# Tag Info

0

At least for me, starting by trying to solve the homework questions, even if I hadn't fully grasped some proofs (or even full grasped the concepts involved) usually worked out better than trying to understand everything first and only then starting with the homework problems. As long as I found a solution (where found means found it myself, though. I tried ...

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I will second Serge Lang's book "Basic Mathematics". It is definitely challenging though for those used to traditional high-school textbooks. But as originally asked, AOPS has something called the "Alcumus" which you might find useful. As mentioned on the website http://www.artofproblemsolving.com : Art of Problem Solving's Alcumus offers students a ...

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I don't know anything about programming but you mentioned "3D" and to me that screams Linear Algebra. However, you only mention doing Gr.7 level math so you have lots of work to do. When growing up I learned a lot from the site Purple Math. It seems to have a lot more topics now then when I used to use it but I remember it being quite good. Recently I have ...

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If you've enjoyed the Khan Academy for coding, then check out it offerings with respect to math! That's my "starting point" recommendation. (Math:...that's where Sal Khan got going, before branching off into other areas.) See also Paul's Notes: Click on course notes: you'll see a drop down menu: Algebra, Calculus I, II, III, Linear Algebra, etc. Many ...

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The obvious answer is the math section of Khan Academy! More advanced courses can be found here and here, a couple of nice ones on analysis and functional analysis by Joel Feinstein here and some brilliant ones on linear algebra/systems/optimisation by Stephen Boyd here. It is also worthwhile to check for courses here and (in the future) here. See also ...

0

I find this interesting myself as I also have messy writing. For me I always try to keep equal signs aligned and leave equal spacing. I also prefer using paper landscape as opposed to portrait but this is all just personal preference. As for speed vs. neatness, I think it really is just about finding a fine line between them, neatness is important but you ...

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It may just be my fanatical opinion, but I think that clarity is one of the most important attributes of performing mathematics, regardless of level. For your purposes, let me present an example: Regardless of what course I am teaching, whether it be first year calculus or fourth year topology, I always have students who submit messy, poorly structured, ...

3

My handwriting is pretty bad, I love $\LaTeX$, I've lectured on chalkboards/blackboards a couple of times, and given computerized presentations. My general feeling is that you should make the general direction of everything you write in exams crystal clear, and keep letters/symbols distinct, but otherwise don't waste too much time on lovely handwriting. ...

0

I.M Gelfand's books on trigonometry,algebra,functions and graphs and calculus of variations(and much more) are comparable to Feynman Lectures. He has even stated his effort to write a book like feynman's in the book's preface. I strongly recommend the books. You can search the books in amazon for user reviews.

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Search by web for how to understand mathematics. Read for instance: http://en.wikipedia.org/wiki/Lists_of_mathematics_topics - many topics of mathematics http://www-history.mcs.st-andrews.ac.uk/Biographies/Lang.html - this is about Serge Lang Good Luck!

2

I posted this first in a comment, but it is really worth status as an answer.: A middle school teacher gifted me an old book: "How to Lie with Statistics" by Darrell Huff: "Darrell Huff runs the gamut of every popularly used type of statistic, probes such things as the sample study, the tabulation method, the interview technique, or the way results are ...

1

It is a common misconception in regard to math, but this kind of misconception is common everywhere. It's caused by the recursive properties of knowledge. The more you learn, the more you realize how little you know.; i.e: Since it is known that less informed individuals see fields as more finite than informed individuals; While laymen may relate to the ...

1

Probability with dice and coin tosses seems like a standard starting point. Discussion of normal distributions (percentiles, standard deviation) also is pretty fundamental. The great thing about both of those is that they're easy to illustrate. Probability is great to discuss in terms of games of chance (and while talking about dice and coin tosses is great ...

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Maybe you can look into: Share My Lesson High School Statistics Resources Statistics and Probability Books Activities and Projects for High School Statistics Courses by Ron Millard, John C. Turner See this list of books on Google Teaching Stats in High School Search out other such books and resources You might also want to try some of the ...

0

The number looks small enough to be brute-forced on a computer. Just try every possible factor, starting with 2, 3, 4, ... and keep dividing them out as long as the division comes out even. Then continue looking for factors of the quotient. You don't even need to explicitly restrict to primes, because any composite number you try simply won't divide the ...

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The thing with Project Euler is that there is usually an obvious brute-force method to do the problem, which will take just about forever. As the questions become more difficult, you will need to implement clever solutions. One way you can solve this problem is to use a loop that always finds the smallest (positive integer) factor of a number. When the ...

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Project Euler problems (at least the ones that I have done) tend to deal with a lot of number theory topics. So, reading an introductory number theory book could be helpful. With regards to your particular situation, I suggest finding primes first, then testing the primes for divisibility. That is, to find prime factors of $25$, don't test $1, 2, 3, 4, 5, ... 0 The important aspect in these kinds of permutations and combinations questioins is the direction with which you are approaching the problem. For some questions, starting from the left most digit and then moving towards right may be a good strategy. Ex-- How many numbers are greater than 500 or less than 800 kind of questions, where hundred's digit has a lot ... 1 On the wiki page I noticed this appeared: Lisa Glendenning (May 2005). Mastering Quoridor (B.Sc. thesis). University of New Mexico. The link to it does not work, but if you can get to it, it probably at least contains references to published work done on the game. 0 I agree with the author of the question that inferring the domain from the definition of the function is backward. There are a couple of points that I think would be worth adding to the previous answers. Firstly, it is not quite clear to me that the one right setup in which all of mathematics is happening is$\mathbb{R}$. There seems to be no particular ... 1 We have a museum at our university, with an installation of morenaments, an application I wrote myself. We find visitors of all ages spending hours drawing wallpaper ornaments with these. For iOS devices like the iPad, there is a newer development called iOrnament. These applications can be great to use an aesthetic faszination and turn it into curiosity ... 2 My two cents: if you are not sure about where exactly in mathematics you want to go, then vector calculus is a pretty good way to make more paths viable. If you do any sort of analysis, chances are you'll need to know vector calculus forwards and backwards; this is especially true if you want to work in PDEs. If you want to do anything on manifolds ... 4 Yes. "Studying math is as much about specific examples as it is about general theorems." My teacher Mike Artin used to expound on this concept (and then assign ridiculous and tedious homework assignments) and I would not get what he was saying. But the more I read math, the more I've come to realize that understanding specific examples is what allows you ... 2 The answer depends on your interests, and on the place you continue your education. In some areas in the world, PhD's are very specialized so any course that is not directly related to the subject matter is not necessary. One could complete a pure math PhD and not know vector or multivariate calculus. However, this is increasingly rare; more and more PhD ... 2 Necessary but not sufficient condition: recognizes the importance of good definitions. 4 I second the notion of Intuiton and Instincts, though in a different context. Many hard research problems are hard simply because there is no cookie cutter method for solving them. After trudging though core curriculum like Calculus, Algebra, Analysis, PDE's, etc, you acquire a vast array of problem solving techniques albeit for specific classes of problems. ... 5 A controversial and easy answer is "intuition". I know you don't like it, but sadly it is true. All that we know about Calculus started with Newton and Leibnitz's intuition about limits,continuity and derivatives and integrals. And for many decades Calculus stuck to be an "intuitively correct" idea, and along came Augustin-Louis Cauchy who defined it ... 3 $$I=\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \int_0^1\frac{4x-5}{\sqrt{4-(x-1)^2}}dx$$ Let$x-1=2\sin\theta\implies dx=2\cos\theta d\theta$If$x=0, \sin\theta=\frac12, \theta=\frac\pi6$If$x=1, \sin\theta=0, \theta=0$$$\text{So,}I=\int_{\frac\pi6}^0\frac{4(2\sin\theta+1)-5}{2\cos\theta}\cdot2\cos\theta d\theta$$ $$=8\int_{\frac\pi6}^0\sin\theta ... 1 Note first that the derivative of the quadratic in the denominator is -2x+2. It would be great if the numerator were 4x-4, because then we could set u=3+2x-x^2, du=(-2x+2)dx, and write the indefinite integral as$$\int\frac{-2}{\sqrt u}du=-2\int u^{-1/2}du\;.$$Unfortunately, the numerator is actually$4x-5=-2(-2x+2)-1$. The trick is to split the ... 0 My first think of infinity was square diagonal vs. orthogonal stepping. Start with route (0,0) -- (0,1) -- (1,1), then take (0,0) -- (0,½) -- (½,½) -- (1,½) -- (1,1) and so on. Form of later will come closer to diagonal, but lenght will not. 0 I have been going back to review my calculus and differential equations for myself. I was using Lyx and LaTex for a while but it is slow and tedious to learn(for me). I will say that if you are willing use Microsoft OneNote install the math plugin which you can download from MS(free) also install the Microsoft Graphing Calculator software (also free) where ... 0 If you're writing a children's book on mathematics, please start by reading some excellent children's books dealing with mathematics. Here are some books I have fond memories of: The Man Who Counted The Phantom Tollbooth Flatland Alice in Wonderland / Through the Looking Glass Everything by Martin Gardner Godel, Escher, Bach: An Eternal Golden Thread 0 For me it was Monty Hall problem: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your ... 0 The fact that you can't divide by zero always amazed me. I once read the following analogy: Imagine you go to a shop with 100 dollars in your pocket, and imagine that everything in the shop costs 1 dollar. How many things can you buy? 100. What if instead of 1 dollar, each thing costed$0.5? How many things can you buy? 200. Now imagine that ...

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Rosenlicht's Intro to Analsysis was an awesome read, but the real learning took place in the excersises. It was cheap, and just as rigorous as the introductory analysis course I took the following semester!

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To be able to solve the problem I think a student would need to know basic algebra some trigonometry some geometry, including the concept of circles, triangles, and equidistance At what education level... Education level is a strange concept, but I think a bright student in an advanced secondary mathematics class should be able to solve this. It's ...

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I might be being silly but it appears to me that the question is not written very well and it is mathematically inaccurate. My main issue is that Dana might not be able to stand anywhere. How do we know that there is a place that is one rod away from both Bob and Carl? They might be at a distance of more than two rods away from each other, in which case ...

2

By the way, just to give a much simpler answer (which indeed does not really explain the issue but might help if you're not studying calculus yet): The problem here is that, in reality, $\infty$ is not a number. It is used to represent an unimaginably big number, but you obviously can't tell which. Therefore, infinity itself is not a defined number. That's ...

5

When your teacher talks about $0/0$ or $\infty/\infty$ or $1^\infty$ he/she's not talking about numbers, but about functions, more precisely about limits of functions. It's just a convenient expression, but it should not be confused with computations on simple numbers (which $\infty$ isn't, by the way). When $1^\infty$ is referred to, it is to mean the ...

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What $1^\infty$ is, or is not, is merely a matter of definition. Normally, one would only define $a^b$ for some specific class of pairs of $a,b$ - say $b$ - positive integer, $a$ - real number. When extending the definition of exponentiation to more general pairs, the key thing people keep in mind is that various nice properties are preserved. For ...

1

The Sieve of Eratosthenes code on Wikipedia is intended to generate a list of all primes up to $n$. Suppose $k\le n$ is not prime, so we have two factors $a,b$. We can't have $a,b$ both larger than $\sqrt{n}$, as then $k=ab$ would be larger than $n$. Thus in order to show that $k$ is prime, we only need to check that it is not divisible by a number up to ...

0

I was always good at maths as a child, and took to reading extension maths books for fun (other kids thought I was weird). When I was about 10 I was completely hooked when I saw Euclid's proof for an infinity of primes. I had been given it as a question in one of the books I was reading. I spent about an hour desperately trying to prove it . . . then I ...

2

As a high-school student, who studies Euclidean Geometry for the purpose of Mathematics Olymnpiads, I would recommend the following, not as high-powered as Coxeter, books. The Geometry of the Triangle - Gerry Leversha Plane Euclidean Geometry - AD Gardiner and CJ Bradley Introduction to Geometry (2 book set) - Richard Rusczyk These are all fairly basic, ...

4

I'm not sure I like the subspace topology for this. I think the torus bit is good, though; perhaps expand that to flipping over two cards, one for the space and one for the topology, where the space is given by an identification diagram, which would yield the cylinder, moebius strip, torus, klein bottle, sphere, and the real projective space on R2 (that I ...

0

To add my 2 cents-part of what's hindering and scaring a lot of people who have to teach "college geometry" these days is the utter collapse of the American high school system. As a result,it's no longer a given that your students are comfortable with what used to be "high school" geometry-something that used to be a given for any student at any university. ...

1

I was hooked on math by a small side note in a kid's book of mathematics about perfect numbers, numbers that are twice the sum of their factors. For example, 6 is the smallest perfect number because 1 + 2 + 3 + 6 = 2 × 6 and 28 is the next one because 1 + 2 + 4 + 7 + 14 + 28 = 2 × 28. The next perfect numbers are 496, 8,128, 33,550,336, and 8,589,869,056. I ...

0

The most wonderful thing I've recently seen is this (sorry it's in French) form of the sieve of Eratosthenes and of course your question too.

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I have my students play almost exactly this game at the start of a course in College Geometry, through GeoGebra. Of course, it lacks the video game style interface you're describing (and which, I agree, would be awesome), so I would be excited to see something like this polished up nicely. I'll tell you briefly what I do in class and a little about how ...

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