 2+2 What operations have represented 2^2? ? = 3+3+3=9 ; 3.3.3=81 and continue is . what is representation for operations ? I assume to have a operation, it is called f. I ...
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50 views

How did Euler prove the partial fraction expansion of the cotangent function?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I've seen several modern proofs of it and they all ...
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Using the given digits 0,1,2,3,4,5, tge numerals indicated Q.

1.Write the largest numeral w/o repetation in figures & in words. 2.write the smallest number without reptition in figures and words. 3 write the largest 4 digits number without repetition in ...
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67 views

In mathematics, what is a placeholder?

Google defines the word placeholder in the following image: My question is: Is this the only definition of the word placeholder in mathematics? I am thinking along the lines: If $M = p^k m^2$ ...
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3answers
265 views

Have historians responded to Raju's critique?

C. K. Raju has made some outrageous criticisms of the traditional take on Euclid in particular and Western history in general. Yes he has a book published on the subject with an apparently respectable ...
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1answer
58 views

When did Euler find his formula for $\zeta(2n)$

Does anybody know when Euler found his famous formula $$\zeta(2n)=\frac{(-1)^{n-1}(2\pi)^{2n}B_{2n}}{2(2n)!}?$$
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5answers
389 views

Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

My question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
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4answers
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Did Joseph Fourier ever make a pure mathematical mistake?

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001: It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur ...
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45 views

What is the mathematical term describing a pipe or a tube?

I am interested in this unanswered question Pipe-fitting conditions in 3D and so I was trying to find information about it. If the 3D curve $f(x(t), y(t), z(t)) = 0$ is a line I think that the pipe ...
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32 views

Inclusion-exclusion formula and its alias names

I am reading Probability by A. N. Shiryaev. One of the problems refers to "inclusion-exclusion formulas", also known as Poincaré’s formulas, Poincaré’s theorems, Poincaré’s identities. One of my ...
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66 views

Why is matrix multiplication called 'multiplication' if it is non-commutative?

This question begins with the assumption that matrix multiplication was termed 'multiplication' as a form of comparison/parallel to multiplication of integers and real numbers. Why was matrix ...
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89 views

Why weren't “degrees” replaced with a more intuitive angle measure?

$\bf History$ It is speculated that the seemingly arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$ and $60 \times 6 = 360$. ...
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188 views

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
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642 views

How to Axiomize the Notion of “Continuous Space”?

EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") ...
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2answers
187 views

Why $a^2+b^2=c^2$ is named after Pythagoras? It is known by earlier generations before him such as the Chinese. [closed]

Why $a^2+b^2=c^2$ is named after Pythagoras? It is known by earlier generations before him such as the Chinese. It is because he proved it and not other generations? Or Pythagoras put it into ...
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65 views

Origin of Almost Perfect Numbers

Let $N$ be a positive integer. $N$ is called a perfect number if the sum of its positive divisors denoted by $\sigma(N)=2N$. For example $6$ is a perfect number since: $\sigma(N)=1+2+3+6=12=2(6)$. ...
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72 views

Why did Fermat publish so little? [closed]

Fermat published very little during his lifetime. Why is this the case?
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26 views

Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
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3answers
97 views

How (or why) did Topology become so central to modern mathematics?

It is frequently said that topology is nowadays one of the central pillars of modern mathematics (ex. "Because of its central place in a broad spectrum of mathematics") The field has managed to ...
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1answer
139 views

What is the motivation behind the arbitrary union topological axiom?

1. Why is the arbitrary union axiom in the definition of topology necessary? 2. Why is it useful? Why might we expect ("intuitively") that it should be useful? 3. What is the (historical) ...
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1answer
92 views

Was Riemann really the first person to define definite integrals?

So I am doing a study on Riemann, and one of his big things is apparently the "Riemann integral". My understanding is that this concept was meerly the first definition of a definite integral, and ...
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3answers
105 views

There are many numeral systems. Why do we only use the $0-9$ Hindu-Arabic numeral system?

Here is a list of other systems: Babylonian numerals Egyptian numerals Aegean numerals May numerals Chinese numerals These system are far older than the current system. How did it get to be known ...
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33 views

Serendipitous discoveries in mathematics [duplicate]

I have recently been reading about serendipitous discoveries in science and I found them quite inspiring. Most of those discoveries are in Chemistry. I'm looking for examples of these kinds of ...
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2answers
84 views

Who is M. Montel?

From V.I. Arnold's Experimental Mathematics: Not having achieved what they desired, they pretended to desire what they had achieved. –M. Montel Who is M. Montel? Is he related to the ...
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19 views

Origins of Operations Research and original meanings to different terms? [closed]

I am confused by Reliability Engineering to the extent that sometimes the terms used are graph-theoretical: this aspired to be researched here and here. In comparison, terms are sometimes more slack, ...
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61 views

Origin of the Riemann-Lebesgue lemma

When and where did Riemann and Lebesgue give the well known Riemann-Lebesgue lemma? A "lemma" is usually used as a stepping stone to a larger result rather than as a statement of interest by itself. ...
2
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1answer
70 views

What was the “real” first equation? [closed]

The first equation ever written, using a modern equals sign, has TWO versions: Version 1: Wiki link and Another link gives $$ 14x+15=71 $$ But from a UK maths textbook, it gives $$ 14\sqrt{x}+15=...
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1answer
42 views

What is the name for the set of all integers greater than 1?

Ancient Greek Mathematicians such as Euclid defined a number as "a multitude of units." For them, a number was a member of the set of all integers greater than one. Does this set have a name in ...
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20answers
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What are some examples of when Mathematics 'accidentally' discovered something about the world?

I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood ...
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1answer
54 views

What is the motivation behind the Bessel function of second kind

I am studying Bessel function and found the good reference by G.N. Watson At some point in page 58 he introduces the following expression due to Hankel: \begin{eqnarray} \lim_{\nu \to n} \frac{J_{\...
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296 views

What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$?

Is there not any difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$ as long as your function has one variable? $f(x) = x^3\implies \left\{\begin{align}&\dfrac{\...
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700 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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mathematics was recreated on a foundation of number concepts rather than geometrical ones

In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says In modern times mathematics was recreated and vastly expanded on a foundation of number ...
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Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
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38 views

Why did the number of types of integrals got lower from the beginning of the $20^{th}$ to this day?

There is an old $(\text{circa } 1930)$ and interesting book in calculus: Edwards' Treatise on Integral Calculus. This book has a very complete list of cases of integrals, for example, these ...
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1answer
62 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...