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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Choice of $\xi$ [duplicate]

Possible Duplicate: Rational Numbers Suppose $\{x \in \mathbb{Q}|x>0,x^2<2\}$ has a supremum. Call this supremum $c$. In order to show that this cannot be the case, we learned that we ...
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Archimedes Method To Find The Area Under A Curve

I'am reading Tom Apostol's Calculus volume-1 text (page 3 and 4),where he talks about calculating the area under a curve which eventually leads to the concept of the definite integral.In the below ...
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How to think about derivatives in an abstract fashion?

Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on. I, however, have ...
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Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
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Arnold's Trivium problem 52

Calculate the first term of the asymptotic expression as $k \to \infty$ of the integral $$ \int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx $$ May I bother you to explain what the ...
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Intuition for Poincaré duality and Cap product

Can you provide me with any intuition behind the Cap product of a cohomology class and a homology class? What is its geometric meaning? Can you also give me an intuition why the Poincaré duality is ...
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136 views

How can a point have a slope? [duplicate]

Possible Duplicate: Approaching to zero, but not equal to zero, then why do the points get overlapped? You get the derivative of $f(x)$ by getting the limit as $h$ tends to $0$ of ...
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what is the geometrical interpretation to positive definite matrix

What is the geometrical interpretation of positive definite matrix ? (not necessarily symmetric) if $A$ is positive definite, what does it do to a vector $x$ (i.e. $Ax$)?
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Bilinearity: what does it mean?

What does bilinear really mean? Everytime I heard the word, I think it should be "linear in 2 ways?" For example, from the definition of inner product (taken from Appendix A of "Wavelets For ...
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Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
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What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
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Propositional Logic Proof of $\vdash \lnot (p \supset q) \supset (p \land \lnot q)$

$\vdash \lnot (p \supset q) \supset (p \land \lnot q)$ I need to prove the above proposition via intuitionistic logic rules and/or natural logic rules. I guess it is not possible to prove with ...
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Why is the derivative multiplication by frequency in Laplace transform?

Why is the time-domain derivative equivalent to multiplication by frequency ($s$) in the Laplace transform? Why is the time-domain integral equivalent to division by frequency ($\frac{1}{s}$) in the ...
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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 ...
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187 views

Geometric invariants of a scheme

Following my previous question about sheaf cohomology, I'd like to ask about its applications to algebraic geometry. I have now learned a little about homological algebra and I can see that for ...
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928 views

Why graph a function?

Please enlighten me as to how graphing a function helps. I can see a graph's utility with simple functions as they instantly give you value of dependent variable. But ignoring them and considering ...
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441 views

Intuitive results with non-intuitive proofs?

Are there mathematical theorems that sound trivial and are obviously true, but are really tough to rigorously show? For example I stumbled across this question. The author asks how to show that two ...
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644 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
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What is a complex inner product space “really”?

To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. ...
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Plotting $\frac{1}{\ln x}$

I need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?
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How does this image prove the identity $1+2+3+4\cdots + (n-1) = \binom{n}{2}$? [duplicate]

Possible Duplicate: Proof for formula for sum of sequence 1+2+3+…+n? Proof without words: $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad $ How ...
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Why is the area under the curve exponentially greater than the original function?

So I've been a calculus student now for about two years, and I've gone as high as differential equations, but I am still a bit puzzled by the fact that the area under the curve of some function is ...
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Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?

Let $R$ be a commutative ring, $\mathfrak{p}$ a prime ideal of $R$, $M$ a $R$-module, and $N$ a $R_\mathfrak{p}$-module. Why do we have this isomorphism? $$M \mathbin{\otimes_R} N \cong M_\mathfrak{p} ...
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Deepest theorems with simplest proofs [closed]

Which are the deepest theorems with the most elementary proofs? I give two examples: i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function ii) Proof that the halting problem is ...
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Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...
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What is the difference between probability and statistics?

Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?) I'm pretty ...
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Sorting through “algebra of random variables,” vs. “probability space,” etc

I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for ...
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Is it possible to determine that a coin is biased or not, by tossing it a number of times?

Is it possible to determine that a coin is biased or not, by tossing it a number of times ? I am sure that this problem has been studied,I am interested to know about the mathematics behind this ...
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Pattern-matching puzzle (shapes)

some puzzle that can be found online are typically a 3x3 grid where you've got nine spots filled with some images or symbols or geometric figures and you have to predict what the missing one should ...
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Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$ m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy $$ where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$ h(x,y) = x^p y^q f(x,y) dx dy ...
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Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive?

Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive? Whenever $f'$ (the first derivative) is positive the function is increasing,but does that ...
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Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
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396 views

Is there an intuitive interpretation of $ABA^T$?

I see expressions like this all the time in technical literature. The only $A$ and $B$ can be any size matrices as long as the expression is legal. I believe that the transposition is usually ...
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398 views

How many different combinations of $X$ sweaters can we buy if we have $Y$ colors to choose from?

How many different combinations of $X$ sweaters can we buy if we have $Y$ colors to choose from? According to my teacher the right way to think about this problem is to think of partitioning ...
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2answers
281 views

What is the reasoning behind a multinomial coefficient in a practical sense?

If you want to divide a team of 10 people into teams A, B, and C of sizes 3,5, and 2, how many divisions are possible? If you want to divide them into just teams of sizes 3,5, and 2, how many ...
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Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
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140 views

Intuition about the faces in the connected planar graphs

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer ...
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Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
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2answers
3k views

Find the area of the curved shape

How to find area of this curved shape?
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1answer
263 views

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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355 views

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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930 views

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 ...