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25

Here is one example that I find aesthetically pleasing, and which I have found effective in 8th-grade classrooms. Suppose you desire to cut out a triangle from the middle of a piece of paper, not by punching the scissors through and cutting the perimeter, but rather by folding the paper and then cutting straight through the folded paper. The natural ...


21

The short answer is yes, but it is not a very easy concept. The answer lies with the definition of the integral and measures. The idea of a measure is to assign a size to subsets of the reals, and use this idea of size to perform integration (among other things). The first thing to note is that there are different ways to assign a measure to subsets of the ...


11

Your examples can be viewd as particular cases of a general theory of integration, namely, the Lebesgue integration. In fact: If $X=\{m,m+1,...,n\}$, $\mu$ is the counting measure and $f$ is positive, then $$\int_X f\ d\mu=\sum_{k=m}^nf(k).$$ If $X=[a,b]$, $\mu$ is the Lebesgue measure and $f$ is continuous, then $$\int_X f\ d\mu=\int_a^bf(k)\ dk,$$ where ...


7

I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested in Algebraic Geometry and Number Theory, ...


7

Although I am not much of an Algebraic Geometry person, but it is a very important application of commutative ring theory. Just to give you a flavor, here is a brief idea. Consider the ring $\Bbb{C}$. Now we do like to study surfaces/curves (that is an application, right!) , and we define them by equations in rings $R=\Bbb{C}[x_1,x_2,\dots , x_n]$. Now ...


6

I rather like the bean machine as a physical introduction to probability, normal distribution and the concept of randomness: Now that I think of it, I rather like mechanical devices in general that demonstrate mathematical truths/ideas. Leibnitz's mechanical binary calculator is another that springs to mind.


6

A cheeky way of doing it: define $f(i) = \begin{cases} 0 & i \leq 8 \\ 1 & \text{otherwise} \end{cases}$, and plug in $? = f(i)$. But that's lame. One thing you can do that might be what you're looking for but will definitely make whatever you're doing less understandable is to play with the floor function. If you don't know, $\lfloor x \rfloor$, ...


5

Usually, yes, though I prefer Euler's identity. Pretty much every trig identity can be derived from $$e^{ix}=\cos(x)+i\sin(x).$$ However, it is useful to memorize some of the common ones because they will help you a lot in calculus and beyond to quickly identify when an expression can be simplified. I would start with memorizing the angle addition formulas. ...


5

When I was at high school I loved geometrical arguments, especially those which were both simple and profound. Later on, I discovered Combinatorial Geometry, which combines both Euclidean Geometry and Combinatorics. I think that a good idea would be to demonstrate the proof that No matter how the plane is 3-colored, it contains a monochromatic unit distance ...


5

In my opinion Hardy &Wright's book on Number Theory is not the best possible book for someone "who has no prior training in Number Theory", I would suggest the following books. Elementary Number theory by David M. Burton. Number Theory A Historical Approach by John H. Watkins Higher Arithmetic by H. Davenport All the books are ...


5

I think it's pretty hard to find a book which covers martingale theory; usually, books either give just an introduction or they focus on one particular aspect of martingale theory. I'll list some books which might be of interest and sketch (roughly) which parts they cover: David Williams: Probability with Martingales (Basic properties, optional stopping, ...


4

One of the best is An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery.


3

I would recommend An Introduction to the Theory of Numbers By G.H. Hardy and E.M. Wright .


3

Have you checked out Paul Lockhart's "A Mathematician's Lament" and "Measurement"? He describes his approach to teaching K12 math and - in the latter book - walks you through some elementary (but non-trivial) geometry.


3

Cutting a bagel into linked halves is impressive to any audience. Just make sure to make a full twist: cutting along a Moebius band would produce a bigger and thinner bagel rather than linked halves.


3

Let me first note that physicists may be better people to ask this question than mathematicians are. I think it's worth remembering a few, and knowing how to rederive the others. The important ones to remember initially are: (a) The definitions of $\tan$, $\cot$, $\sec$, $\csc$, in terms of $\sin$ and $\cos$, as well as the identity $\cot x = 1/(\tan ...


3

You could use the sign function but you would want to make sure it doesn't land on zero so in your case $$ \DeclareMathOperator{\sign}{sgn} \sum_{i=1}{\sign(17-2i)2x^3} $$ $$ \DeclareMathOperator{\sign}{sgn} \sign(x) = \left\{\begin{matrix} -1 & x<0 \\ 0& x=0 \\ 1& x>0 \end{matrix}\right. $$ $$\DeclareMathOperator{\sign}{sign} $$


3

In a realm of discourse (say, "ordinary mathematics", or ZFC) where both are provably true, or provably false, or each is provably true or false assuming that the other is true or false, they're equivalent. Tacitly assuming that we're in such a realm (since FLT has now been proven), you're correct. They're not semantically equivalent statements, though ...


2

Just to follow on from something inspired by john-mangual's answer: Firstly one might know that the rationals $\mathbb{Q}$ are countable and thus can be enumerated by the positive integers $\mathbb{N}$. Let's pick an enumeration $\mathbb{Q}=\{q_1,q_2,q_3,\ldots\}$. Now let $f:\Bbb R \rightarrow \Bbb R$ such that it is defined on $\mathbb{Q}$ (or restrict ...


2

One might say: Between sums (finitely many summands) and integrals (uncountably many "summands"), we have series (countably many summands).


2

My answer relates less to strictly mathematical proofs of specific concepts, but emphasizes drawing out the math behind stuff. There are 3 main topics which I would split my answer into: Math toys, Cool math tricks, and Math in everyday life Math Toys Spirographs: The mathematical basis of spirographs are simple and definitely within the grasp of high ...


2

I'm not sure this is the kind of intuition you're looking for, but another way to approach the problem is to let $$g(x)=f\left({1+x\over2}\right)-f\left({1-x\over2}\right)$$ It's easy to see that $g(x)$ is an odd function, and $g(-1)=g(0)=g(1)=0$. Moreover, $$g'(x)={1\over2}\left(f'\left({1+x\over2}\right)+f'\left({1-x\over2}\right)\right)\implies ...


2

Dover publishes many number theory titles. At \$10-\$15 each they're a bargain - no need even to look for the Amazon discount. You can get several and jump back and forth among them to get different perspectives on each topic. You can write yourself notes in the margins. Take them to the library to read. This is a standard old undergraduate text: ...


2

I can certainly recommend Elementary Number Theory by Gareth A Jones et al. It will get you started and then you can move onto more advanced texts. It's a very short book (about 300 pages) which means you can easily read through the whole text-a very good choice for self study. As for pre-requisites, a good grasp of algebra will probably do.


2

A delicious classic would be "The Theory of Functions" by Titchmarsh. Despite being old (1939), it is very modern in its rigour, fear not! You may find it freely available on the internet, it is no longer subjected to any copyright.


1

I've just decided to call them "uniands", but I came here hoping there was something standard. I believe you could get away with calling them "summands" and "factors" by analogy of $\cup$ with $+$ and $\cap$ with $\times$ (the first being the sum and product of the Boolean ring of subsets of a set, and the second being the generic terms for sum and product ...


1

Kaplansky's Zero-Divisor Conjecture Let $K$ be a field and $G$ a torsion free group, then is the group ring $KG$ a domain? All the research up until now have been affirmative. This problem has been dealt in the book "The algebraic structure of group rings" by D. Passman. It is one of the toughest and least approachable problem in the whole field ...


1

"Proofs without words: Exercises in Visual Thinking" is a book dedicated to visual proofs. The book has proofs about Geometry, Trigonometry, Calculus and also Sequences and Series. In case you run out of proofs for the class there is also a sequel of this book "Proofs without words II: More Exercises in Visual Thinking". Since, these are "exercises" no ...


1

The proof of Thales' Theorem (that any triangle constructed using the diameter chord of a circle and a third point on the circle that doesn't coincide with the endpoints of the diameter is a right triangle) is a pretty nice one: The triangle $\triangle OAB$ is an isoceles triangle because $OA$ and $OB$ are both radii of the circle and thus by definition ...



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