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1
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2answers
75 views

Conic Sections - Why do I need to know all these terms (foci, latus rectum, directrix, etc)? When will I use them?

I believe in learning something because I want to. If I do not want to learn about a subject or concept, I will not learn it well and master it. I am currently learning about conic sections, and I am ...
5
votes
1answer
156 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
0
votes
1answer
228 views

I want to pursue Computer Science as a major but I don't know if i should…HELP!? [closed]

I'm currently a second semester sophomore who's currently pursuing a Global Affairs major and minoring in Japanese Studies. I plan on going to law school after college and I'm interested in ...
5
votes
1answer
114 views

Why Did You Specialize in X?

For those of you who are researchers or graduate students, why did you choose to specialize in the field of mathematics X (as opposed to some other field Y)? Is it because you think X is important, ...
11
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2answers
496 views

Undergraduate Mathematics Research

I am currently an undergraduate junior. I have taken most of the standard undergraduate math courses and a few introductory graduate courses (measure theory, algebraic topology, complex analysis, ...
0
votes
1answer
49 views

Product Notation for Multiplication in Reverse Order

Is there a standard notation for multiplication in reverse order? For example consider the problem $$x_{k+1} = A_k x_k$$ where $x_i \in \mathbb{R}^n$ and $A_i \in M_n(\mathbb{R})$, ($i=0,1,2,\dots$) ...
0
votes
1answer
48 views

Question regarding the formal definition of limes Inferior/Superior

I have a question regarding the definitions of limes inferior/superior of a sequence $x_n$. Limes inferior for example is defined as $$lim ~ sup_{n \rightarrow \infty} x_n := lim_{n \rightarrow ...
5
votes
1answer
95 views

Uses of integral calculus in discrete mathematics?

I have to do a project in my integral calculus class. But all the topics are too mainstream (parabolic arc calculation,archimedean approzimation of circle are,obtaining $E=mc^2\dots$ However I'm ...
-1
votes
1answer
76 views

Relation/connection between $n!$ or $e$ and $2^n$

What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?
21
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4answers
743 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 & ...
3
votes
2answers
116 views

Time managment while self-studying rigorous mathematical textbooks

Lately I have been introducing myself to Calculus.But I have an issue with time. Namely I have found that on average I need about 2-3 hours per page of material.This of course includes finishing all ...
2
votes
0answers
35 views

Countability of unions versus products

Let $D_{n}$ be a set with $2^{n}$ elements for $n=1,2,...$. Let $A = \bigcup_{n=1}^{\infty}D_{n}$, and let $B = \prod_{n=1}^{\infty}\{0,1\}$. Let $A_{k} = \bigcup_{n=1}^{k} D_{n}$, and let $B_{k} = ...
1
vote
1answer
64 views

Tools or Resources for pictures and visualizations

The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books ...
0
votes
0answers
13 views

How to model decremental frequently ascending value

Eating a tasteful food will initially blow up my mind. But as eating tasteful food gets more frequent, the effect on my mind is getting less and less. Assuming eating 1 tasteful food, will increase ...
1
vote
1answer
134 views

Struggling through Spivak

My friend recommended that I try out Spivak. So I figured what the heck, its just a calculus book, can't be that hard...I was wrong. I'm very slowly working through Chapter 1, not even on calculus ...
0
votes
1answer
78 views

Reference on Infinite Dimensional Manifold

I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold. Now I have two questions or ...
8
votes
4answers
317 views

Linear algebra questions that a high-schooler could explore

Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?
1
vote
1answer
53 views

Standing Generalizations of the Collatz Conjecture?

I remember reading that Erdös once that mathematics isn't matre enough to tackle the collatz conjecture. However the Collatz conjecture seems like a rather specific problem to me. Are there any ...
10
votes
5answers
809 views

Can numbers exist outside of the number space?

I was unsure about how to actually write the question. I know that in the real number axis we have the naturals, integers, rationals, irrationals and reals; and in the imaginary number axis we have ...
9
votes
1answer
150 views

I made a primality test and want to publish it

So I am a high school student, and I am very interested in maths, and I made my own primality test which also expresses all composite numbers with last digits of 1,3,7, or 9 in just 9 simple ...
12
votes
3answers
876 views

Is it usual that a professor never proves a theorem in a class?

I'm taking "real analysis" this year and the lecturer's major is PDE. What he does is literally reading texts except proofs. This drives me crazy, so I asked the professor why he is not giving any ...
6
votes
3answers
105 views

Applications of stochastic processes

What are stochastic processes? What are they used for? How can they be applied to real concepts? What is an example of a "stochastic process" problem?
5
votes
2answers
96 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
29
votes
15answers
3k views

Best way to express 2014 [closed]

In my class, I always write the date using mathematical formulas, or cool little equations. I want to show my students that even the most mundane seeming number often has fascinating features, and its ...
3
votes
5answers
347 views

Modular Arithmetic - Are we allowed to distribute the Modularity?

Assume I have a problem such as "Prove that $\displaystyle103^{53} + 53^{103}$ is divisible by $39$." This would mean I wanted to prove that $\displaystyle103^{53} + 53^{103}\equiv0\pmod{39}$. My ...
7
votes
2answers
200 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
45
votes
10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
0
votes
1answer
1k views

What is the “circle-plus” symbol I see in Abstract Algebra?

Sorry if this comes off as a random or soft question. I keep seeing this symbol in my abstract algebra course where it is a plus sign inside of a circle. I am not sure what it means. Can someone ...
49
votes
14answers
4k views

Do we need to formally teach the Greek Alphabet?

This is a question that I am purely interested in because I think we never thought about this before in Mathematics education... or even so was not discussed. When did we learn the Greek alphabets ...
1
vote
2answers
92 views

Doing for the first time a mathematical essay! Tips [closed]

I am an undegraduate and i though of doing my first essay on number theory!I kinda have the topic and i started selecting info from different references..but i dont want it to be just a copy paste ...
6
votes
3answers
215 views

Definitions which should be propositions/theorems

I am asking for a list of concepts which some sources present as definitions whereas other sources pose them as propositions/theorems. For example, most abstract algebra books will define a group ...
2
votes
1answer
313 views

Famous deaf mathematicians?

There are some really inspiring examples of blind mathematicians. However in my experience I also think problems inside my head using words. So I was wondering if there are some examples of deaf ...
1
vote
4answers
195 views

What is the most awe-inspiring math equation you have come across [closed]

What is the favorite equation of your life? I know this might be a subjective question, and may be not-so-on-topic here, so if anyone decides to close this, could you link me somewhere I can ask this? ...
1
vote
1answer
40 views

Should $\sum_{i=0}^n a + b$ be interpreted as $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$

Should $\sum_{i=0}^n a + b$ be interpreted as $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$ I often see the expression $\sum_{i=0}^n a + b$ in books and wonder whether we take the sum ...
3
votes
1answer
465 views

How can I study efficiently from a textbook that doesn't have answers?

My lecturer has informed us that tutorial questions will be insufficient, and has asked us to practice liberally from the textbook. Unfortunately, none of the exercises have answers supplied and I ...
6
votes
1answer
172 views

More unknown / underappreciated results of Euler

What are some of the more unknown and/or underappreciated things that Euler discovered? The man has done so much that there's bound to be notable results that most people aren't aware of. This could ...
0
votes
1answer
72 views

How to come up with formula for this number growth

I have difficulties to come up with a formula to achieve this in my programming challenge : The input number ranges between 100 to 10.000 (all integers), then the output would be 0.1 - 10, so the ...
6
votes
4answers
653 views

Mathematicians who started late, similar to my situation?

I came here since I know this is the best place to ask a question. I'm a first year student who changed his major to applied mathematics. In middle school I was a garbage math student, but I ...
4
votes
2answers
125 views

When do we write “we are done”?

This may seem like a bit of a silly question, but I notice that in some proofs (a remarkable amount), the author writes: "We are done." after completing a proof. Is this the equivalent of writing one ...
94
votes
32answers
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 ...
5
votes
2answers
640 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
1
vote
2answers
44 views

How could I show that if $a$ is an integer, then $a^3 \equiv 1, 0, 6 \mod 7$?

I've literally tried everything including proofs by cases. If I were to try a an odd integer, then I would get $8k^3 + 4k^2 + 8k^2 + 4k + 1$, which obviously couldn't convert to a $\mod 7$, and I'm ...
0
votes
2answers
57 views

A question about notation

Earlier this week, a friend asked me what the most complicated equation I could think of was that was equal to $1$. The answer I gave was this: Let $G_n$ denote the n$th$ number in the grandi series, ...
4
votes
1answer
144 views

The method of “Mathematical Induction” as explained in my book.

I am reading the book "A Treatise on Advanced Calculus" by Philip Franklin. I found this book in our city's central library and liked it at the first reading, am continuing this book as a reference. ...
11
votes
3answers
256 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
2
votes
3answers
141 views

Study of polynomials and root approximation algorithms

I am currently beginning a long term investigation (I will have 6 months of time). I wish to better understand fundamental properties of polynomials and eventually come up with a good root finding ...
2
votes
0answers
27 views

Motivation behind Binet form and its generalization to arbitrary coefficients

Binet form (http://www.proofwiki.org/wiki/Binet_Form) gives a closed-form solution to $n^{th}$ term in a series with recurrence relation as below $U_{n}=m\cdot U_{m-1}+U_{m-2}$ I have two questions ...
0
votes
0answers
41 views

Standardizing the terminology of graph theory

I took a course in graph theory many years ago and remember being frustrated that the terminology was not at all standardized (or didn't seem to be). Various sources used vastly different words or ...
4
votes
0answers
90 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
2
votes
2answers
106 views

From Engineering-Style to Proper Mathematics

I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not ...