Tag Info

0

Make your notes beautiful: use lots of colours to highlight stuff, re-write good versions after a lecture, and liberally sprinkle with your own annotations. Love them dearly, for when exam time comes around you will be spending a lot of time with them. You have to remember that your notes are your porthole into the course. They are also personal to you. I ...

0

I've just started university and my advice is to found out what modules you are doing and see what is involved in them. If you haven't studied further maths (like myself), read up on various topics that you haven't covered such as matrices, complex numbers etc. If you do, you won't be as far behind. You're likely to be doing theoretical modules, which ...

2

As mentioned in the comments, it would be hard to get all math teachers to agree on a common notation. Some prefer $(a,b)$ for the open interval and will probably argue vigorously for that use while others will argue that we should use $]a,b[$ for the interval. This is not just a "problem" with simple things, but there is even disagreements in advanced ...

0

Thanks to some useful comments and an answer (and the downvotes)-: I think now that one of the above definitions is wrong and should be replaced by the following (also mind the additional '.') : $$P_{c,k}(x) \; :\Longleftrightarrow \; \mbox{" symbol at position k in the decimal representation of x is c "} \\ \mbox{where} \quad c \in ... 1 Are you sure that you are not mixing two different concepts ? When you consider decimal representation of real numbers, you are speaking of : 1) rational numbers approximating the real number : in this case 1.000 and 0.999 are different rational numbers 2) two names for the same object (the real number 1) : in this case you are dealing with names as ... 5 Keep asking questions, as others have said. More specifically ... mathematicians have an annoying habit of presenting only the final "beautiful" results, without showing you all the ugly hard work and false starts that led up to them. As far as I can recall (it was 45 years ago) most things were presented this way. Everything looked like the magical ... 5 I could give you many advices but most of them would be quite generic. The most important one which I unfortunately realized only when it was a bit too late is that you have to study continuously. Maths is a skill - you have to work it, bit by bit, everyday. Don't expect superb results straight away - but they will come with persistence. Also try to keep ... 1 I don't know. Was it my ‘feminine intuition’ that helped me realize that \displaystyle\sum_{n=0}^\infty\left[\frac{(2n-3)!!}{(2n)!!}\right]^2=\frac4\pi, or merely plucking n=\frac12 into Vandermonde's formula \displaystyle\sum_{k=0}^n{n\choose k}^2={2n\choose n}, and using the fact that \Gamma\left(\frac12\right)=\sqrt\pi ? :-) 3 You have several directions you can go here. A. You will of course have to go on to more topics. The reason linear algebra is a good next choice is that it sits right on the intersection of so many different branches of mathematics, such as multivariate calculus, differential equations, numerical methods, and of course algebra and geometry. It's just ... 1 Linear algebra is next. Let me know if you need some books. 0$$\text{arithmetic} : \text{mathematics} \quad::\quad \text{spelling} : \text{ literature}$$0 When I was doing my undergrad your experience is the same as mine, so I guess things are kind of similar in undergrad programs. The motivated student can understand and do the proofs of his/her own but the exams are designed to only know whether the student understood the material covered in the class and whether (s)he could apply them for a given problem. ... 1 This is a "chain" that I discovered recently. The first element in the chain is Fermat’s little theorem which can be proven by the combinatorial necklace argument. However I believe that this idea can be generalized and better understood after learning it through Euler's Theorem, which can be proved by the argument that multiplying two distinct relative ... 9 One nice example is solving quadratic equations. If you know only about natural numbers or integers then it is difficult to give a concise universal solution for quadratic equations. But if you abstract your notion of "number" to allow negative, rational (possibly even imaginary) numbers, then one can give a single universal formula - the well known ... 4 A beautiful number-theoretical example is David Cox's book Primes of the form x^2 + n y^2. Below is an excerpt from the introduction. Most first courses in number theory or abstract algebra prove a theorem of Fermat which states that for an odd prime p,$$ p = x^2 + y^2,\ x,y \in \Bbb Z \iff p \equiv 1 \pmod 4. This is only the first of many ...

2

The best book on combinatorics of this sort is Lovasz's combinatorial problems and exercises.

0

Warning: flawed argument below. (is it better to delete it? dunno how it works here) As already pointed out by User Some Number in his answer, $f$ is clearly a self-homeomorphism of the unit disc $D$. By contradiction, assume $f$ is not the identity. Hence there are two distinct points $p$ and $q$ (necessarily in the interior of $D$) which get exchanged, ...

1

The Kiss Precise by Frederick Soddy For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. ...

1

I don't think there is a single reason why the US lags behind Russia in math, but there are several possible explanations, each of which has its strengths and weaknesses. I will list them in the order of plausibility as it seems to me, and also pointing out their possible weaknesses. The cultural atmosphere in the US does not look up to math and the ...

1

A mathematician named Klein Thought the Möbius band was divine Said he, "If you glue Together edges of two, You'll get a weird bottle like mine".

2

I forget the joke (I always do) - I am sure you could make one up - but the punchline was: ... because the squaw on the Hippopotamus hide is equal to the sons of the squaws on the other two hides. There seems to be a reference to one version of the joke here.

2

Your professor, who is Russian, likely grew up and did and his undergraduate maths education in the Soviet Union. What your professor said may have been true then, though I don't think it is still the case today: if it were true, graduate departments everywhere would be filled with Russian students. In either case, it sounds like your professor made a ...

1

Being multilingual helps a lot. It really does not matter which languages one has, though. What this does is to help the mind realise that words like 'always' actually is several different ideas, and that one can not effotlessly jump from one to another. In maths, you see infinity as meaning 'anything that is unbounded', where the Indians had no fewer ...

10

How about $\color{blue}{\frac{12 + 144 + 20 + 3 \sqrt{4}}{7} + (5 \times 11) = 9^2+0}$ Read: A dozen, a gross, and a score Plus three times the square root of four Divided by seven Plus five times eleven Is nine squared and not a bit more. Source: http://www.lhup.edu/~DSIMANEK/mayhem.htm

1

You might look at "Worms: Georg Cantor, from Halle Sanatorium, 1884", by Adam Vines, in his book The Coal Life: Poems

0

Consider the fact that $f(f(z))=z$. This shows that $f$ is its own inverse. Since $f$ is invertible, it is a bijection; a bijection from the compact space closed disk into the Hausdorff space of the closed disk, so that it is a homeomorphism (continuous bijection from compact to hausdorff is a homeomorphism). Since the closed unit disk is contractible, any ...

1

As pointed out in the comments on the MO question, this follows immediately from a 1931 theorem of M.H.A. Newman, which is Theorem 2 in this paper. Notice that Theorem 2 is basically a lemma (used to prove the main theorem 1) whose proof takes two pages, and uses nothing other than definition of manifold.

3

In his poem A Valediction Forbidding Mourning, John Donne metaphorically compares love to a compass (in the geometry sense): ... If they be two, they are two so As stiff twin compasses are two ; Thy soul, the fix'd foot, makes no show To move, but doth, if th' other do. And though it in the centre sit, Yet, when the other far doth ...

2

If you work it out, the equation is correct Integral z squared dz From one to the cube root of three Times the cosine of three pi over 9 equals log of the cube root of e Found another one here http://blog.drscottfranklin.net/2009/01/13/another-calculus-limerick/

3

On the book "Cyberiad" Stanislaw Lem writes this poem, called "Love and Tensor Algebra". Of course it is translated from polish by some dude, but you'll get the spirit of it. I Love it personally, you can find it here: http://wonderingminstrels.blogspot.com/2003/03/love-and-tensor-algebra-stanislaw-lem.html

4

Poe, E.: Near a Raven: http://www.uoguelph.ca/~pjf/Pi/pi.mnemonic.extrordinaire.html Poe, E. - Near a Raven Midnights so dreary, tired and weary. Silently pondering volumes extolling all by-now obsolete lore. During my rather long nap - the weirdest tap! An ominous vibrating sound disturbing my chamber's antedoor. "This", I whispered ...

1

Corollary, by Ian W. Gouldstone An animated poem about "the atrophy of a theoretical math education".

2

acko.net uses the quote below after which the guy proceeds to explain limits and construction of rational numbers from integers by taking limits and ultimately explains how one can take infinite number of steps to go to infinity and take a single step back to come back home It is known that there are an infinite number of worlds, simply because there is ...

2

Life is like math. Come to: Add our friendships. . . Subtract our enemies. . . Multiply our jubilation. . . Division our sorrows. . .

2

Depending on what you are after it might be worth to look into the ancient Indian treaties on mathematics. Much of which where written in verse form. For example From infinity is born infinity. When infinity is taken out of infinity, only infinity is left over. From the Vedas, source http://www.hindupedia.com/en/Mathematics_of_the_Vedas

1

Here's the third stanza of Lewis Carroll's Four Riddles (taken from page 872 of The Complete Illustrated Works of Lewis Carroll (Bounty Books, 2004)): Yet what are all such gaieties to me Whose thoughts are full of indices and surds? $x^2+7x+53$ $=\frac{11}{3}$

1

Zelazny's Doorways in the Sand has a... well, math-related poem: http://en.wikipedia.org/wiki/Doorways_in_the_Sand#Allusions

22

Roses are red. Violets are approximately blue. A paracompact manifold with a Lorentzian metric, can be a spacetime, if it has dimension greater than or equal to two.

15

Imho what you need is not a poem - these can be pretty boring. So just as a counter-suggestion: Add a few fancy chords and check out Youtube: "I will derive" - http://www.youtube.com/watch?v=P9dpTTpjymE

9

And there's always topological rhyme doggerel: A mathematician confided That a Mobius band is one-sided, And you'll get quite a laugh If you cut one in half, For it stays in one piece when divided!

2

Not a poem, but a quote from one of my favorite books, "We" by Yevgeny Zamyatin. " . . . by means of the glass, the electric, the fire-breathing integral to integrate the indefinite equation of the universe. It is for you to place the beneficial yoke of reason round the necks of unknown beings who inhabit other planets-still living, as it may be, in ...

6

I had a prof who gave us this bit by Jacob Bernoulli in an attempt to get us excited about delta-epsilon proofs. Treatise on Infinite Series Even as the finite encloses an infinite series And in the unlimited limits appear, So the soul of immensity dwells in minutia And in narrowest limits no limits inhere. What joy to discern the minute in infinity! ...

7

How about Gilbert and Sullivan's Modern Major General: And I am well acquainted with methods mathematical, I understand equations both the simple and quadratical, About binomial theorem I am teeming with a lot of news, With many cheerful facts about the square of the hypoteneuse! Well, that's pretty close, I'll warrant! Hope this helps. Cheerio, and ...

7

It's not a poem (exactly) but there's always the Mandelbrot set song. Oh: and "Reel," by Vi Hart - a poem about $i=\sqrt{-1}$.

1

For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1 I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of 3" and was also featured in a Harold ...

1

For studying math for the first time ? No ! For deepening one's understanding of math not usually taught in school ? Yes ! E.g., I've tried once, for instance, to find a parametric form for the sum of two cubes, only to find out later, (using Google, no less!), that Euler has already done it $300$ years ago! And the list of pointlessly wasted time and effort ...

3

I don't think so, here are the reasons why I think it's not: First reason: mathematical methods for pedagogy have advanced through time. A lot of effort has been put to develop mathematical content which is useful for learning. Second reason: efforts during the last centuries have developed new tools which are useful for analysing mathematical knowledge, ...

1

Studying mathematical theories which were invented centuries ago is not necessarily a bad thing. In particular, Euclidean Geometry is still, in my opinion, the best way to be introduced to the notion of the mathematical proof, and of course, what is axiom, theorem, and why are all these important. Mathematical proof was born more that 2,500 years ago in the ...

1

There is little intrinsic value to studying mathematics chronologically, unless you are interested in the history of mathematics, as opposed to purely mathematics itself. You reasoned that learning the history of physics is interesting and can help 'flavor' the learning. Of course, the history of physics is not the same as that of mathematics -- I would ...

1

$a^2+b^2+ab=b-a-1 \implies (a+b)^2+(a+1)^2+(b-1)^2=0 \implies a=-1,b=1$

Top 50 recent answers are included