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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Definition of trignometric functions

Why is the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse? How this definition came? Similar question for other five trigonometric function? Why they ...
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Convergence of alternating nested radicals

Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found ...
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101 views

Exponent Upon Exponent? [duplicate]

I can't understand this. Can you please make a clearer explanation?
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Intuitive understanding of Lemma of Jordan

Can somebody give me an intuitive understanding of the Lemma of Jordan, which is: $$\lim \limits_{R\rightarrow \infty} \int_{\gamma} \exp(i \omega z)\,\rm dz=0 $$ if: $$\lim \limits_{z\rightarrow ...
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Finding the derivative of a piecewise function of a complex variable

I am working on an assignment that involves finding the derivative evaluated at zero of this piecewise function of a complex variable: Let $g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by: ...
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248 views

When does infinite intersection preserve a closed property?

There are two statements well known in Math and Computer Science: Intersection of infinite number of regular languages is not regular. Intersection of infinite number of convex sets is convex. ...
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69 views

How to reason about congruences? If $x^2 \equiv a$ (mod $m$) and $y^3 \equiv a$ (mod $m$), then $\gcd(a,m) = 1$

Generally, I have no high level conception of what is going on in my number theory class. It feels like a loose collection of theorems and techniques that you can use on some problems, but I have ...
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144 views

Intuition for Geometric Transformations

I've been making a lot of effort over the past few hours to gain some intuition into the art of geometric transformation but to little avail. I would really like to be able to look at a transformation ...
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71 views

Solving limits of form $\infty \times \infty$

I need help with solving $f(x) \lim_{x \to \infty} e^x\times x^3$ I know that $\to \infty$ is the answer but I don't know how to get there. Using since $e^x \to \infty$ and $x^3 \to \infty$ I get ...
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On subspace verification

I am struggling with the following Problem: \begin{align}Y= \lbrace (x^4-y^4,0,0,0) \mid x,y \in \mathbb{R} \rbrace \subset \mathbb{R}^4 \end{align} Question, is the given Set a subspace of ...
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55 views

Significance of the DEGREE of Differential equation

Can anybody gives the idea why degree of a differential equation is important?. Every differential equation book writes the definition of "degree of a differential equation". But, why do we care about ...
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If $A$ is the adjacency matrix of a graph, why does the $(i,j)$ entry of $A^n$ give the number of $n$-step walks from $i$th vertex to $j$th vertex?

Let $A$ be the adjacency matrix of some directed graph with $m$ vertices labeled as $v_1, v_2, \ldots, v_m$. So here $A_{ij} = 1$ if there is an edge from $v_i$ to $v_j$, and $A_{ij} = 0$ otherwise. ...
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How to think about the change-of-coordinates matrix $P_{\mathcal{C}\leftarrow\mathcal{B}}$

I've taken a linear algebra course in the past, but I feel my understanding of coordinate change is very superficial. For example this exercise (4.7.1 from Lay's "Linear Algebra and its Applications" ...
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235 views

Geometric interpretation of the addition of linear equations in general form

I have a very simple question: suppose I have two 2D linear equations in general form $$ a_1x + b_1y + c_1 = 0$$ $$ a_2x + b_2y + c_2 = 0$$ I'd like to know what's the (intuitive) geometric ...
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Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
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270 views

Problem with definition of regular surface in classical differential geometry

I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this: I have few doubts about this definition: 1) Why we need to find ...
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349 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
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200 views

Geometric Intuition of Gaussian Curvature

Curvature of a curve at a point can be understood as how rapidly the curve tries to move away from the tangent of the curve at that point. And for curved surfaces we have defined the Gaussian ...
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150 views

Visualizing Markov and Chebyshev inequalities

I am helping a class on introductory probability covering Markov and Chebyshev's inequalities. I would like to give the students a nice visualization for why they are true or at least to show what ...
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What's the idea behind the covariant derivative?

I'm learning differential geometry from what I find on the Internet (to eventually find a grasp on General Relativity too). Right now I playing with a sphere. I have 3 functions ($x$, $y$, $z$) that ...
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260 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...
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Visual proof ot the distributive property in $\mathbb{Z}$

Is there a intuitive/visual (not formal) "proof" that the distributive property holds in $\mathbb{Z}$? For the natural numbers $\mathbb{N}$ I know something like this: There are two ways to get ...
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Explanation for $\lim_{x\to\infty}\sqrt{x^2-4x}-x=-2$ and not $0$

I am trying to intuitively understand why the solution to the following problem is $-2$. $$\lim_{x\to\infty}\sqrt{x^2-4x}-x$$ ...
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Visualising finite fields

I'm interested in finding visual and/or physical approaches to understanding finite fields. I know of a few: V. I. Arnold has a few pictures of 'finite circles' and 'finite tori' in his book Dynamics, ...
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Proof of an odd function plus an even function

I was looking at the wiki page http://en.wikipedia.org/wiki/Even_and_odd_functions#The_sum_of_even_and_odd_functions and it says that to prove an even function plus an odd function, we first have to ...
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How would one define a “manifold” object in prose writing?

I have a question that I fear may raise some objection to the fact that it has been posted here, but I cannot think of a more appropriate place to pose it. I am not a mathematician; I'm a historian, ...
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105 views

Is there a way to mathematically describe “surprise”?

Let's say that there are ten people entered into a random drawing, the winner gets some large prize. If I were one of those ten people, and I were to win, then I would be pleasantly surprised. If ...
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54 views

What comes first, a vector base or orthogonality?

Pick any three vectors of a vector space randomly (but linearly independent). Then we assign them coordinates: $$e_1=[1 0 0]$$ $$e_2=[0 1 0]$$ $$e_3=[0 0 1]$$ Therefore now they are orthonormal ...
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Intuition: Power Set of Intersection/Union (Velleman P77 & Ex 2.3.10, 11)

Source: How to Prove It, 2nd Ed by Velleman. $\mathcal{P}(...) =$ power set of ... & $A, B$ are any sets: Ex 2.3.10: $\qquad \qquad \qquad \qquad \qquad \qquad \mathcal{P}(A \cap B) = ...
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Explicit Formula for a Recurrence Relation for {2, 5, 9, 14, …} (Chartrand Ex 6.46[b])

Consider the sequence $a_1 = 2, a_2 = 5, a_3 = 9, a_4 = 14,$ etc... (a) The recurrence relation is: $a_1 = 2$ and $a_n = a_{n - 1} + (n + 1) \; \forall \;n \in [\mathbb{Z \geq 2}]$. (b) ...
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Finding a mapping such that its kernel equals the image of another non bijective mapping

For an $a \in \mathbb{R}$ let $\phi_a: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear mapping such that $\phi_a(x) := \begin{pmatrix} 1 & 2 & 2 \\1 & 3 & 5 \\ 1 & -1 & a ...
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72 views

Expectation of $[Y g(X)]$

Firstly, how do I interpret $\mathbb{E}[g(X)]$. I understand $\mathbb{E}[X]$ is like the most likely outcome of a set of experiments (loosely speaking at least - not really a very maths person)? But ...
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If $X\sim \exp(\lambda)$ and $Y\sim \exp(\mu)$ then $P(X\leq Y)=\frac{\lambda}{\lambda+\mu}$. Is there an intuitive interpretation for this fact?

I can verify this via double integrals, but I'm wondering if this can be put in the context of a Poisson process or something to give it an obvious meaning. I can't think of exactly how it would work. ...
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Why is only the first (highest) term of the divisor in polynomial long division used to divide?

There is one small matter that has always stumped me with polynomial long division. In the example from the Wikipedia on Polynomial long division, why is the equation only divided by the first/highest ...
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The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
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Minimal Counterexample for False Prime-Generating Quadratic Polynomials (Chartrand Ex 7.66)

Factor the quadratic: $n^2 \pm n + 41 = n(n \pm 1) + 41 = n\left[(n \pm 1) + \cfrac{41}{n}\right]$. So if we find at least one $n$ such that $\frac{41}{n}$ is an integer, or equivalently an $n$ ...
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St. Petersburg Paradox

A fair coin will be tossed until a heads results. You will then be paid $2^{n-1}$ dollars where $n$ equals the number of flips. Now why is the expected pay out infinite? $$ \sum_{n \geq 1} ...
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Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function ...
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Equality and order in sets

Just started Baby Rudin and got struck in this. While defining order in sets, $<$ was introduced as a relation and for a set to be ordered the condition was: for all $x,y$ belonging to an ordered ...
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158 views

Is it correct to think about homeomorphisms as deformations?

The definition of homeomorphism is that of a continuous bijection with continuous inverse. Because we can think of continuous functions as functions that maps nearby points to nearby points, we could ...
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102 views

Which of these topological properties imply which?

I am going through the chapter on compactness and completeness from Sternberg's Advanced Calculus and trying to build an intuition for what many of this topological properties mean, and which imply ...
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Geometric Mean limit of $\ell_p$ norm of sums

My analysis professor introduced the $\ell_p$ norm to our class as: \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} We are asked to prove the following: ...
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How does a geometry-oriented mind learn analysis?

I find it very difficult to understand analysis, because I can't find a way to learn it geometrically. To make my point clearer, let me take calculus as the example in contrast. I find calculus very ...
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Calculating cardinal numbers of subsets in $\mathbb R\times\mathbb R$

Calculate the cardinal numbers of the following subsets of $\mathbb R\times\mathbb R$ : a.$X=\left\{ (a,b)\in\mathbb{R}\times\mathbb{R}\mid a+b\in\mathbb{Q}\right\} $ b.$Y=\left\{ ...
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What are some ways of showing that a structure is not minimal?

The question is really in the title. My background in model theory is very limited. Basically nothing past the definition of minimal structures and minimal subsets. I am interested in some ...
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185 views

Intuition behind the Jacobi triple product

Jacobi's triple product identity states that: $\displaystyle \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = \prod_{n = 1}^{\infty}(1 - q^{2n})(1 + zq^{2n - 1})(1 + z^{-1}q^{2n - 1})$ I've seen a messy ...
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189 views

non-archimedean in lay terms

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ...
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Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
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Can we perform this operation on block matrices?

We have a block matrix: $$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$ Here $A$, $B$ and $C$ are all permutation ...