# Tagged Questions

For questions related to the teaching and learning of mathematics. Note that Mathematics Educators Stack Exchange may be a better home for narrowly scoped questions on specific issues in mathematics education.

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### Is Vector Calculus useful for pure math?

I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering ...
6k views

### What makes a good mathematician?

A soft question but I believe important to help get maturity in maths. Not until I got admitted to a graduate applied math program that I started to learn math. Before that, I was a life science ...
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### Integrating a school homework question.

Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found. Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$ Thank you in advance!
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### Why is the definition of “limit” difficult to understand at first?

Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
314 views

### learning/teaching approach to rigorous math with the goal of improving

I will state this now: yes, this is a subjective question. But I feel the answers people give may benefit students. I want to get better at doing non trivial proofs. Real analysis is standard ...
172 views

### At what education level is this “simple” problem adequate?

Given the following problem: Alice, Bob and Carl stand on a straight line. Alice is one rod away from Bob. Dana stands one rod away from both Bob and Carl. Carl is as far from Alice as Alice ...
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### Optimal partition for a riemann integral

I am a statistician tasked with teaching an elementary calculus course. I am about to teach Riemann sums. The breakpoints for the rectangles (the partition) that make up the Riemann sum need not be ...
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### Why is 1 raised to infinity Not defined and not “1” [duplicate]

$1$ square is $1$, so is raised $1$ to $123434234$. My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this? I know that any number raised to ...
553 views

### Graduate schools in set theory

What options for graduate schools are there for an aspiring set theorist? In the US, the only top 50 schools with active research in set theory seem to be Berkeley and Carnegie Mellon. Are there any ...
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### Chi square independence test

How to work out chi square independence in the following table? Below is the observed and expected data concerning 7 themes displayed in a newspaper over a period of 3 months. I understand how to ...
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### Understanding the pseudocode for the Sieve of Eratosthenes

The outer loop on the Wikipedia page for the Sieve of Eratosthenes ends at √n: for i = 2, 3, 4, ..., √n : Is this because if n has a square root it wont be prime? From what I understand this ...
3k views

### About Euclid's Elements and modern video games

Update (6/19/2014) $\;$ Just wanted to say that this idea that I posted more than a year ago, has now become reality at: http://euclidthegame.com/ 12.292 users have played it in 96 different ...
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I know it is hard to ask for the perfect method for doing mathematics, but I hope there are some paths that are more preferable over others. I am a engineering student who has switched to mathematics ...
17k views

### Self-study Real analysis Tao or Rudin?

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here come another one. Recently I've began to learn real analysis ...
870 views

### Using the Constant Function Theorem to prove the Increasing Function Theorem

I quote Thomas W.Tucker: ... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
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### Turning an ellipse into a parabola

Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
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### Mathematical Journey text

Hi I am looking for a HARDCOVER book I read when I was younger. It is fairly modern and published probably around 2006 - 2007. It is a hardcover with a cheesey title like: Mathematical thinking ...
4k views

### How to find multiples of numbers under a certain range

I recently found a 'question' that requires me to find the sum of all multiples of 3 and 5 under 1000, I sadly cheated and found some code online to help build a code in python: ...
14k views

### The Best of Dover Books (a.k.a the best cheap mathematical texts)

Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books ...
202 views

### what are the rules/rationale for “simplifying” negative denominators

So I'm losing my mind trying to understand the rules and rationale behind "simplifying" expressions and equations in Algebra. It's been decades since I've had to study for it (going back and ...
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### Does learning logic and set theory before arithmetic, algebra, and geometry have an advantage?

I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic ...
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### Explaining why we can't “find” an antiderivative of $f(t) = e^{t^2}$.

We can't find $$\int e^{t^2} \; dt$$ using basic tools from a calculus class. That is, we can't express an antiderivative of $f(t) = e^{t^2}$ using the basic operations. We can of course just ...
264 views

### Learning Combinatorial Species.

I have been reading the book conceptual mathematics(first edition) and I'm also about halfway through Diestel's Graph theory (4th edition). I was wondering if I was able to start learning about ...
3k views

### Is this way of teaching how to solve equations dangerous somehow?

Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline. There I learned a new way of solving equations, which is related to the principle that states that any ...
758 views

### Is it possible to learn mathematics right from the source instead of reading textbooks. By studying the masters and not their pupils

I was wondering if mathematics learning process require the use of textbooks. When I was a high school student, I read as a preparation for university, Legendre book on Elements of geometry and ...
5k views

### Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
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### Learning Mathematics using only audio.

Are there any mathematics audio books or other audio sources for learning mathematics. I ask this because I make about 1 hour from my house to the school and staring at a screen on the car makes me ...
497 views

### The Game of Nim alternative solutions

Nim has a mathematical solution which uses binary number system and addition modulo 2.I was wondering if there is an alternative solution to this game or at least another interpretation of the ...
1k views

### How can I make sure I never forget.

I am currently refreshing my Elementary Algebra using Schaum's Outlines. I find them useful as they are choc full of exercises (Which I now realize is the only way to master algebra). I am worried ...
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### Course/homework management systems for computational math courses [closed]

I will be teaching a course which will be based on a mathematical software, (as in MATLAB, MAPLE, Mathematica). The question is how to manage homework. Ordinary email is not efficient it seems. I am ...
187 views

### Darboux integral too sophisticated for Calculus 1 students?

I strongly prefer Darboux's method to the one commonly found in introductory level calculus texts such as Stewart, but I'm worried that it might be a bit overwhelming for my freshman level calculus ...
1k views

### Complex division: polar form vs complex conjugate

The original problem In an electricity course which I volunteered to help with, the students solve circuits using phasors. Using phasors requires a good knowledge of complex numbers arithmetics, ...
115 views

### What is $0^0$? Should we define $0^0$ on its correctness or convenience? [duplicate]

What is $0^0$ ? I have read many debates about this question. The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that $0^0$ = 1, until in ...
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### Why are variables typically given names like x in math?

Why do equations typically have variable names like x and y? In computer programming, we value meaningful names for variables. For example, if I were trying to calculate a square root, I would call ...
440 views

### High-School Level Introduction to Dynamical Systems

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough ...
189 views

### The shape of mathematical self-tutelage

I am interested primarily in physics, and I am generally self-taught in mathematics. However, this implies an inaptitude for rigorous proof. While I am confident that I can grasp the concepts and ...
1k views

### How to deal with exercises with no solutions given?

Probably most people will acknowledge the importance of doing exercises when reading a mathematical textbook. Here I am talking about a textbook of similar level as those ones listed in GTM. However, ...
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### Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
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### teaching concept of “subspace” to linear algebra students

I'm currently a T.A. for an introductory combined linear algebra & differential equations course geared toward engineering students. One major problem I've run across is that most of my students, ...
1k views

### How does one visualize a function with a discontinuous second derivative?

Let us assume that all functions are continuous. I was teaching my calculus students the other day. We were talking about what points of non-differentiability look like. Two ways a function can fail ...
153 views

### Proof vs Practice

I've been brushing up on a lot of basic arithmetic, algebra, and logic as I work towards a review of calculus (and beyond), and I keep noticing that in order to fully understand many principles in the ...
455 views

### Name of odd powered polynomial graph (Opposite of parabola(ic))

I am writing an assignment and have to describe the graphs for when the powers are even and when they are odd. I described the even power graphs as being parabolic or parabolas. The only problem is, I ...