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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671 views

Intuition Building: Visualizing Complex Roots

I can graph the parabola $y = x^2 + 1$. I see that it does not intersect the $x$-axis, and therefore it must have complex roots, namely $+i$ and $-i$. I can plot these roots on an Argand diagram at ...
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2answers
208 views

Improper integrals question: why does this work?

So, I'm solving $$ \int_0^3{ \frac{dx}{ \sqrt{ 9 - x^2 } } } $$ The catch here is $f(x)$ is defined on ( -3, 3 ) only. And I know the way to solve this is to integrate (yielding $ \arcsin{ ...
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2answers
904 views

Intuitive meaning of Pearson Product-moment correlation coefficient Formula

I can't understand the intuition behind Pearson Product-moment correlation coefficient Formula for bivariate data. The formula is : $\rho$ = cov(X,Y)/($S_x$ * $S_y$) where cov is covariance. $S_x$ and ...
4
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1answer
177 views

Visualizing Operators on $\mathbb{C}^n$

I am trying to get some better intuition about operators on complex inner product spaces. When we identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, is there a nice geometric interpretation for the ...
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3answers
266 views

Some basic problems of group theory

1.Prove the every group G of order 4 is isomorphic to either Z4 or 4-group V,that is {1 (1,2)(3,4) (1,3)(2,4) (1,4)(2,3)} 2.If G is a group with $|G|\leq 5$ then G is abelian. I have learned ...
4
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1answer
759 views

Explain what is a linear transformation

My textbook says "unique linear transformations can be defined by a few values, if the given domain vectors form a basis." However, that is all it says. So can someone explain what a unique linear ...
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15answers
1k views

List of Local to Global principles

What are some of the local to global principles in different areas of mathematics?
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1answer
993 views

Why is the ratio test for $L=1$ inconclusive?

One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or ...
4
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1answer
316 views

What's the intuition behind and some illustrative applications of probability kernels?

Given measure spaces $(X, \mathcal{X})$ and $(Y, \mathcal{Y})$ we define measure kernel $\pi : \mathcal{X} \times Y \to [0,\infty]$ such that $\pi(\cdot|y)$ is a measure on $\mathcal{X}$ for every $y ...
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10answers
15k views

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it ...
3
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1answer
3k views

Connection between chain rule, u-substitution and Riemann-Stieltjes integral

I think I understand these concepts ok: chain rule u-substitution Riemann-Stieltjes integral But there seems to be a layer that I miss: They all seem to be connected, alas I don't know how ...
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2answers
2k views

How to think deeply in mathematics way? [closed]

Most of the time, I solve a problem by doing the following steps: 1. Look for a related theorem. 2. Look for a related formula. 3. Find a similar problem or sub-problem. 4. Try to adapt the one that I ...
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7answers
1k views

Why in an inconsistent axiom system every statement is true? (For Dummies)

I would like to know if someone can explain in a somehow down to earth (almost logic free) way why is it true that in an axiom system where there is some statement $P$ such that $P$ and its negation ...
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3answers
3k views

Why do 4 circles cover the surface of a sphere?

Is there a geometric explanation for why a sphere has surface area $4 \pi r^2$ ? Ie equal to 4 times its cross-section (a circle of radius r).
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2answers
1k views

hypergeometric distribution problem

I am looking for some insight into a problem: Consider a group of $T$ persons, and let $a_1, a_2, ..., a_T$ denote the height of these $T$ persons. Suppose that $n$ are selected from this group at ...
4
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2answers
192 views

Counting and Ordering of Numbers

Are there differences between 'counting' and 'ordering'? As such, the whole of rational number is countable, or they order-able too? In what cases counting and ordering are same or not?
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2answers
729 views

Intuition behind arc length formula

I understand the arc length formula is derived from adding the distances between a series of points on the curve, and using the mean value theorem to get: $ L = \int_a^b \sqrt{ 1 + (f'(x))^2 } dx ...
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8answers
2k views

Why do we restrict the definition of Lebesgue Integrability?

The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.) Why is it we ...
3
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1answer
709 views

A question on finding the intersecting line between two planes

According to my math book, in order to find the intersecting line between two planes we need to: Find the vector product of the direction normals of the two planes Write the equations of the planes ...
26
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5answers
3k views

What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?

I am aware of an intuitive explanation for $\operatorname{curl} \operatorname{grad} F = 0$ (a block placed on a mountainous frictionless surface will slide to lower ground without spinning), and was ...
2
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3answers
571 views

Solutions to interesting problems with elegant and unintuitive methods

I am looking to build a repertoire of olympiad type problems which have non-intuitive elegant solutions, If possible instead of a resource, I think problems would be the best. (i.e. select the best ...
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6answers
7k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
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15answers
3k views

Intuitive Understanding of the constant “e”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say ...
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3answers
439 views

Nasty examples for different classes of functions

Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming ...
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2answers
679 views

Amalgamated Free Product

Suppose $H$ is embedded in $G$ and $H'$ is isomorphic to $H$ and embedded in $G'$. Then we can simultaneously embed $H$, $H'$, $G$ and $G'$ into a single object (the amalgamated free ...
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2answers
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Matrix multiplication: interpreting and understanding the process

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
33
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4answers
2k views

Understanding the intuition behind math

I'm currently a Calculus III student. I enjoy math a lot, but only when I understand its beauty and meaning. However, so many times I have no idea what it is I am learning about, althought I am still ...
4
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1answer
490 views

Largest known integer

Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number ...
29
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13answers
11k views

Formerly good at math, but after 12 years I've lost most of my skills. Now I need them once again. Any advice to grow them back?

I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the ...
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1answer
355 views

Intuition Behind Geometric Means

Suppose $w \in L^{1}(G)$ and $w \geq 0$. The geometric mean of $w$ is defined by $$ \Delta(w) = \exp \int_{G} \log w(x) \ dx$$ where $\Delta(w) = 0$ if the integral is $-\infty$. What is the ...
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2answers
129 views

Continuous scalar fields on spheres

I've been thinking about this for awhile now (I am trying to find a method of proving the Borsuk-Ulam theorem in 2 dimensions without resorting to the usual, and not so intuitive to non-mathematicians ...
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3answers
465 views

Infinitesimals - what's the intuition?

When considering an infinitesimal distance/interval/in calculus, what is the intuitive interpretation? Is it too small to be measurable but still has some distance on an unattainable scale? Are there ...
5
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1answer
107 views

Diffusions - global and local

Suppose $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$ is a diffusion. Is there a sense in which the dynamics are "dominated" locally by the diffusion term, and dominated globally by the drift term? If $\mu$ ...
8
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3answers
2k views

Intuitive explanation of the Fundamental Theorem of Linear Algebra

Can someone explain intuitively what the Fundamental Theorem of Linear Algebra states? and why specifically it is called the above? Specifically, what makes it 'Fundamental' in the broad scope of the ...
9
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2answers
281 views

geometric argument for van-kampen?

I've seen Van-Kampen's theorem presented algebraically many times; and although it provides a useful method of calculation; I don't have a very clear picture for "why" it should be true. Does anyone ...
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1answer
273 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
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3answers
2k views

What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
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9answers
971 views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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5answers
321 views

how to explain that Prob[heads, tails] = 2 * Prob[heads, heads] to a student?

I throw two coins (simultaneously). A student (very much a beginner in both math and probability theory) thought that the following 3 outcomes are equally likely: "two heads", "two tails", "a head and ...
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0answers
114 views

Intuition for Calderon-Zygmund operator?

What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
2
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3answers
2k views

What does rate of change actually mean?

If we have a function, say, $y=x^2$, then $dy/dx=2x$. Now $y=1,4,9,16...$ for $x=1,2,3,4...$ and $dy/dx=2,4,6,8...$ for $x=1,2,3,4...$ Now as $dy/dx$ represents rate of change of y w.r.t x but I can't ...
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5answers
3k views

Dominoes and induction, or how does induction work?

I've never really understood why math induction is supposed to work. You have these 3 steps: Prove true for base case (n=0 or 1 or whatever) Assume true for n=k. Call this the induction ...
11
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4answers
2k views

Intuitive explanation of Nakayama's Lemma

Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$. I've read the proof, ...
8
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0answers
488 views

Combinatorial reasoning for the identity $\left ( \sum_{i=1}^n i \right )^2 = \left ( \sum_{i=1}^n i^3 \right ) $ [duplicate]

Possible Duplicate: Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ There is the interesting identity: $$\left ( \sum_{i=1}^n i ...
3
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1answer
687 views

Similarity between subtraction and division

I would like to hear some intuition about difference between subtraction and division. For binary subtraction operator the standard development is introduction of unary operation of taking negative ...
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3answers
872 views

What is the intuition behind Gauss sums?

Let $ \chi $ be a character on the field $ F_p $, and fix some $a \in F_p $. We define a Gauss sum to be: $g_a (\chi) = \sum_{t\in F_p}\chi(t)\zeta^{at}$ where $\zeta$ is a primitive $p^{th}$ root of ...
9
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1answer
3k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
3
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1answer
811 views

Seminorms and norms

Suppose we have the following lemma: Lemma If $E_0 \hookrightarrow E$, and $E_0$ is a closed subspace then $E/E_0$ is a normed space and for $[x] \in E/E_0$ its norm is given by $||[x]|| = ...
3
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2answers
87 views

What is this question asking for?

I have a question that is: Find a degree 3 polynomial with real coefficients whose leading coefficient is 5 that has -2, 1, and 4 as zeros? I do not want the question answered for me what I want it ...
15
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7answers
1k views

What is the intuition for the point-set topology definition of continuity?

Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is defined as continuous if for each open set $U \subset Y$, $f^{-1}(U)$ is open in $X$. This definition makes sense to me when ...