Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Find the area of the curved shape

How to find area of this curved shape?
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Generalisation of dualities, what concept do dualities represent?

Duality is a concept that pops up in different areas of mathematics as well as other science, but besides being a "woo isn't that nice?", is there anything more to duality (than loosely stated some ...
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Factorials and Combinations

I understand that $$n! = n (n-1) (n-2)\cdots 2 \cdot 1.$$ My book says this can also be written as $$n (n-1)!$$ Without telling me why My question is How and why is that? Why can't we leave it as it ...
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Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
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Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements

Let $X$ be the set of natural numbers $k_i$, $k_i \geq 1$, with the property that at least one of the equations $p_i = $6$ k_i \pm 1$ gives the $i$-th prime number (disregarding $2$ and $3$), and ...
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In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...
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Difference between Logarithms of different bases

Every time i see a logarithmic function and if it so happens that i'am required to take the derivative or the integral of that particular function i get stumped and i tend to avoid that problem. What ...
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where did determinant come from? [duplicate]

Possible Duplicate: What's an intuitive way to think about the determinant? I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$ I ...
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Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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Ten soldiers puzzle

This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each ...
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Intuition behind looking at permutations of the roots in Galois theory

What I find after reading books is that they explain only the conceptual definition and no one mentions the explanation behind it; I have been reading the Galois theory as many people told me to read, ...
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Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
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262 views

An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
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What is a physical “dimension” - in the sense of “dimensional” analysis?

Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ...
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Definition of e

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form Is $dy/dx$ not a ratio? I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as ...
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Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
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What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
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880 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
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What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?

What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ? $\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation ...
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Complex graphs in 3D

Does anyone have red-green 3D software for plotting 4D graphs in 3D with 3D glasses? I've seen a 4D hypercube done this way and it's very revealing...
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I need explanation for this solution for the proof. (Perfect square ends with 0,1,4,5,6,9)

Give a proof to the sentence: "The final decimal digit of a perfect square is 0, 1, 4, 5, 6 or 9." Solution: A integer $n$ can be expressed as $10a+b$, where $a$ and $b$ are positive integers and $b$ ...
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Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
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What are the Axiom of Choice and Axiom of Determinacy?

Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple ...
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Looking for an article on general principles of discrete mathematics

In his article 2 cultures Timothy Gowers states that the structure in combinatorics is there in the form of somewhat vague general statements that allow proofs to be condensed in the mind, and ...
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Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
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What is the intuition behind the “par” operator in linear logic?

I'm $\DeclareMathOperator{\par}{\unicode{8523}}$ trying to wrap my mind around the $\par$ ("par") operator of linear logic. The other connectives have simple resource interpretations ($A\otimes B$ ...
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Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)? ...
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“Contradiction-free” in logic vs. “Contradiction-free” in plain mathematics

In our course we have defined a theory $T$ to be contradiction-free, if there are no formulas $\alpha_1,\ldots \alpha_n\in T$ such that $\neg ( \alpha_1 \& \ldots \ \& \alpha_n )$ is provable ...
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Intuition behind the scaling property of Fourier Transforms

I had a course in PDE last year where we used fourier transforms extensively; I understand the rules of manipulation and can prove the scaling theorem directly from the definition using a ...
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The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
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I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
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Why is it that Complex Numbers are algebraically closed?

I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
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Four men, hats and probability

I encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...
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Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$

I came across the following problem in my self-study of real analysis: For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = ...
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Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
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One divided by Infinity?

Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me. I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - ...
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Intuition behind elliptic curves and $K$-rational points

I find myself becoming confused whenever I try to think about this. In the following, $K$ is a field. An elliptic curve $\mathcal{C}$ is defined to be a nonsingular projective cubic curve over $K$, ...
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A way to see that $\int_{0}^{\infty}\exp(-x)dx=1$?

One can easily find the integral $\int_{0}^{\infty}\exp(-x)dx$. It is equal to 1. But is there a way to understand this geometrically without integration? If i rotate the picture i see that ...
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What is the mathematical process behind fully homomorphic encryption?

I've often wondered about how to compute encrypted data, and appears that a "hack" to do so has been found: http://www.technologyreview.com/computing/37197/ Is anyone able to offer an better, yet ...
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practical uses of matrix multiplication

The use of matrix multiplication is usually given with graphics initially (scalings, translations, rotations, etc). Then there are more in-depth examples such as counting the number of walks between ...
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Intuition behind Sobolev norm

This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional $$F(u)=u(1)-u(0).$$ $F$ is not continuous on $L^2(0, 1)$ ...
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Signs in the tensor product and internal hom of chain complexes

Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a ...
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How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
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A question on logic - where intuition can fail

Suppose I have two predicates $P(x)$ and $Q(x)$, such that $\overline{P(x)\wedge Q(x)}$ holds for all $x$. Now, if $\displaystyle \bigwedge_{x\in A}P(x)$ for a set $A$, it must be certainly true, ...
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The simplest $\Delta$-complex structure on $S^2$

I think my reasoning is correct, but I want to run through it here because having the right intuition will make similar problems easier in future. A 2-simplex is homeomorphic to a closed disc, and a ...
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The Mayer-Vietoris sequence

If $X$ is a space with a pair of subspaces $A, B \subset X$ such that $X$ is the union of the interiors of $A$ and $B$, then there is a long exact sequence of homology groups $\displaystyle \ldots\to ...
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Demonstrating the value of abstracting away from elements/subsets to maps

Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction: an actual element, e.g. $s\in S$ an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$ an ...
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Are there any generic thinking approaches for providing mathematical proofs to a given theorem

To produce mathematical proofs for theorems we should have the required knowledge in that area. But even having adequate knowledge, people like me struggle a lot for writing down the proofs for any ...
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...