For questions about gaining and achieving motivation.

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6
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1answer
46 views

Motivation for and differences between properties of measure, outer measure

This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the big picture relationship between properties of measure and outer measure. I ...
7
votes
3answers
229 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
0
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0answers
28 views

Pairwise Independent Events that are not Independent

Let a ball be drawn from an urn containing 4 balls, numbered 1,2,3,4 . let E = {1,2}, F = {1.3}, G = {1,4} . If all outcomes are assumed equally likely, then P(EF) = P(E)*P(F) = 1/4 P(EG) = ...
4
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0answers
51 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
0
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1answer
32 views

Uses of Mersenne primes in math

There is an international search for Mersenne primes. The project is huge. But what are the uses of Mersenne Primes in math? Do they have any other properties other than being of the form $2^n-1$?
0
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2answers
53 views

Where do I start learning and how do I stay motivated?

I just finished intermediate algebra in college. I liked it and breezed through it. I feel like at this level of math I can only expect dull and unenthusiastic teachers (which has been the case ...
0
votes
1answer
198 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
1
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0answers
88 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
7
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1answer
47 views

Motivation behind parameters

This article shows a technique of evaluating a definite integral by introducing a suitable parameter. This however doesn't throw light on motivation for introducing that particular parameter. For ...
2
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0answers
34 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
1
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2answers
67 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
1
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5answers
91 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
0
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1answer
61 views

Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
5
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3answers
99 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
2
votes
2answers
68 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
5
votes
5answers
418 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
10
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3answers
396 views

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
17
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4answers
747 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
17
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4answers
1k views

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ ...
2
votes
1answer
160 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
16
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5answers
1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
1
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2answers
77 views

Motivation of vector bundle of a manifold

I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it? Thanks! Also, i have to ...
2
votes
1answer
43 views

Why do we require $X$ to be Hausdorff when defining $C(X)$?

In Efton Park's "Complex Topological K-theory", he begins a section by defining the Banach algebra $C(X)$ (continuous functions $f: X \rightarrow \Bbb C$), presupposing that $X$ is Hausdorff and ...
7
votes
2answers
191 views

Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
7
votes
4answers
772 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
0
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1answer
54 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
2
votes
1answer
248 views

Is RCA-Rudin one of the worst textbooks? [closed]

Someone told me that "Real and complex analysis - rudin" is actually rated a bad textbook among researchers, since it gives no motivation. Is it true? I agree that this text provides less ...
2
votes
2answers
66 views

Practical motivation for analysis at school

I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job. I could not remember an impressive example for most topics. Let's assume ...
1
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2answers
67 views

What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
0
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0answers
26 views

What's an application of that the set of invertible linear operators is open?

Let $V$ be a finite-dimensional normed space. Let $\Omega$ be the set of all invertible linear operators on $V$. Then, $\Omega$ is open in $(\mathscr{L}(V),||\cdot||_{op})$ and $f:\Omega\rightarrow ...
20
votes
4answers
599 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
2
votes
2answers
120 views

What is the importance of the spectral theorem?

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...
2
votes
1answer
64 views

For what reason, the surface measure represents the surface area?

Let $||\cdot||$ be a norm on $\mathbb{R}^n$ (It's an arbitrary norm, not 2-norm) Define $S^{n-1}=\{x\in\mathbb{R}^n : ||x||=1\}$ Let $\mu$ be the n-dimensional Lebesgue measure. Define ...
0
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0answers
29 views

What kind of norm is it in the definition of $S^{n-1}$?

Definition (wikipedia) $S^n\triangleq\{x\in\mathbb{R}^{n+1}: ||x||=1\}$ is said to be a 'n-sphere' What norm is it referring to ? I have proven that ...
5
votes
1answer
78 views

Reference request: Introduction to Finite Group Cohomology

I don't know anything about group cohomology and I'd like to. What is the best text to learn this subject? I'd prefer as soft an introduction as possible - that is, lots of motivation, lots of ...
4
votes
1answer
87 views

What is a motivation for this theorem and what is an example this theorem is applied?

If one doesn't know a motivation, it's hard to memorize such a theorem. So do I. Rudin RCA p.30 Let $(X,\Sigma,\mu)$ be a measure space such that $\mu(X)<\infty$ and $f\in L^1(\mu)$. ...
3
votes
1answer
85 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
8
votes
1answer
105 views

Representation theory approach VS Module theory approach?

Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left ...
2
votes
5answers
408 views

Why - not how - do you solve Differential Equations? [closed]

I know HOW to mechanically solve basic diff. equations. To recap, you start out with the derivative $\frac{dy}{dx}=...$ and you aim to find out y=... To do this, you separate the variables, and ...
2
votes
1answer
169 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
3
votes
0answers
130 views

What makes Beal's conjecture “beautiful” enough to make people offer a million dollar prize? [closed]

What makes Beal's conjecture "beautiful" enough to make people offer a million dollar prize? Is it just a challenge or does it have real applications?
3
votes
1answer
85 views

Motivation for definition of Mobius function

Why is the Mobius function defined the way it is? \begin{align*} \mu(n) = \begin{cases} (-1)^r & \text{ if $n$ is square-free and is of the form }n=p_1p_2\ldots p_r\\ 0 & \text{ if $n$ is not ...
5
votes
6answers
365 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
11
votes
3answers
153 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
6
votes
1answer
200 views

Can we capture all domains of discouse in the predicate logic within categorical logic?

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding ...
3
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2answers
112 views

Motivation for abstractness

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case ...
2
votes
1answer
149 views

What motivates the study of Abelian groups?

Monoids arise naturally as endomorphism monoids, and groups arise naturally as automorphism groups. These are among the primary motivators for their study, in my opinion. What are the (main) ...
25
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9answers
717 views

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
-2
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6answers
683 views

What purpose does the use of functions serve in mathematics?

Ok, so I know the overview a bit. I would like to know why one should use one, what they're used for, and maybe even the history behind their purpose. X = X (Y) + 7 ... 5 = 5 (20) + 7 ... 32 is the ...
0
votes
3answers
79 views

Mean Value Theorem Motivation

I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a ...