Questions tagged [mathematics]

For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.

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74 views

History of $\sin(nx) = 2^{n-1} \prod_{0}^{n-1} \sin\left(x + \frac{\pi k}{n}\right)$

What is the name of this identity? Who discovered this identity? What is the history behind this? I have looked up on Wikipedia with little documentation of the identity see finite product of ...
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4answers
169 views

What problem was solved by introducing the dimension of a vector space?

In linear algebra, we care a lot about dimensions. I get why it’s useful but not why it’s such a big deal. So I wondered what problem was solved historically by introducing a rigorous definition of ...
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How was mathematics used in World War II to “act on the right amount of intelligence”?

In the movie "The Imitation Game", Alan Turing along with his team crack the German encryption machine Enigma but advises his superiors to not act on all decrypted intelligence, as that might lead to ...
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1answer
158 views

What mathematics did Isaac Newton learn at school?

Since Sir Issac Newton invented a lot of modern mathematics, what mathematics did he already know? Since he was standing on the shoulders of giants which giants was he speaking of? I presume he knew ...
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73 views

Handbook of proofs

Do you know any handbook where original proofs of mathematicians' of the past theorems and facts are in modern notation? For example, for the Archimedean spiral etc
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51 views

How did the definition of each “ordered set” come about?

I could get a little intuition about preset, poset, and toset. e.g. A toset is, in effect, a linearly ordered set, and a poset is a set in which no more than one element in the same order exists, ...
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1answer
127 views

When and WHY did mathematicians start turning their attention to imaginary exponents?

When and WHY did mathematicians start turning their attention to imaginary exponents? I read on Wikipedia about Euler's correspondences with Bernouille and such, but it doesn't answer what exactly ...
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1answer
42 views

What did Noether contribute to the theory of integral invariants?

What did Emmy Noether contribute to the theory of integral invariants that wasn't already done by, e.g., Sophus Lie in his 1902 Über Integralinvarianten und Differentialgleichungen?
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Where can I find Lagrange’s equilateral triangular solution for arbitrary masses?

This answer to What kind of triangle is formed by three unequal masses in a circular restricted three body orbit? explains that In the Newtonian limit, an equilateral 3-body solution exists for any ...
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70 views

Literature on Mayan mathematics

I asked this question on math.se and they sent me here. It is well known that Mayan people used vigesimal (base 20) numeral system, and had had an advanced calendar system. Except for these facts, I'...
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387 views

The origin(s) of the word “elliptic” [closed]

The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ...
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Who first called the Brouwer Fixed Point Theorem “the crumpled paper theorem”?

Wikipedia attributes the remark to Brouwer himself, but I am extremely skeptical. Their citation goes to a webpage of a ? French educational TV show, where the remark appears to be a fictionalized ...
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77 views

Origin of the term 'index of a subgroup'

The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$. Why did someone decide to call this an 'index'?
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65 views

First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
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101 views

Origin of Fourier Transform (1878)

I located Joseph Fourier's book, Analytial Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?
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55 views

Who discovered Napier's number?

Who discovered Napier's number? I read Bernoulli calculated it, Napier discovered it but it is e because of Euler. I know how Bernoulli calculated it but where did Napier see it?
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2k views

What was Kolmogorov’s point of view in the philosophy of mathematics?

Today the standard interpretation of intuitionistic logic is the Brouwer-Heyting-Kolmogorov-interpretation which was presented independently by Arend Heyting and Andrei Nikolajewitsch Kolmogorow. ...
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1answer
95 views

Were $\sigma$-algebras defined for probability?

If you want a crash course in $\sigma$-algebras and probability spaces, you should absolutely read this amazing answer by @Sycorax on Cross Validated. Sycorax says something in particular though that ...
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1answer
80 views

What letters are used in this paper by Halin (1976)?

I was trying to read an older paper about treewidth by Halin (1976). He used different hand-written labels, obviously Sütterlin: I assume the hand-written letters to be (from top to bottom and from ...
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1answer
85 views

Significance level $\alpha$ values - who devised to use $\alpha = 5 \%$?

In a statistical hypotheses testing a significance level $\alpha$ has to be set. The most often, $\alpha$ is set to be 5 %, sometimes 1 % and 10 % values are used. Value of $\alpha$ tells us what is ...
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40 views

First historical register of an improper fraction [duplicate]

I'm looking for the earliest known written register of an improper fraction, that is, a numerical fraction in which the numerator is greater than the denominator (like 3/2). By the way, who invented ...
2
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1answer
97 views

Laplace's Treatise of celestial mechanics

I am interested in nineteenth-century astronomy and Laplace's Treatise of celestial mechanics is often mentioned as one of the most significant contributions to science in this period. The more than ...
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59 views

Did the mathematician Garrett Birkhoff ever work with or mention Feynman's path integrals?

Did Garrett Birkhoff ever work with Feynman's path integral? Did he ever work in his Many-Histories interpretation? Or at least, did he mention it in any of his articles?
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1answer
105 views

Is it fair to say that the Space Race incentivized research in pure mathematics?

I've only heard personal anecdotes about perceived stimuli and economic support to research in pure mathematics in the 1960s, presumably tied to the space race of the time. Is this something we can ...
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1answer
150 views

What is the basis of the claim that $F_5$ was fully factored in 1732?

The Wikipedia Page on Fermat numbers states that $F_5$ was "fully factored" in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form $$2^{n+...
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865 views

Arithmetic calculation before the 17th Century

Apparently Dijkstra wrote in an article in Datamation1 in 1977: It's very illuminating to think about the fact that some – at most four hundred – years ago, professors at European universities ...
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1answer
100 views

Were epicycloids from astronomy acceptable curves in Greek geometry?

My simplified historical understanding is as follows. Euclidean geometry accepted a limited number of geometrical objects (straight-edge and compass constructions, conics). Descartes' Géométrie ...
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1answer
97 views

Who discovered the indeterminate forms like 0/0?

Who discovered the indeterminate forms and how did they discover them? How did someone come to know that a particular form (fraction, product, sum/difference, exponent) is indeterminate? For example, $...
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1answer
100 views

When and where did scientific publications become the norm in mathematics?

In other words, how old is the practice of submitting mathematical work for peer review to specialized magazines? When/where it started to become the norm? My question is oriented toward the ...
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1answer
79 views

When did the study of the rate of convergence of algorithms begin?

I was reading a book about computational complexity theory and the author made a claim that the study of time complexity of algorithms started with a result on the upper bound on the number of ...
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2answers
92 views

When did the concept of probability density explicitly appear in mathematics?

I was reading the wikipedia article about the normal distribution and it's attribution by some historians of science to De Moivre(althought he lacked the concept of the probability density function) ...
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Apéry’s mysterious recurrence relation

A fairly detailed (14 page) account of Apéry’s original proof of the irrationality of $\zeta(3)$ is given in Julian Havil’s book The Irrationals which states that Apéry’s starting point is the ...
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172 views

For many years, were Emmy Noether and Helene Braun the only female mathematicians to obtain habilitation at Göttingen University?

Emmy Noether was the first woman in Germany to obtain habilitation in 1919. But I remember to have heard in the debate concerning the situation of women in academic mathematics that took place on the ...
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Books on elliptic functions

In the end of his address to the Mathematical Association in 1933 titled "The marquis and the land agent: a tale of the 18th century", G. N. Watson says: My final task is to express my gratitude to ...
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66 views

Why do Thai numerals look so different than Arabic numerals?

The Arabic numerals I am referring to are “1234567890”. I have read that Thai numerals, “๑๒๓๔๕๖๗๘๙๐”, are actually distantly related. Both descend from the numeral system invented by the Phoenicians, ...
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57 views

How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis?

Although I do not doubt in Riemann˙s originality, I would like to know how much complex analysis was developed up to the day when Riemann conjectured what is today called Riemann hypothesis and how ...
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81 views

Why did Jordan introduce his canonical form?

Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear ...
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Conjectures mentioned (and proposed) at International Congresses of Mathematics

I guess that every International Congress of Mathematics (Mathematicians) brings up some unresolved issues of various types but that also that at least some (if not all) ICM´s present some new ...
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1answer
121 views

When is the first use of Newton's method for root finding?

I saw this from Wikipedia. The name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in ...
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Where did the Delannoy numbers make their first appearance?

I am writing a short exposition on the central Delannoy numbers and would like to find the year which Henri Delannoy first introduced them in a formal setting. I believe Delannoy's initial ...
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146 views

Origin of $\ll$ notation

Vinogradov introduced the notation $$f(x) \ll g(x)$$ to denote that for some $C>0$, we have $|f(x)|\leqslant C\,g(x)$ for all $x$ under consideration; usually for all $x$ larger than a fixed ...
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1answer
77 views

How did Weibull derive the three parameter Weibull distribution?

How did Weibull or any other mathematician arrive at the above expression? I saw the 1951 paper, but it is not clear to me. In 1939 he had published a book called "A Statistical Theory of the Strength ...
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1answer
55 views

When was the Laguerre's method first used to approximate roots?

Is there a specific date when Laguerre published his root finding method? I found his 1880 note Résolution des équations numériques, but I am not sure if this is the source because I can not read ...
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1answer
289 views

Alternatives to «Lost in math: how beauty leads physics astray» by Sabine Hossenfelder?

Recently, I came across this book «Lost in math» that aroused my interest. Having read about half of it, I have to admit that I am not a big fan of Mrs. Hossenfelder's informal popular style of ...
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121 views

The Mathematical Principles of Natural Philosophy

Please, advise a good review of Isaac Newton's work "The Mathematical Principles of Natural Philosophy" with a detailed analysis of his mathematical ideas. Smth about 20-25 pages. Thank you in ...
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71 views

What did Dieudonne mean by “Theories in a state of dilution”?

In "A Panorama of Pure Mathematics" by Dieudonne, he said The history of mathematics shows that a theory almost always originates in efforts to solve a specific problem (for example, the ...
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231 views

Who pioneered the study of the sedenions?

I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (...
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What was Littlewood's quip about Hardy and plagiarism?

I'm searching for a quote by Littlewood about Hardy not giving proper credit. The story (as I remember it) is that Littlewood claimed uncredited authorship of something Hardy wrote, Hardy claimed it ...
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228 views

What did G.H. Hardy teach Ramanujan?

Ramanujan didn't know modern mathematics. he lacked idea regarding analysis. I found in Wikipedia- Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for ...
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128 views

Did Newton invent convex hulls?

The convex hull of a set of points appears recognizably in a 1676 letter from Newton to Henry Oldenburg describing Newton polygons. Is there an earlier precedent for convex hulls or is this their ...

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