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What are some of the Hardest Unsolved Mathematics Problems? [closed]

At the moment, are there any major unsolved mathematical problems yet to be solved, and do they have any prize associated with the solving of them? Furthermore, is there any particular reason that ...
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2answers
165 views

How did the name “The Calculus” come about, was there a reason or just good marketing?

This is a historical and lighthearted question about etymology. The area of mathematics that deals with limiting processes over real numbers (Real Analysis) or real vector spaces, or even complex ...
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Understanding a medieval approximation

A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part: … that the ...
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German translation needed - final sentences of a paper by Hilbert

I am translating a paper by Hilbert into English. I am finished except for the last few sentences, which are confusing me. If anyone can give me a rough/quick translation it would greatly appreciated. ...
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1answer
43 views

How has the teaching of (undergraduate) Set Theory changed over time?

I'm writing an Essay on Set Theory, and realized it was formulated quite recently, so I thought it might be cool to have some first person accounts. Russell's Paradox was discovered just around a ...
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Is there any way to retain Russell's original proof of induction in Appendix B of PM 1925?

Recently I was reading this question again and the following question occurred to me, Can there be some new interpretation of the system of PM $1925$ so that Russell's proof of $^\ast89.16$ is not ...
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343 views

The three unsolved problems of antiquity

In Sidelights on the Cardan-Tartaglia Controversy (Apr., 1938) by Martin A. Nordgaard in the National Mathematics Magazine, Vol. 12, No. 7, pp. 327-364, it is written on the first page The ...
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0answers
26 views

history of holomorphic implies analytic and goursat theorem

I'm studing complex analysis and am curious about its history. Did Cauchy know that holomorphic functions (to have derivative in every point of an open set) are infinitely derivable? and that they ...
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2answers
63 views

why soh cah toa is right?

i am confused by the sine of an angle, (it might appear evident for some of you but please i am not an expert ). sine of an angle is says to be the half of the magnitude of the chord of 2 time the ...
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1answer
48 views

How to graph in hyperbolic geometry?

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$....
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1answer
33 views

What is the example called, where someone was wrongly convinced of a sequence function because of naive induction.

I remember I have seen a classical example of a mistake, where someone was convinced that a sequence defined somehow had a close form, which did in turn work until some very high $n$. I think the ...
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1answer
44 views

How are gigantic primes actually defined in the 1992 article by Samuel Yates?

The Prime Glossary states: In a 1992 article, Samuel Yates coined the name gigantic prime for any prime with 10,000 or more decimal digits (he had also coined the term titanic primes a decade ...
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214 views

Why do we use degrees? [closed]

I see a lot of people who ask why we use radians instead of degrees. But why do we use degrees instead of radians. In the cases we use degrees instead of radians, what convenience does it bring? The ...
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35 views

Weak convergence in probability and functional analysis

Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have $$\int\...
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60 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
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1answer
62 views

What is the first absolutely normal number to be discovered?

What is the first absolutely normal number to be discovered? Is it the Chaitin's constant? $$\Omega_F = \sum_{p \in P_F} 2^{-|p|}$$
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9answers
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Why are the Trig functions defined by the counterclockwise path of a circle?

My understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I understand ...
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0answers
23 views

Oka's coherence theorem and Cartan's theorem A and B

So reading up on some articles I've found that whilst attempting to characterize Oka's coherence theorem in sheaf-theoretic language, Cartan was ultimately led to formulating his theorem A and B in ...
3
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2answers
88 views

Mathematic books with historical and original view

I am looking for books along the lines of history of mathematics but I have some conditions; History must not be the main aim of the book, the main aim of using historical context should be ...
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2answers
390 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
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1answer
150 views

Personal notebooks of a Fields medalist

I once read that some Fields medalist published all of his personal handwritten notebooks, and that they are freely available somewhere on the net. I can't remember whose mathematician it was, so I ...
4
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1answer
92 views

Why isn't '&' used for logical conjunction?

There is a beautiful and well-established logogram for "and" that is known to virtually every more or less educated person in the world - it's the ampersand '&'. It's completely unambiguous, as ...
3
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1answer
39 views

Why can we identify complex numbers as points on a plane?

Modern mathematicians seem to define the complex number $a+bi$ as the ordered pair $(a,b)$, with the usual rules for complex addition and multiplication. I'm reading a book on the history of the ...
3
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1answer
60 views

Gauss: The study of Euler's works…

I keep coming across this quote by Gauss but I haven't actually been able to locate the original source: “The Study of Euler’s works will remain the best school for the various fields of mathematics ...
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2answers
83 views

Cauchy's contribution

Sometime, I believe perhaps 2 years, ago I asked a question about breakthroughs, such as those within mathematics and physics which may lead a whole discipline forwards in many ways. One example from ...
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7answers
371 views

On a definition of manifold

In the book Mathematical Masterpiece, on page 160, the authors wrote that A manifold, in Riemann's words, is a continuous transition of an instance I know a manifold is something glued by ...
3
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1answer
109 views

why can't quintics be solved by radicals and the relevance of permutations of roots of polynomials

I am seeking to learn about the motivation in the development of group theory. It has been a few years since algebra, and we got as far as rings and fields. I am aware that there were several ...
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4answers
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Why weren't continuous functions defined as Darboux functions?

When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like "Continuous functions are those you can draw without lifting your pen" With this in ...
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0answers
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Historical use of k in proof by induction

Does anybody know the history of why the symbol k is used in proof by induction? As an example, in physics the symbol p is used for momentum because Newton called it impetus, and the letters i and m ...
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2answers
190 views

Andrew Wiles' Abel Prize for FLT - delayed or not? [closed]

Andrew Wiles was recently awarded the Abel Prize for his work proving Fermat's Last Theorem (FLT). The Abel Prize has existed for 14 years. From my layperson's perspective, it would seem that he ...
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1answer
60 views

Mersenne numbers fail primality test at 2047 itself. How could we believe Mersennes are primes?

M$_{11}=2047$ is a composite number. How could one, not check the primaility of such a small number and believe that all Mersenne numbers are primes?
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1answer
48 views

Why is it called the *Inverse* Galois Problem?

This is just a very quick question and hopefully not poorly received. Question: Why is it called the inverse galois problem? The very brief statement given on wikipedia says Is every finite ...
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0answers
63 views

Largest prime known to ancients

As is well known, Fermat couldn't check the primality of $F_{5} = 2^{2^{5}} + 1$. This raises an interesting question : what was the largest prime number that was known to ancients (particularly ...
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Why is the Weierstrass test called the M-Test [duplicate]

Is there any reason why we call this test an $M$-test? The presence of $M_n$'s in the standard formulation?
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4answers
134 views

Who found the expression $n^2 - n + 41 $ for generating prime numbers?

I am doing some research and I cannot seem to find the answer anywhere so does anyone know who found the expression $n^2 - n + 41 $ for generating prime numbers?
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148 views

Who is the mathematician “Jacques” in this anecdote?

Who is the mathematician "Jacques" in this anecdote, which I read on p. 260 of The Mathematical Magpie by Clifton Fadiman, who quotes it from the 1942 memoir The Last Time I Saw Paris by Elliot Paul? ...
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1answer
30 views

On applications of Alexander's Theorem

I would like to know a bit about applications of the Alexander Theorem from Knot and Braid Theory. I would be very interested in learning about possible applications for the description of everyday ...
0
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1answer
74 views

Developments from Charles Peirce's logic diagrams?

These last weeks I have been revisiting Charles Sanders Peirce's logical or thought diagrams (what he called, alpha, beta and gamma diagramms) and I found many of them highly interesting. Some ...
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5answers
2k views

Why are turns not used as the default angle measure?

Why is $2\pi$ radians not replaced by $1$ turn in formulas? The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?
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1answer
49 views

Order of operation in math

Who decides order of operation in any math calculation. Is it scientific or arbitrary? e.g. 1+4-6x7+7
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0answers
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About Special and Extra-special $p$-groups

A $p$-group $G$ is said to be special $p$-group if $Z(G)=[G,G]=$ elementary abelian. A $p$-group $G$ is said to be extra-special if $Z(G)=[G,G]=$ elementary abelian of order $p$. The questions ...
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2answers
101 views

Differentiation and integration

Which came first : Differentiation or Integration? If one of them was developed to solve certain types of problems, was the other developed for backward compatibility, or was it an independent ...
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2answers
98 views

How do people calculate values for trig functions?

This may sound like a stupid question, but I'm wondering how people originally calculated specific values for trig functions before calculators existed. Did they just draw circles and manually measure ...
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3answers
69 views

Fermat's Challenge of composition of numbers

In his letter to Carcavi (August 1659), Fermat mentions the following challenge There is no number, one less than a multiple of $3$, composed of a square and the triple of another square. ...
3
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1answer
64 views

Are there any instances of significant progress deriving from mathematical 'silliness'? [closed]

Last night I thought I'd be silly finding the eigenvalues of a $2\times2$ matrix $A$ with real components. Instead of calculating $\det(A-\lambda I)=0$ I tried to compute the determinant by ...
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About the domain of the Gamma function

I started to read about the history of the Gamma Function. There are three places I liked most, The early history of the factorial function (p. 239 - 243) Leonhard Euler's Integral: An Historical ...
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5answers
235 views

Is $22/7$ an often used approximation for $\pi$?

It is $\pi$-day and the internet is full of stories about $\pi$. One story mentions that "an approximation -- $22/7$ -- is used in many calculations." I have never actually used $22/7$ as an ...
5
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2answers
133 views

What will mathematicians do when they run out of letters in the Greek and English alphabets? [closed]

Like x,y,z are commonly understood to be dimensions and theta is an angle and Pi is a specific irrational constant, and Tau is half of Pi, etc. etc. etc. They must be running out of letters by now. ...
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Define the word 'smooth' in explict mathematical language?

Note in the comments of this question: Create a formula that creates a curve between two points ... that someone was asking me to define what is meant by the word "smooth". Given that one of the ...
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Ignoring the lack of rigor, is this a fair argument to make when considering if 0^0 should be equivalent to 1? [closed]

The Professor of Mathematics argued that 0^0 is undefined because the limits $0^x$ and $x^0$ as x approaches 0 don't agree. That seemed logical to me, but then Scott pointed out in the comments that ...