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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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 ...
2
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1answer
149 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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238 views

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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2answers
238 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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1answer
98 views

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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1k views

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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4answers
688 views

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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1answer
2k views

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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5answers
207 views

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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1answer
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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1answer
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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4answers
1k views

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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2k views

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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1answer
146 views

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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1answer
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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3answers
126 views

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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1answer
93 views

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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1answer
888 views

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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2answers
171 views

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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528 views

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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0answers
194 views

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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0answers
89 views

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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1answer
86 views

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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1answer
150 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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2answers
101 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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2answers
178 views

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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2answers
186 views

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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1answer
131 views

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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0answers
44 views

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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1answer
184 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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3answers
186 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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2answers
2k views

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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0answers
52 views

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 ...
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12answers
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Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
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0answers
250 views

Intuition behind the definition of a measurable set

This week I saw the definition of a measurable set for an outer measure. Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + ...
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1answer
584 views

Intuition for Absorption and Distributive Laws in Elementary Logic

$P ∧ (Q ∨ R) \equiv (P ∧ Q) ∨ (P ∧ R) \tag{Distributive Law 1}$ $P ∨ (Q ∧ R) \equiv (P ∨ Q) ∧ (P ∨ R) \tag{Distributive Law 2}$ $P ∨ (P ∧ Q) \equiv P \tag{Absorption Law 1}$ $P ∧ (P ∨ Q) ...
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60 views

Clarification on Formal Definition of Functions

In my Abstract Algebra class, the professor defined a restriction as Given $ X\xrightarrow{f} Y $ and a non-void subset $S$ of $X$ define $ f \mid S\xrightarrow{S} Y $ by $(f \mid S )(s) = f(s), ...
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2answers
228 views

Matrices Intuition

I am currently studying matrix algebra. The axioms and theorems of this form of algebra are a bit different from the high school algebra I did. However one knows that one is dealing with real numbers ...
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0answers
359 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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1answer
107 views

Monty Hall vs. Card Example

In class, while illustrating the topic of conditional probability, my professor presented the following card example: You have 3 cards that have been randomly shuffled: card1, card2, and card3. ...
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2answers
170 views

Why does it make sense to pair existence and uniqueness?

Edit. September 8th, 2013. I've added another another section to the question, which should hopefully constrain its scope. Please see below the horizontal line!! Lets write the statement 'there is at ...
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1answer
58 views

Finding Similar Sequences

Can we find two sequences: $$\{a (b^0), a (b^1), a (b^2), a (b^3), \dots, a(b^n)\} \bmod p_1$$ $$\{c (d^0), c (d^1), c (d^2), c (d^3), \dots, c(d^n)\} \bmod p_2$$ that differ by only one number? ...
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1answer
169 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
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1answer
986 views

Do all vectors have direction and magnitude?

I go by Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah! For ...
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2answers
547 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
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1answer
305 views

Foundation of category theory

In the first pages of "category theory for the working mathematician" Saunders claims that category can be introduced, without set theory, as objects and arrows without some "operations" satisfying ...
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0answers
149 views

Wilson's theorem intuition

Wilson's Theorem: $p$ is prime $\iff$ $(p-1)!\equiv -1\mod p$ I can use Wilson's theorem in questions, and I can follow the proof whereby factors of $(p-1)!$ are paired up with their (mod $p$) ...
3
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1answer
92 views

What's the intuition behind this definition of ordered pair in the $\lambda$-calculus?

On this page, we have the following definitions. pair = λabf.fab first = λp.p(λab.a) second = λp.p(λab.b) So I tried computing "first (pair a b)," and sure ...
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3answers
178 views

Are statements like “Every time I've done X, Y has happened” (vacuously) true if I've never done X?

I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following ...
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1answer
60 views

$|f(x)|<|g(x)|$ and $\int g(x)<\infty\Rightarrow\ \int f(x)<\infty$

Let f,g continious functions on $[0,\infty)$ s.t $\forall x\in[0,\infty), |f(x)|\le|g(x)|$. Prove or give counterexample thtat if $\int_ 0^\infty g(x)dx<\infty$ then $\int_0^\infty ...