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How much Math do you REALLY do in your job?

I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied ...
54
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8answers
8k views

How to debug math?

May seem strange as I'm good in programming, but I just started diving into math. ATM I'm learning combinatorics at Khan Academy, and here's an example of a question that I struggled with (that's not ...
54
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9answers
9k views

How to effectively study math?

Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble ...
54
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17answers
3k views

Good “history of mathematical ideas” book?

All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living ...
54
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7answers
11k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
54
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8answers
6k views

Is linear algebra laying the foundation for something important?

I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear ...
54
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5answers
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What is “ultrafinitism” and why do people believe it?

I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone ...
54
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6answers
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Am I too young to learn more advanced math and get a teacher?

I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics ...
54
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7answers
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How to explain to the layperson what mathematics is, why it's important, and why it's interesting [closed]

A mathematician walks into a party. No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. ...
53
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19answers
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Theorems' names that don't credit the right people

The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) ...
53
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10answers
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Why are primes considered to be the “building blocks” of the integers?

I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers. I've always had ...
53
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7answers
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Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra

I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of ...
52
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10answers
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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 ...
51
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51answers
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What was the book that opened your mind to the beauty of mathematics? [closed]

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...
51
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17answers
5k views

What is $-i$ exactly?

We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything? Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen ...
51
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11answers
3k views

Is it not effective to learn math top-down?

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving ...
51
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3answers
2k views

What is this pattern called?

Back-Story I became interested in the patterns in multiplication tables for different base number systems a while ago. Specifically, the pattern made by the last digit of each number in the ...
51
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2answers
2k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
50
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15answers
7k views

How do I motivate myself to do math again? [closed]

I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating. Still, I am currently in a university studying mathematics as an undergrad. I took ...
50
votes
5answers
2k views

How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
49
votes
24answers
11k views

“Negative” versus “Minus”

As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$? The so called "textbook answer" regarding this question reads: ...
49
votes
13answers
23k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
49
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3answers
12k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
49
votes
8answers
2k views

How to maintain enthusiasm and joy in teaching when the material grows stale

I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same ...
49
votes
14answers
2k views

How to entertain a crowd with mathematics? [closed]

I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, ...
48
votes
7answers
6k views

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
48
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14answers
5k views

Do we need to formally teach the Greek Alphabet? [closed]

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 ...
48
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4answers
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Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
47
votes
18answers
6k views

What's the goal of mathematics?

Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what? I've already read G. H. Hardy Apology but I didn't get an answer from it.
47
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17answers
2k views

What are some math books written in dialogue or story form, e.g., a teacher explaining to a student?

Good examples would be The Square Root of 2 by David Flannery or Math Girls by Hiroshi Yuki.
47
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6answers
5k views

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
47
votes
5answers
3k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
47
votes
10answers
2k views

Have there been efforts to introduce non Greek or Latin alphabets into mathematics?

As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic ...
47
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6answers
10k views

What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail ...
47
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6answers
2k views

What to answer when people ask what I do in mathematics

This is not really a math question, but I think that every mathematician and student in math (specially pure) struggles with this at some point. Inevitably at some point when we're talking with ...
47
votes
8answers
3k views

Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
47
votes
5answers
6k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
47
votes
3answers
6k views

Motivation and methods for self-study [closed]

First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started ...
46
votes
19answers
6k views

What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It ...
46
votes
17answers
5k views

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results [closed]

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, ...
46
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7answers
9k views

How to tell if I'm good enough for graduate school?

I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I ...
46
votes
14answers
4k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
46
votes
14answers
5k views

Nobody told me that self teaching could be so damaging…

Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed. For starters, I'd like to mention that I'm 21 ...
46
votes
9answers
1k views

Examples of group-theoretic results more easily obtained through topology or geometry

Earlier, I was looking at a question here about the abelianization of a certain group $X$. Since $X$ was the fundamental group of a closed surface $\Sigma$, it was easy to compute $X^{ab}$ as ...
46
votes
4answers
3k views

How to understand and appreciate the prime number industry?

Why would I want to buy prime numbers? There is a website (found it!) selling a table of 400 digit primes for twenty dollars. Like an updated version of this. I have a layman's idea that prime numbers ...
45
votes
15answers
4k views

Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
45
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
45
votes
14answers
2k views

Surprising Generalizations

I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same ...
45
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6answers
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How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
45
votes
5answers
1k views