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9
votes
2answers
167 views

Soft question: Unconventional proofs

I'm not sure if I understood it correctly, but one of my professors told us that one theorem was proved this way: A mathematician assumed the truth of the Riemann hypothesis and was able to prove a ...
3
votes
1answer
56 views

Do I need to prove $f(x)$ is convergent before defining it?

For example, I define $\sin x=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$, should I show that $\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$ is convergent for all real $x$ first, or I can ...
4
votes
1answer
132 views

Why is “Amenable Group” a pun?

"The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in ...
-3
votes
1answer
186 views

The Irrationality of 2

I am sorry it is not 'research level'. A quick answer will do. When I attempt using the Square root of 2 method to prove the rationality of Square root of 4 according to how it was done in a book, 2 ...
2
votes
2answers
144 views

Alternative model of Euclidean geometry

I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of ...
1
vote
2answers
2k views

What is a standard precalculus syllabus?

I'm about to start teaching a calculus I class next week and I was wondering what I can expect from my students. I'm a Brit teaching in the US so I am unfamiliar with the system. I am hoping that ...
3
votes
1answer
729 views

What does “rigor” mean in mathematics? [duplicate]

I spend a lot of time on math.se and even though I don't understand many of the questions posed, I try to understand what is being said or atleast wiki something to get some gist of the question and ...
5
votes
1answer
220 views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
0
votes
2answers
3k views

what is the best book for calculus? [duplicate]

I am looking for the best calculus book to use it to teach myself calculus. I have already had a bit of a search and these are what I have come up so far, But I have no idea which one is truly the ...
3
votes
4answers
191 views

Is math capable of predicting social evolution?

By the way math and statistics are evolving, it seems possible to me that with time, and as the population increases, we are going to be able to create mathematical models that predict social ...
25
votes
5answers
934 views

Level of Rigor in Mathematical Physics

I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
4
votes
3answers
1k views

Comparing Hilbert spaces and Banach spaces.

Why do we say Hilbert space is better than Banach space? I know that a Hilbert space has inner product (and so a norm) but Banach space has just norm.
0
votes
1answer
64 views

Permutation,Combination and Probability [closed]

I have been doing Permutation,Combination and Probability form quite a long time...even now I am not so confident about these...I can only tackle those problems which have concepts similiar to one tht ...
2
votes
0answers
175 views

Making Math a Job [closed]

As of now, I am pursuing a degree in math with my concentration being mostly on the applied side. I got into a grad level differential equations course over the summer and got an A. That being said, I ...
8
votes
1answer
449 views

Quick questions about studying math and physics?

Just curious about how people usually self study these subjects. 1) Is it the most efficient to read through the chapters, and write down theorems, definitions, and take notes of important parts and ...
20
votes
1answer
476 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
7
votes
4answers
396 views

Mere coincidence? (prime factors) [closed]

Whether some things in mathematics are mere coincidences might keep philosophers busy for 100,000 aeons, but maybe when such a coincidence gets exploited then it's not a "mere" coincidence any more. ...
1
vote
0answers
241 views

Question about the mathematics in actuarial studies

I tried Google but there isn't much information on this and I would really like some insights into actuarial studies, the mathematics involved and how it compars to the mathematics in a bachelor of ...
19
votes
1answer
461 views

Is there any mathematical meaning in this set-theoretical joke?

Recently I heard a joke: If an object exists, mathematicians call it a set and study it. But if an object does not exist, mathematicians call it a proper class and study it anyway. I wonder, ...
4
votes
3answers
209 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
7
votes
3answers
348 views

How was $e$ first calculated?

I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed. What mathematical principles are behind the value of $e$?
4
votes
1answer
160 views

Is 4D visualization necessary? [closed]

Is 4D visualization necessary in order to be successful at math (complex analysis for example)?
5
votes
1answer
88 views

Masters in mathematics with bachelors in different subject [closed]

I have a bachelor's degree in electronics engineering, and I'm fairly interested in discrete mathematics, is it possible to study masters in mathematics? If so would it be worth pursing it?
40
votes
11answers
2k views

What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
1
vote
0answers
45 views

''Technique manuals'' recommendation

In my studying of generating functions I got hold of this article and I liked it very much. Why? It's concise, does a good job of introducing you to the main ideas and shows a lot of uses of this ...
1
vote
0answers
432 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
6
votes
4answers
311 views

How do you retain your ability for contest-like math as you take more and more advanced math courses?

From my limited experience, it seems like many people loose their ability to do contest-like math as they go trough college. For example, before I posted it here I asked a math undergrad student (who ...
2
votes
2answers
218 views

Are there any other interesting functions such as $e^x$ whose derivative and integral are the same?

$e^x$ is interesting, but does anybody know if there are other functions that behave in an interesting way when taking the derivative/integral?
3
votes
6answers
291 views

What is the significance of logarithms in higher mathematics? [closed]

I search for an answer but nobody really is answering or understanding what I mean by "significance" and I really don't know how to explain so I'm hoping somebody out there could give me a decent, ...
8
votes
2answers
187 views

What is a fiber bundle? (for non-mathematicians)

How can I explain the concept of a fiber bundle to someone with no mathematical background?
0
votes
1answer
142 views

Discrete mathematics for someone from a non-mathematical background

I have been a software programmer for over six years and I'm from a non-mathematical background. Though I had some limited exposure to discrete mathematics in my college years it didn't leave any ...
8
votes
4answers
159 views

Dimension of an object?

One simple way to define dimension is "the number of numbers required to describe an object." If we consider the set of circles, we can describe each of them by one number -- radius, or ...
3
votes
2answers
274 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
4
votes
4answers
411 views

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a ...
6
votes
1answer
116 views

Who introduced the term “norm” into mathematics?

I've always been curious about the motivation behind the use of the word norm, as used in linear algebra and functional analysis, for a function that assigns a positive number to a vector. Who ...
49
votes
16answers
3k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
10
votes
1answer
496 views

Why are there mathematicians that do not use computers?

I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood ...
4
votes
2answers
190 views

Training to actively learn, think and use mathematics studied and construct everything on ones own

Someone said this: "Memorizing the entire Swedish dictionary will never make you mastering Swedish. Same to Maths, the only way is to actively think & use basics you absorbed so far to practice ...
2
votes
1answer
167 views

How is it shown that quintic equations can be solved by radicals and ultraradicals?

See this article from wikipedia: http://en.wikipedia.org/wiki/Bring_radical George Jerrard showed that some quintic equations can be solved using radicals and Bring radicals, which had been ...
4
votes
1answer
204 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
37
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
5
votes
1answer
144 views

Why does sum of squares appear in so many mathematical applications?

I have some little background in statistics, where in many applications summing of squares is an important calculation. Recently, I came across a mention that summing squares is involved in ...
5
votes
1answer
84 views

Why is one way easier than the other for the same object?

As I exemplify below, one methodology/strategy can be easier than the other for a given mathematical definition or object. Is there something deeper and more expansive, beyond the examples below, ...
1
vote
1answer
36 views

Is there any reason to prefer one phrasing over the other?

The following two sentences in the language of $\mathbb{N}$ are logically equivalent, in the sense that first-order logic alone is enough to get from one to the other. For all $a,b;$ if there exists ...
16
votes
6answers
809 views

What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
7
votes
7answers
200 views

Mathematical Games suitable for undergraduates

I am looking for mathematical games for an undergraduate `maths club' for interested students. I am thinking of things like topological tic-tac-toe and singularity chess. I have some funding for this ...
8
votes
4answers
381 views

What are applications of number theory in physics?

I was reading Goro Shimura's The Map of My Life. He wrote the following quote in the book. It made me come up with the title question. In particular, is there any application of the theory of modular ...
2
votes
0answers
208 views

The most fundamental papers in stochastic analysis

I have soft a question. What papers will be good to on start and allow me to make little step into research, without harm for reader. I am interested in an stochastic analysis. I am looking for ...
2
votes
2answers
384 views

Pure mathematics for engineers

I have recently completed my first year of Eng. Physics taking the standard math courses: Calculus, Linear Algebra 1 and 2, Multivariable Calculus and Numerical Analysis. Recently though I have been ...
3
votes
1answer
600 views

How to prepare for Integral Calculus (Calculus 2)

I'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I've ...