For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

learn more… | top users | synonyms

2
votes
0answers
96 views

What can you do with rational solutions to linear equations?

I'm currently doing a project and for part of it I've been looking at rational solutions to linear eqautions in two vaiables. ie. ax+by=c. I'd like to add a bit about what we can use these types of ...
4
votes
3answers
1k views

Can the word “derive” be used to mean “take the derivative of”?

Back when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be ...
7
votes
1answer
305 views

Finding 'verbally smallest' element of a finitely generated group

Let $G = ({\Large\ast}^n\mathbb{Z})/K$ be a group, and for each $g \in G$ define $l(g)$ as the smallest positive integer $m$ such that $g = g_1 \ldots g_m$, where each $g_i$ is a generator of $G$. Now ...
17
votes
6answers
3k views

What does Khan Academy have to offer? Depth? Rigor?

Khan Academy - http://www.khanacademy.org/ - is often cited as a great online resource for learning mathematics and other subjects. I have heard many good things about this website and was wondering ...
20
votes
4answers
1k views

A question about the definition of a neighborhood in topology

Let $X$ be a topological space, and $x \in X$ be a point. There are two prevalent conventions on how to define a neighborhood of $x$: 1) A neighborhood of $x$ is any open subset $W \subset X$ such ...
22
votes
9answers
2k views

Mathematics and Music

I have heard that, in recent years, many mathematicians as well as music theorists have applied different branches of mathematics to music. I would like to know about some books/resources relating to ...
9
votes
4answers
6k views

Difficulty level of Courant's book

I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John.I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the ...
1
vote
0answers
856 views

Working through Math 55 problem sets as self-study

I am not a professional mathematician, but have learnt Engineering Mathematics in college and worked through parts of maths textbooks myself. The latter include the first few chapters of include Real ...
5
votes
3answers
885 views

Why graph a function?

Please enlighten me as to how graphing a function helps. I can see a graph's utility with simple functions as they instantly give you value of dependent variable. But ignoring them and considering ...
9
votes
2answers
2k views

Category theory vs. Universal Algebra - Any References?

After seeing the answer to the question, "Category theory, a branch of abstract algebra", I would like to ask a question, Are there books/papers discussing the difference/indifference/comparison ...
14
votes
1answer
846 views

Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
8
votes
2answers
437 views

Different standards for writing down logical quantifiers in a formal way

What are standard ways to write mathematical expressions involving quantifiers in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic ...
2
votes
2answers
190 views

How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g. "Let $X$ be a normed vector space. Then $X$ is called a ...
2
votes
1answer
99 views

What is the “conjugacy problem for differentiable maps”?

A couple of days ago our professor reviewed some of the exercises we had to do and one of them involved giving an example of a conjugacy class in a group. Someone gave an example that involved ...
4
votes
2answers
587 views

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),......)=1$ generates for natural ...
9
votes
3answers
712 views

A Concrete Approach to Category Theory

Is there a way to learn Category Theory without learning so many concepts of which you have never seen examples?
21
votes
6answers
1k views

Do online lecture recordings hurt or help math students at university? [closed]

Continuing with my series of soft questions on teaching practice: My university uses a system whereby all lectures (given via computer slides or hand-writing on a sort of overhead projector called a ...
4
votes
1answer
522 views

How many letters of recommendations is too many [closed]

I am planning to apply for post-doc positions soon. Most universities require three letters of recommendation and most of peers suggested that I should have at least three research letters and one ...
3
votes
1answer
2k views

Why do some people like analysis while others like algebra? [closed]

In general, what traits do you think may cause people to favor one mathematical field over another? I have a friend who swears by algebra but hates analysis, and I like analysis better than algebra ...
34
votes
6answers
1k views

Attitude towards exercises in mathematics

I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too ...
3
votes
2answers
90 views

Pronunciation of $M(x)$ and $m(x)$

Suppose I use two functions and I denoted them by lowercase and uppercase letters $m(x)$ and $M(x)$. Of course, I have to distinguish them somehow. How do I read this? Is capital/uppercase $M$ of ...
6
votes
4answers
441 views

Intuitive results with non-intuitive proofs?

Are there mathematical theorems that sound trivial and are obviously true, but are really tough to rigorously show? For example I stumbled across this question. The author asks how to show that two ...
15
votes
3answers
632 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
8
votes
2answers
230 views

I feel the need to prove every result for myself

I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I ...
8
votes
4answers
836 views

How To Reach The “Next Level” of Mathematics

I am a junior-high pre-algebra student. I feel that my class is holding me back, so I wanted to learn "higher-level math". So what should I learn now? What do you believe is a "next step"?
8
votes
2answers
486 views

Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original ...
18
votes
3answers
956 views

“Best practice” innovative teaching in mathematics

Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering. The existing ...
2
votes
2answers
164 views

Weak Lower Bound in Apostol's “Number Theory”?

In Apostol's "Introduction to Analytic Number Theory" on page 7 he introduces Fermat numbers of the form $F_n = 2^{2^{n}} + 1$ where $n$ is a non-negative integer. He then states that The ...
4
votes
0answers
657 views

Topic appropriate for undergraduate research program

This may be not appropriate to ask this kind of question here. But please feel freely helping me. In our Maths faculty, there is not a program like Undergraduate Research Program(URP), however, the ...
2
votes
1answer
302 views

Calculating BMI (Body Mass Index)

If I'm given the following values: Weight = 15st 9lbs Height = 5.8 ft If we want to calculate the BMI, would it be: 33.30? ...
12
votes
1answer
430 views

Mathematicians who started out studying something else

I did not know much about modern mathematicians's biographies. One of my friend has just told me Edward Witten was a history student when he was undergradute. Are there any different modern ...
3
votes
3answers
788 views

Folland & Functional Analysis

I'm reading Folland's Real Analysis to learn some basic functional analysis. I read through his section Normed Vector Spaces and could make my way through most of the exercises I attempted. I am ...
7
votes
3answers
1k views

Experiences with Kumon

We have enrolled our 5 year old son in Kumon which is an after school math and reading enrichment program of Japanese origin. While he is learning lots of things (currently learning how to add i.e., ...
1
vote
1answer
202 views

Algebraic Topology in Simple Terms

I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So ...
6
votes
1answer
316 views

Concrete Categories Where Epis are Just Surjections

Before I begin let me provide some background to fix notation/make the post more readable to interested outsiders. In a category $\mathscr{C}$ we say that a morphism $X\xrightarrow{f}Y$ is an ...
18
votes
2answers
394 views

Mathematical suggestions for a long commute

I have quite a long commute on my way to my university (about 75 minutes) so I was wondering if there's a math-related podcast or something of the sort you'd recommend. Exercises are appreciated as ...
23
votes
5answers
770 views

relaxing maths papers

I am looking for some relaxing, funny maths article that one can read when tired. For instance, I found Bouton's paper "Nim, a game with a complete mathematical theory" very funny. If you know ...
2
votes
2answers
100 views

What are equivalence classes that would not have (1+1 and 2 ) or (2 x 3 and 6) in same classes?

Are there any ordered equivalence classes that can be used to distinguish 1+1 and 2 , On one side there is an operation and on the other side just a single number. This becomes more obvious when ...
3
votes
3answers
532 views

What Does the Associative Property Mean Intuitively Across All Notational Schemes?

You can find descriptions of associativity as intuitively meaning that the order of operations performed does not matter, e. g. such as that of Wikipedia. However, if you write what associativity ...
6
votes
5answers
988 views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...
17
votes
5answers
1k views

Do mathematicians Switch Fields of Expertise?

I'm curious if it is common or quite rare for mathematicians to change their area of expertise, whether it be after their PHD or mid-career, how radical this change might be, whether it's often for ...
4
votes
2answers
716 views

Deepest theorems with simplest proofs [closed]

Which are the deepest theorems with the most elementary proofs? I give two examples: i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function ii) Proof that the halting problem is ...
34
votes
2answers
1k views

How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in ...
8
votes
2answers
222 views

The role of writing in understanding concepts

I am an engineer student from Norway who is not that fascinated with "engineering maths", so I am trying to work through Spivak's Calculus book on my own. I know that there is no "how to" manual for ...
6
votes
3answers
469 views

Is there any good reason for a programmer to study geometry?

I'm a programmer and I've recently come back to math hoping to sharpen some of my skills. I did well in math in high school. I also did math competitively. I majored in music in college, so I stopped ...
30
votes
8answers
5k views

How can I learn to “read maths” at a University level?

When I look at math, it's like my mind goes fuzzy. The only way to describe it is in terms of what I can relate it to. You know how when you read, you see the letters and words, but your brain picks ...
3
votes
4answers
351 views

Getting Help in Abstract Math Classes

This semester I am taking abstract algebra, and I am finding the homework to be much more difficult than calculus or linear algebra. The book my class uses is Fraleigh's Abstract Algebra. Although I ...
1
vote
4answers
807 views

Totally lost in calculus [closed]

I currently have Calculus and am half way through the semester. My problem is i really didnt pay attention. We are now upto Implicit Differentiation. And i am totally lost. Do i still have time to ...
18
votes
6answers
3k views

What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
5
votes
2answers
173 views

differentiability and continuity

I have a question which is quite "candid" and for which I can hardly formalize properly every concepts but nonetheless, I was wondering if there were some topoligical and/or algebraic structure ...