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7
votes
2answers
350 views

Why does this text insist on changing the variable name here?

In What is mathematics? by Courant, Robbins, and Stewart, "5. An important inequality", the authors change $n$ in this example: $$(1+p)^n\geq1+np$$ to $r$ in this example: $$(1+p)^r\geq1+rp$$ In ...
6
votes
1answer
133 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
6
votes
2answers
200 views

Logarithms of logarithms of Graham's number, is the result ever handy?

The other day I was asked how to represent really big numbers. I half-jokingly replied to just take the logarithm repeatedly: $$\log \log \log N$$ makes almost any number $N$ handy. (Assume base ...
6
votes
3answers
816 views

How to fill gaps in my math knowledge?

Just finishing highschool, even though I am doing "well" (in the context of the math course itself), I have significant holes in my actual math knowledge. As I think many people who explore math ...
6
votes
5answers
561 views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
6
votes
6answers
942 views

Purpose of Linear Algebra

How much emphasizes should be on proof on a first course in Linear Algebra? I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra. However I also think a central theme of ...
5
votes
2answers
118 views

Is the exclusion of uncountable additivity a drawback of Lebesgue measure?

A friend and I were having a discussion about Lebesgue measure. I attempted to be profound by making the following points: Analytic geometry has been a fantastic tool, but the concept of ...
5
votes
5answers
224 views

Why isn't there a fixed procedure to find the integral of a function? [duplicate]

Since the integration of a function is the opposite of a the derivative of a function, and there are clear steps to follow when we want to find the derivative of a function, I thought there would be ...
5
votes
3answers
288 views

Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
5
votes
2answers
210 views

The complement of a torus is a torus.

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can ...
5
votes
1answer
121 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
5
votes
3answers
190 views

Dealing with many entities that need a symbol

What does one do when one needs a lot of symbols and one has exhausted the useful symbols of the latin and greek alphabets? (I say useful symbols because letters like iota (ι) and upsilon (υ) seem too ...
5
votes
5answers
1k views

Side-stepping contradiction in the proof of ; ab = 0 then a or b is 0.

Suppose we need to show a field has no zero divisors - that is prove the title - then we head off exactly like the one common argument in the reals (unsurprisingly as they themselves are a field). ...
5
votes
1answer
404 views

Is it ever really Pi Time?

Walking with my son at 3:14pm the other day, I mentioned to him, "Hey, it's Pi Time". My son knows 35 digits of $\pi$ (don't ask), and knows that it's transcendental. He replied, "is it exactly ...
4
votes
4answers
300 views

The set of all things. A thing itself?

If the universe is the set of all things. Does it contain itself? In other words is it a thing itself? I know its a stupid question, but it really grinds my gears. Thanks! Edit 8.12 Okey, someone ...
4
votes
2answers
350 views

What is Ramsey Theory ? what is its own importance in maths?

3 days ago , i had a discussion with a close friend who studies physics - still a student - . and i was telling her about the biggest known numbers in maths , so i told her about numbers such googol ...
4
votes
2answers
375 views

Intuition Of Conditional Probability Equation

I was wondering if any one of you had any intuitive insight regarding the conditional probability equation, $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$. In my textbook, they give a mere definition, ...
4
votes
2answers
211 views

$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?

Update : Should have left the Arithmetic out of this question, the new modified question is posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? ...
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 ...
4
votes
3answers
714 views

What is your favorite proof that $e^{ix}$ has a period of $2\pi$?

as a function of a real variable, apparently. Part of the freedom in choosing a proof is that you get to choose what definition of $e^{ix}$ to start from -- do you use a differential equation? a power ...
3
votes
2answers
57 views

Analogy to the purpose of Taylor series

I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and ...
3
votes
1answer
158 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
3
votes
4answers
93 views

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$?

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$? I was trying my best to do the problem but like I don't know where to start or anything!
3
votes
2answers
442 views

De Morgan's laws in logic and set theory

In logic De Morgan's law means $\lnot (A \land B) \Leftrightarrow \lnot A \lor \lnot B$ In set theory De Morgan's law means $(A \cap B)^C = A^C \cup B^C$ I'm surprised that the same idea is true in ...
3
votes
4answers
312 views

How to explain Fractional and Negative Exponents

My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me... Is there any way to explain it clearly to them?
3
votes
0answers
110 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
votes
3answers
120 views

Basis of a basis

I'm having troubles to understand the concept of coordinates in Linear Algebra. Let me give an example: Consider the following basis of $\mathbb R^2$: $S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and ...
3
votes
2answers
237 views

Zorn's Lemma $\equiv$ Axiom of Choice

I'm confused a little bit about this, I've been told many times that Zorn's lemma is equivalent to the axiom of choice. Is it an axiom or is it lemma, I mean is there a proof of Zorn's lemma or we ...
3
votes
0answers
87 views

References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
3
votes
2answers
253 views

Why are only the first four alternating groups are non-simple?

I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
3
votes
5answers
257 views

Why do you need to specify that a coin is fair?

This sounds like the kind of etherial question that generally gets dropped from stack exchange sites, but I don't know of a better venue to ask so I'm hoping this question will help other folks with a ...
3
votes
2answers
339 views

Interesting or non-obvious finite subsets of the natural numbers

I was recently explaining to someone how to prove that there are infinitely many prime numbers, and I mentioned to them that it's not immediately obvious, upon first encountering the natural numbers, ...
3
votes
1answer
134 views

summing series using circles inside curves

After watching the infinity elephants video http://www.youtube.com/watch?v=DK5Z709J2eo and seeing how a geometric series could be represented by drawing a circle between a pair of lines, then the ...
3
votes
3answers
455 views

What are some good examples for suggestive notation?

Motivation: Today I first wondered about and later remembered why the set of all functions from a set $X$ to $Y$ is denoted $Y^X$. They wikipedia page gives the explaination "The latter notation is ...
3
votes
3answers
510 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 ...
2
votes
0answers
24 views

Identifying a function that involves combinations of terms

I need to know if a function exists that partitions terms in such a way as seen below $$ \frac{d^n}{dx^n}[\frac{(x)_c}{n!}] $$ Note that $(x)_c$ is the falling factorial of x and $c \geq n$, This in ...
2
votes
2answers
110 views

Mathematicians average in student life but later became significant

What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians. Up until ...
2
votes
1answer
146 views

Book to prepare for university math?

Can you suggest some books to prepare for university math?
2
votes
1answer
73 views

What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
2
votes
1answer
50 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
2
votes
3answers
142 views

Mathematician's names in structures.

I would like to know how it is that mathematical objects come to receive the name of a mathematician. Do these mostly happen through the author's proposal, or is it a process that takes more time? ...
2
votes
2answers
206 views

Fundamental problem of Linear Algebra

What is the fundamental problem of linear algebra? I understand it is a big question and not easy to explain completely, and seems no way to prove an answer is correct. I just wanna listen to you ...
2
votes
2answers
84 views

Conditional events that are not in the event algebra?

The Wikip. page on conditional event algebra states that: David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any ...
2
votes
1answer
104 views

Finding a logical expression (under some constrains) s.t. it is equivalent to another one

In this question, it was made clear, when $\bullet$ some statement $A$ is stronger than another statement $B$, namely if $A\Rightarrow B$ holds; and when the statement $A$ is weaker than another ...
2
votes
3answers
464 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
2
votes
1answer
1k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
2
votes
3answers
347 views

Should I write out stuff? [closed]

When I go through textbooks should I write out solutions to the exercises? Or is it fine if I just do it in my head? I mean either way you are still doing the problems right?
1
vote
0answers
31 views

Site to find no popular papers/books

I'm studying this article and in a particular proof the author advices to look for this paper/book: Buchweitz,R.-O.: Über Deformationem monomialer Kurvensingularitäten und Weierstrasspunkte auf ...
1
vote
1answer
61 views

Useful techniques of experimental mathematics (reference request)

I am searching for papers or books that explain thoroughly useful interesting techniques of experimental mathematics that can be understood and profitably applied by an undergraduate student.
1
vote
1answer
89 views

Where to post discovered formulae? [closed]

I have discovered an alternate formula for the Fibonacci sequence and I would like to find a way in which I can present this. Please could you give me suggestions on how I can go about posting this ...