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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Finding/Recognising non-cyclic proper subgroups.

$Q$ is a multiplicative group of order $12$. You are given that two elements of $Q$ are $a$ and $r$ and that $r$ has order $6$ and $a^2=r^3$ You are also given that $a$ has order $4$, $a^2$ has order ...
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P entails Q implies P

I have been looking at the following: P entails Q implies P And developed the proof as follows: ...
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202 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
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On prime(less)ness and composite(less)ness of 1

I was sitting in my room when suddenly my cousin came and asked me, "Why is $1$ neither prime nor composite". Well ofcourse, i was never given an explaination of that in school, it was just a ...
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Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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What are the advantages/disadvantages of integration vs. summation?

If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't. ...
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The relationship between the intercepts and the remainder in the remainder theorem

The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is: $$\begin{array}l If ...
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When is $c_1 \cdot f(g(x+c_2)) = f'(x)g(x)$?

We are allowed to pick and $c_1, c_2$ that helps make this question easier. So when is $$c_1 \cdot f(g(x+c_2)) = f'(x)g(x) \tag{1}$$ Also, separately, I'm wondering: $$c_1 \cdot f(g(x+c_2)) = ...
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138 views

What is the significance of integrating a function?

Now i understand how important these things can be in terms of very small changes or finding area under curves and otherwise. However, when we integrate a function such as y = x we get (x^2)/2, and ...
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65 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
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202 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
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60 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
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67 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
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140 views

Why is the expected number coin tosses to get $HTH$ is $10$?

Can someone please explain why is the expected number of coin tosses to get the sequence of $HTH$ is $10$? What is the intuition and formulas behind this?
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Can't see the intuition behind the validity of this formula: $\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$

I know that $$\vdash_{\mathcal G}\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$$ (I have read and done a syntactic proof of this.) And therefore also $$\models \exists x(\exists yP(x,y) → ...
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When do Entries Remain, after and despite Matrix Multiplication? [Strang P92 2.5.41]

Suppose $E_1, E_2, E_3$ are 4 by 4 identity matrices, except $E_1$ has $a, b, c$ in column 1 and $E_2$ has $d, e$ in column $2$ and $E_3$ has $f$ in column 3 (below the $1$ s). Multiply $L = ...
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Is there a deeper meaning when a number is squared? [closed]

In my opinion, math is about more than just memorizing equations, it's about numbers that are built in a way that represents our understanding of something. So I ask this, what does it mean ...
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115 views

How to understand blowing up a submanifold

I am trying to understand the idea of blowing up a submanifold of a smooth real manifold. The definition I know is replacing the submanifold by its unit tangent bundle (however, in the place I read ...
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28 views

What is a complete intersection?

I was reading and I encountered something that goes: We have degree $d$ polynomials in $s$ variables $F_1, ..., F_n$ with coefficients in integers. Let $X$ be the complete intersection defined by the ...
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44 views

Scale-invariance of $\int_0^\infty \frac{f(x)}{x} \ dx$

Let $f$ be some non-negative, measurable function on $[0,\infty)$. The quantity $\int_0^\infty \frac{f(x)}{x} \ dx$ is scale-invariant in the sense that, if one puts $f_c(x) := f(cx)$ for $c > 0$, ...
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Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a picture, and no proofs please. $1.$ Intuitively, in ...
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35 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
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What is an intuitive extension of extreme-values and critical points in one variable to multiple variables?

While it is simple to grasp limits in multiple variables, since the formal definition extends in the obvious way, I am having a harder time grasping the same concept with critical points and extreme ...
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90 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
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Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
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49 views

What is $R$-algebra and do I need to understand $R$-modules for it?

I was given the following definition of $R$-algebra: Let $R$ be a commutative ring. An $R$-algebra is a ring $A$ (with $1$) together with a ring homomorphism $f : R \to A$ such that ...
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Line Integrals and Surface Integrals

Can someone please explain what surface integrals and line integrals are measuring? Is a line integral the arc length along a surface, and a surface integral is the surface area? Also, why is a line ...
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73 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
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Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed ...
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Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
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If $f(2\alpha-\theta) = f(\theta)$, then $\theta=\alpha$ is a line of symmetry of $r=f(\theta)$. How do you derive $f(2\alpha-\theta) = f(\theta)$?

For Polar Coordinates I know that for x-axis symmetry $f(-\theta)=f(\theta)$, for y-axis symmetry $f(\theta)=f(\pi-\theta)$, and for symmetry about the origin $f(\theta)=f(\theta+\pi)$. The big ...
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Why $\dfrac{d}{dt} \dfrac{dy}{dx} = \dfrac{d}{dx} [ \dfrac{dy}{dx} ] \quad \dfrac{dx}{dt} $ ? [Stewart P206 3.4.95, BDP P165 3.3.34]

If $y=f(x)$, and $x = u(t)$ is a new independent variable, where $f$ and $u$ are twice differentiable functions, what's $\dfrac{d^{2}y}{dt^{2}} $? By the chain rule, $\dfrac{dy}{dt} = \dfrac{dy}{dx} ...
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If $z = f(x, y)$, then why are $\partial_x z$ and $\partial_y z$ functions of x and y also? [Stewart P905]

This is Figure 5 from P905 which appears to show this, but Stewart doesn't write this explicitly or explain. I'm interested in an informal, intuitive explanation please. I'm not interested in a ...
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What is a smooth curve in $\mathbb{R}^2$ intuitively?

While studying for my exam, I've run into some problems understanding what a smooth curve in $\mathbb{R}^2$ is. I first thought that, intuitively, I could think of a piece of string on a piece of ...
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120 views

Why is $\cos(x)$ the derivative of $\sin(x)$?

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Is there a simple proof of this, preferably using pictures?
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rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
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1answer
41 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
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A graph of all of mathematics

In mathematics, one often makes (proves) statements on the basis of: Previously proven statements Axioms I like to think of these dependencies as a directed graph, with edges from the accepted ...
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1answer
22 views

Find the area bounded by the hypercycloid

Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$ If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable ...
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61 views

understanding Green's theorem Intuition

The idea of it is to find the area of a region, yet I keep seeing vector fields popping up all over the place. Take this example from my text book: Find the region enclosed by the two graphs: $y = ...
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What, fundamentally, is the reason for the shape of a sin curve?

Say we have a metal bar in space aligned horizontally and we start rotating it counter-clockwise about its left end. Then, the sin of the angle from between the horizontal and the bar is the y ...
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170 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
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116 views

What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
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91 views

Gaining Linear Algebra Intuition — Subspaces

So I aced linear algebra over the fall semester, though I'm deeply troubled in that I struggle to really describe what I did. I cannot say with confidence what it all meant, nor do I have any sort of ...
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1answer
61 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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29 views

Finding Cos of an angle between matrices

I have two $2\times 2$ matrices and it's asking me to find the Cos of the angle between them. Firstly, how do yall visulize matrices and the angles between them? I think that's my first problem. ...
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40 views

Why must an inner function of a compound function be linear in order to integrate it using the power rule?

This is from my previous thread: $\int(1+x^2)^4\mathrm dx$ $\ne$ $\frac{(1+x^2)^{5}}{5(2x)}+C$? because differentiating back gives ...
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Intuition - Linear Congruence Theorem

Let a and b be integers (not both 0) with greatest common divisor d. Then an integer $c = ax + by$ for some $x, y \in Z$ $\iff d|c$. In particular, d is the least positive integer of the ...
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Fermat's Little Theorem fails for composite instead of prime numbers.

I know Fermat's Little Theorem = Fermat-Euler's Totient Theorem when $n$ is prime. Elementary Number Theory, Jones, p83 writes if we simply replace p with a composite integer n, then the ...
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Ground plan of Forward direction - Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Prove by contradiction. Thence suppose NOT $p\equiv 1 \; (mod 4)$. Thence 3 possibilities remain: $4|p, 4|(p - 2), 4|(p - 3)$. But $p > 2$ is prime, thence $4 \not | p$. (1) How can you ...