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
23 views

Very Basic Numerical Methods Book for Freshman students

To cut a long story short; the nature of this degree (it's not a college degree) is such that numerical methods is treated shortly after Calc I (single-var) and linear algebra, but before multi-var ...
-2
votes
2answers
33 views

General results on the change of the parity of a number by repeatedly dividing by 2 [closed]

I know this question may seem open, but I'm a bit interested in figuring out, getting some ideas, or at least getting some sources on how the parity of a number is affected by repeatedly diving it by ...
12
votes
0answers
221 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
2
votes
4answers
205 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 ...
-1
votes
1answer
23 views

Functional analysis as a prerequisite [closed]

Can someone give me an example of a mathematical field in which functional analysis is a prerequisite?!
0
votes
1answer
54 views

Interpretation help: Showing that Riemann Hypothesis holds “almost surely”

I was perusing this textbook on algorithmic number theory, where I came across this page where they appear to prove that the Riemann Hypothesis holds almost surely. This seems like an odd statement ...
0
votes
0answers
40 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
1
vote
1answer
35 views

Why are alternating divergent series generally easier to evaluate? [closed]

Why is it that alternating divergent series tend to be easier to evaluate or that there are more ways to evaluate them? Is there a particular reason for the difficulty to evaluate series that don't ...
21
votes
10answers
2k views

Is basis change ever useful in practical linear algebra?

In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?
1
vote
2answers
86 views

a healthy perspective on “knowing everything” [closed]

This is a question about attitude, but related to math studies. I have trouble with two things: 1. making "normal" progress in my learning and 2. having the satisfaction that I understand what is ...
23
votes
4answers
5k views

Real life examples of commutative but non-associative operations

I've been trying to find ways to explain to people why associativity is important. Subtraction is a good example of something that isn't associative, but it is not commutative. So the best I could ...
-6
votes
2answers
211 views

What is the most complex mathematical topic? [closed]

I'm a simple man living my simple life and often I like to learn more about math and science. Today my daughter asked me about how many numbers are there and I explained that there are infinite ...
0
votes
2answers
44 views

How to prove that Lebesgue outer measure is monotone?

It is very clear that if $A \subset B $ then $m^*(A)\leq m^*(B)$. But how to prove it ? Most of the books says that it is obvious. But what is the proof ?
2
votes
0answers
31 views

Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
3
votes
1answer
85 views

How to figure out the “idea behind” proofs in analysis?

I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty ...
0
votes
1answer
15 views

Applying a vector of functions on vectors

Is there an adequate mathematical representation (operator $\star$) to apply a vector of functions to another vector of values, element by element? Something like the following: $$ \left[ f_1, ...
4
votes
1answer
59 views

How Do You Check Your Computations?

I am already a grad student, but sadly I still have problems with careless computations. In my most recent mid-term (multivariable analysis), I lost 21 points out of 100 because of calculation errors ...
0
votes
0answers
13 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 ...
0
votes
1answer
27 views

Difference in Notation for Vectors in Linear Algebra & Multivariable Calculus

Often in Linear Algebra we see vectors depicted either in Column or Row Form as : Linear Algebra : Vector in Row Form $$ \vec{V}^{\,} = \left[x_1,\ldots,x_n\right]$$ OR Linear Algebra : Vector in ...
4
votes
2answers
655 views

What's the most effective ways of teaching kids - times tables?

I'd like to help a $6$ year old who already has a pretty good grasp of $2$, $5$, and $10$ times tables.
2
votes
0answers
40 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
2
votes
1answer
36 views

Reading a Matrix

This is a softer question, but I'm having trouble keeping straight all of the information that a matrix provides you with straight in my head. All I know is that the rows correspond with the codomain ...
6
votes
1answer
101 views

How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
16
votes
5answers
2k views

Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
0
votes
0answers
11 views

Spanning Spaces by Different Basis

I have a query related to spanning space by two bases $S_1=\{V_1+V_2, V_3, V_1-V_4,V_3-V_2\}$ $S_2=\{V_1, V_2, V_3, V_4\}$ Can we consider spaces generated by $S_1$ and $S_2$ to be equivalent?? Or ...
1
vote
0answers
22 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
3
votes
1answer
82 views

What is the most general way to think about Integrals?

Given a single-variable scalar function, $f : \mathbb{R} \to \mathbb{R}$ The "area under the curve" (of the graph of the function $f$ in $\mathbb{R^2}$) is given by $$\int_{a}^{b} f(x) \ dx = ...
11
votes
2answers
156 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
3
votes
0answers
72 views

Most complicated proof of Pythagoras

Usually a mathematician aims for clarity and elegance when conducting a proof. However, the antimathematician buries all hope of assimilating intuition and reasoning. To illustrate this, I seek the ...
5
votes
4answers
735 views

How important are inequalities?

When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important ...
1
vote
1answer
39 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
1
vote
2answers
53 views

Linear Algebra Trivia: Can anyone identify this class of matrix?

Consider a matrix: \begin{pmatrix} 0 & -y & x \\ x & 0 & -y \\ -y & x & 0 \\ \end{pmatrix} where $x,y$ are positive real numbers I wish to identy the most "specific" class ...
6
votes
2answers
219 views

“There is no set containing everything”? [duplicate]

I was reading this question regarding codomains, and I found something interesting in User134824's answer: "On the other hand, owing to the set-theoretic fact that "there is no set containing ...
1
vote
1answer
37 views

What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
2
votes
3answers
56 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
616
votes
160answers
38k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
1
vote
1answer
54 views

Should I remember the proof of mathematical theorems(every step)?

The problem is, that when I am reading the proof of mathematical theorem(in my case - it is calculus), U understand the idea and every step of proof. But i can't prove the theorem individualy even if ...
0
votes
0answers
25 views

Difficulty during self-studying unique set proofs

I have been following Velleman's How to prove it and working through it on my own. I am working full time now so I can only study after work without any other help. It's been going fairly ok until I ...
66
votes
30answers
31k views

Best Maths Books for Non-Mathematicians

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
3
votes
1answer
309 views

Why is trigonometry important in calculus? [closed]

I need to write short note why trigonometry is important is calculus and engineering mostly for presentation. I am not focusing on on what topic it specifically it appears (because I am guessing the ...
2
votes
0answers
48 views

Geometric derivative, existance, interpretation and usefulness.

What if one was to define the limit $$\lim_{h\to 0} \sqrt[h]{\frac{f(x+h)}{f(x)}}$$ If we play around with h=1 and for the gamma function this would be a linear function for positive x: $$\Gamma(n+1) ...
44
votes
25answers
3k views

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus. My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have ...
19
votes
6answers
6k views

Is memorization a good skill to learn or master mathematics?

I sometimes spend inordinate amounts of time memorizing math articles or theorems/proofs or formulas. My question is "am I wasting time?" and will 'active thinking' or 'working out problems' be faster ...
3
votes
1answer
59 views

How should one characterize mathematical conclusions? [closed]

I have posted this in Philosophy SE as well because I feel that it is appropriate both here and there. As practiced, mathematical proof seems not to be an explicit formal deduction within a formal ...
103
votes
34answers
17k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
1
vote
1answer
38 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
0
votes
1answer
48 views

Sum of integers [duplicate]

I cannot accept that $\sum_{n=1}^\infty n = -\frac{1}{12}$. It should be that such a sum is divergent. That it is divergent is useful for the Test for Divergence in many such problems. I feel like ...
4
votes
3answers
75 views

How useful are non-square matrices in maths or sciences?

I know that a matrix will be either square or rectangular matrix. I know that square matrices are used to solve a system of linear equations. But what's the use of rectangular matrices, why do we ...
3
votes
1answer
58 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 ...
3
votes
4answers
313 views

Definition of an Algebraic Objects

How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of ...