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4
votes
0answers
32 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
11
votes
0answers
56 views

When are two proofs “the same”?

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude ...
2
votes
2answers
62 views

How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily?

How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is ...
6
votes
2answers
214 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
1
vote
0answers
92 views

Is Euclidean geometry really a “dead” subject? If so, why? [on hold]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
1
vote
0answers
17 views

Application of theorems 'about free groups'

What consequences have theorem Any nonzero subgroup of free group is free or some another similar theorems? P.S. especially not-group-theretic applications.
2
votes
0answers
48 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
1
vote
0answers
34 views

Intuition for Entropy over Fractals

Is there intuition for "mathematical" entropy. I know that physical entropy tracks the order in a dynamical system, for thermodynamics. As entropy goes up, general randomness and disorder goes up. ...
1
vote
1answer
38 views

Understanding and teaching the concept of derivative

I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years ...
0
votes
0answers
17 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
-5
votes
2answers
103 views

How to practically make use of Mathematics? [on hold]

How to practically make use of Mathematics ? I have a basic question.How to use Mathematics in our modern day lives? Are there any ways by which we can make Mathematics come out of our classrooms and ...
1
vote
1answer
52 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
1
vote
1answer
31 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...
3
votes
0answers
67 views

How to understand if one is eligible for research?

What are the eligibility criteria for one to undergo research in Mathematics? Or should I place the question as what virtues of a student are given importance when one is interviewed for a PhD ...
0
votes
4answers
111 views

Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
1
vote
0answers
36 views

Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
1
vote
1answer
50 views

How is problem solving ability on an olympiad level relevant to mathematical graduate study/research?

I am starting on math later than many of my peers and hence have little to no experience in competitive problem solving. Is this a disadvantage during the study of the more abstract mathematics that ...
0
votes
0answers
39 views

Suggestions(Anything) regarding GRE Math Subject test [duplicate]

I am to appear for GRE Math Subject test probably this year or next year .I have basic knowledge of calculus 1,2,3 ,group theory ,Linear Algebra , Small part of real analysis . I haven't yet studied ...
0
votes
0answers
17 views

I hear that some operators don't have analytical properties. What does that mean?

The floor, ceiling, and mod functions are very useful operators, but in general discussion I've heard their usefulness called into question because of their lack of analytical properties. For instance ...
0
votes
0answers
38 views

if you had to choose a degree again, would you pick math? [on hold]

Pardon me if this isn't the right place to ask, but I wonder whether mathematicians on this website would study it again and choose the same career, if they could go back. Have you been disappointed? ...
-1
votes
0answers
45 views

I want to study Maths in France but have no idea about anything University related, let alone Uni abroad?! [on hold]

I'm 17, I live in the UK and I'm really passionate about Maths, I'm studying Maths and Further Maths A-levels (as well as some others) in my first year of what we call "Sixth Form College". I've ...
1
vote
1answer
46 views

Building a training program… for mathematics

I want to ask for your advice building my one-year training program for mathematics. Objectives: Keep 'mathematically fit' Improve for the pleasure Get competent at high-level economics and ...
1
vote
1answer
22 views

Expressing a line as a linear combination of two points on the line.

I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and y is the set of linear combinations $sx + ty$ where $s + t = 1$. I'm satisfied that this is true, ...
4
votes
3answers
91 views

Difficulty faced in solving maths problems

I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often ...
1
vote
0answers
37 views

Ideas for math problem solving class for undergraduate students in university

In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ...
5
votes
4answers
743 views

Albert, Bernard and Cheryl popular question (Please comment on my theory)

Here is the problem, I think that there is one point that makes the question ambiguous, I think they should explicitly say the reason why Albert knows that Bernard does not know the date. Case 1: ...
1
vote
0answers
40 views

A mathmatical question about Wolfram Alpha's loading graphic [closed]

It seems as though Wolfram Alpha's loading graphic is a cellular automata of some sort? If so, what is the rewrite rule? A reference would be appreciated.
1
vote
1answer
44 views

Do equations that rely on a fractional number of variables exist?

In statistics, data is usually fitted with trend lines. Usually you can get statistics back that say things pertaining to how correlated one variable is with another. For instance if a variable $x$ ...
1
vote
2answers
43 views

Numeric system without “zero”, how to explain importance of zero to average person?

As we all knew that Aryabhata (http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero) invented zero ($0$) in our number system. I have few questions about it. How did the numeric system ...
3
votes
2answers
37 views

Concept of random sample? I have a truly problem understanding it.

I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). Suppose that we have a box with ...
0
votes
0answers
44 views

Collatz algorithm generalization try-out (Collatz k-algorithm)

Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of professor Lagarias about it. Everything was so interesting (and I ...
1
vote
1answer
28 views

Question about optimization

I have a question about maximization/minimization problems. I have noticed that for almost all the practice problems that I have had that ask to find the sum of numbers and minimize product or ...
4
votes
2answers
94 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
0
votes
2answers
36 views

How should trigonometric expressions be simplified?

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to ...
0
votes
0answers
11 views

Formula to convert currencies

I know this currency exchange rates: 1.000000 USD = 0.943837 EUR 1.000000 USD = 0.683463 GBP What would be the formula to find: 245 EUR = ??? GBP
32
votes
10answers
2k views

How do mathematicians find formulas?

How do mathematicians find formulas? For instance, the area of a triangle is $$\mathrm{area}=\frac{\mathrm{base}\times \mathrm{height}}{2}.\tag{1}$$ When I study maths, the book I am using tells ...
1
vote
2answers
38 views

Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

In the preface of Munkres's Topology, he writes, Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let ...
3
votes
2answers
45 views

Particular case of an Implication

Let's take the following propositions : 1 - "If Bill Gates is poor then Bill Gates is rich". 2 - "If Bill Gates is poor then the moon is made of cheese". Both propositions are inevitably true ...
-1
votes
1answer
41 views

Pre Requisites for understanding Ramanujan's Notebooks [closed]

I have basic knowledge of high school in Maths . I also have studies (though not rigrously) Calculus ,Linear Algebra , Real Analysis (just small part ) .But i want to redo again because i want to ...
1
vote
1answer
64 views

Learning mathematical concepts

Our teacher loves to test us on pure concept based questions and test if we really know what we are doing when learning a particular lesson. For example, when we first started learning about ...
2
votes
0answers
45 views

How does Godel Escher Bach support Artificial Intelligence? [closed]

Typically, Godel's Incompleteness theorems have been used to argue against the possibility that the human mind is essentially equivalent to a formal system. However, in Daniel Dennett's book "Darwin's ...
1
vote
0answers
39 views

How to remember a proof for a long time

A very basic question of mine: Whenever I read a proof I am able to remember it only for a couple of months. But I really want to remember it for at least one year or so if not more. Is it the ...
3
votes
1answer
31 views

Is the closure axiom necessary for algebraic structures defined via a binary operation?

Numerous algebraic structures are often defined as a set $X$ equipped with a binary operation $f:X\times{X}\rightarrow {X}$ that satisfies some set of axioms. Since the image of $f$ is always in $X$ ...
0
votes
0answers
50 views

What is the general consensus on Non-standard analysis where you live? [closed]

Most mathematicians/math professors I've met seem to think non-standard analysis is not a real branch in mathematics, that is was sort of conceived to give a "more intuitive" understanding to things ...
1
vote
0answers
26 views

What is the value of an Infinitesimal?

In the regular type of math "0.999..." is the same thing as the value 1. In some other different kind of math they say that "0.999..." is not the same thing as the value 1. Where 1 is > "0.999..." ...
44
votes
4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
4
votes
1answer
96 views

Why is it called the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers. What I don't ...
3
votes
1answer
103 views

Mathematics books that tell you what is really happening? [closed]

Many book I've read teach you symblobic manipulations instead of pointing out what's really happening. So my question would be: what mathematics textbooks don't do that? Books that rather than listing ...
8
votes
1answer
71 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
2
votes
0answers
52 views

How much math was “Broken” by Russell's Paradox?

As you know, the phrase "the set of all sets that don't contain themselves" caused a paradox that "broke" (made trivial) Naive set theory. How much mathematics had to be redone because of this? Most ...