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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Proving integration techniques with intuition.

I've recently competed my A levels and now that I'm in the university I finally found the time to understand calculus on a intuitive level. So I've been reading up on books such as "Calculus with ...
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Develop intuition in trigonometry problems?

I'm going to write the JEE-Advanced exams, that has problems on trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple ...
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32 views

Why do repeated linear factors have to be dealt with in this way?

When dealing with partial fractions, and your denominator has a repeated linear factor, the way to solve is this: $\frac{2x+3}{(x-2)^2}=\frac{A}{(x-2)^2}+\frac{B}{(x-2)}$ $2x+3=A+B(x-2)$ and so on. ...
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219 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
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163 views

Sources for mathematics outside the mathematics world

In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate ...
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Let $z_1$, $z_2$ and $z_3$ be complex vertices of an equilateral triangle. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.

Prompt: Let $z_1$, $z_2$ and $z_3$ represent vertices of an equilateral triangle in the complex plane. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$. Question: I hope the following ...
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63 views

Why is this intuitive method valid?

Problem. There are $2$ white and $3$ black balls in the urn. A person randomly picked $2$ balls and put $1$ white ball. What is the probability of the event that the next randomly-picked ball would be ...
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100 views

How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook. The skillset I am least satisfied with is the application of the Category Theory to Algebraic ...
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54 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
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53 views

How to visualize(inside ones brain) the Four-dimensional_space

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans. Does the professional mathematicians can do this ? If so what are the things to ...
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315 views

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
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16 views

Intuition for using vectors in sale related problems

I am reading Linear Algebra from David Lay's book. He gives one example to showcase use of linear combination of vectors : I understand the solution, but I am completely clueless about how to ...
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5answers
145 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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3answers
247 views

Is there a notion in mathematics saying that, in a sense, all finite dimensions are actually infinite dimensional?

So then every ordered pair or triplet and so on would be actually represented by an infinite sequence of numbers, and what we think of as 3 dimensions would mean that the point has an infinite number ...
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19 views

Steady state state distributions.

I am looking for a less "proofy" explanation of how a finite, irreducible, aperiodic Markov chain has a unique steady state $\pi$. No need define terms or include proofs of Bezout's lemma or number ...
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21 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
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51 views

Non- intuitive connected space.

There exist knowing examples of connected spaces such that its picture is a counter intuitive for us?. I mean a topology on a set who makes see the space as connected (no connected) but it is no ...
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80 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
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Linear programming and their geometry: cones

This is a snippet of a solution in my notes. Why can we conclude that from figure 1? Why can we say that the arrows are between and do not go all the way around the axis?
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3answers
240 views

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...
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1answer
54 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
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20 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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134 views

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
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2answers
140 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
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136 views

Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...
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101 views

Self-studying Russell's Paradox

I'm self-studying and having trouble wrapping my head around Russell's paradox, even after looking here. I'd really appreciate a more intuitive explanation of the concept before I move on to ...
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57 views

Intuitive understanding of path integral formula

I have learned a general formula for a path/line integral $$ \int_a^b f\left(\mathbf{r}(t)\right) \|\mathbf{r}'(t)\|\ dt \tag{1} $$ and I'm trying to better understand it. Specifically, I'm ...
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48 views

Proof and interpretation of $\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]$

First, I understand that $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$, but how to prove that $$\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]?$$ Second, for ...
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30 views

Why is $x$ restricted this way? (limits of functions)

Here is a corollary from Ross' Elementary Analysis: Why is $x$ restricted this way?
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365 views

What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
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73 views

Why is conic isomorphic to $\mathbb{P(C^2)}$?

Given a quadratic form $C(x)=x_1 ^2+x_2 ^2 + x_3^2$ in $\mathbb{C}[x_1,x_2,x_3],$ we have a conic $$C=\{C(x)=0\} = \{[x_1:x_2:x_3]: x_1 ^2+x_2 ^2 + x_3^2 = 0\}$$ in $\mathbb{P(C^3)}$, given in ...
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Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: 20.3 Definition (a) For $a\in\mathbb R$ and a function $f$ we write $\lim_{x\to a} f(x)=L$ provided ...
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144 views

Two plus two equals four when earth has one moon?

As is well known, we have the least intuitive of basic operations, the 'implication' or '=>'. Consider 'A => B'. Most beginners get stumped on the vacuous truth, that implication could be true even ...
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95 views

Quick Question on a Proof of Artin-Wedderburn Theorem

Question [Edited]: [See below.] Are the isomorphisms in $(1)$ and $(2)$ (additive) group homomorphisms? If I'm right, $\text{End}_R(M)$ is a ring, but ...
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Intuition behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
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27 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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50 views

How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $?

So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its ...
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65 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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Developing intuition for the modulus of continuity $\omega$?

I am currently in a course which is going through rigorous definitions of continuity, convergence, integrals, etc. I am trying to develop an intuition in understanding the modulus of continuity, ...
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If f(x)dx is a rectangle with height f(x) and width dx, what is f(z)dz in complex analysis

I am trying to intuitively understand the multiplication $f(z)dz$ in complex analysis. For instance, $f(x)dx$, we are all aware, is a rectangle with height $f(x)$ and width $dx$ so its multiplication ...
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1answer
48 views

Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.
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F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?
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65 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
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34 views

Extensions Fields and Zeros of Polynomials

I've just digested the proof for Kronecker's theorem that states every polynomial $f(x)$ over a field $F$ has a zero in an extension field of $F$, mainly, the factor ring of polynomials ...
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69 views

Building intuition in group theory

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense. Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is ...
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152 views

Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
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30 views

Intuitive understanding of relationship between unit vectors and position vector

On Wolfram Mathworld they give a unit vector in the $\textbf{x}_n $ direction as: $$ \hat{\textbf{x}_n} \equiv \dfrac{\frac{\partial \textbf{r}}{\partial x_n}}{\lvert \frac{\partial ...
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244 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
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57 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
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Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...