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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On the origins of the (Weierstrass) Tangent half-angle substitution

The Weierstrass substitution is great for transforming complex trig integrals into simpler rational functions. Wikipedia suggests that it wasn't invented by Weierstrass, since Euler was already ...
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Can we prove that this formula correctly describes the $n$th least significant digit?

So if we take a look at the binary representation of numbers, we can see that the digits follow a pattern: $$0 = 0000$$ $$1 = 0001$$ $$2 = 0010$$ $$3 = 0011$$ $$4 = 0100$$ $$\dots$$ i.e. the least ...
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How to Garner Mathematical Intuition

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain ...
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55 views

Geometric $k$-blades

What are the purposes of $k$-blades? Why is it important to have a oriented area, or an oriented volume? I'm referring to $k$-blades in such a way that $$\hat{v_1} \wedge \cdots \wedge \hat{v_k} $$ ...
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860 views

Calculating limits using the $\epsilon$-$\delta$ definition.

Suppose you have a function $f(x)=( x^2-4)/(x-2)$. How then do we find the limit as $x\to2$ in accordance with the epsilon delta definition? I mean suppose we don't know how to calculate limit and we ...
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510 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
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58 views

Gauge fractions with exponents - No Calculator

How does one (without the use of Calculator) determine that $5/6$ is less than $(35/36)^6$? How is this done mentally?
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245 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
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What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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1answer
2k views

What is the difference between a discrete function and a continuous function

Intuitively it seems that both concepts should be disjoint because if a function is discrete then it has some holes on it and if a function is continuous then it doesn't have holes. But now I'm not ...
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1answer
166 views

What does it mean for a coalgebra to be cogenerated by a subspace?

The usual definition of an algebra being generated by a subspace is as follows: Let $A$ be an algebra, $X \subset A$ a subspace, $\mathrm{Alg}(X)$ the free algebra generated by $X$. Then $A$ is ...
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1answer
68 views

Type $\sim$ Minimal Polynomial & Orbit

In Model Theory by Wilfrid Hodges, he gives an intuition of what a type is in the following way: "One can think of types as a common generalisation of two well-known mathematical notions: the ...
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Intuition on Wald's equation without using the optional stopping theorem.

The Wald's equation even at its simplest form as stated below simplifies many problems of calculating expectation. Wald's Equation: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of real-valued, ...
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arranging letters without repetition

In how many ways can one arrange all of the letters in the word INFORMATION so that no pair of consecutive letters appears more than once. Eg ININformota is not acceptable as IN appears twice. So ...
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2k views

not divisible by 2,3 or 5 but divisible by 7

The question is to determine the number of positive integers up to $2000$ that are not divisible by $2,3 or 5$but are divisible by $7$. The answer is supposed to be 76 but not sure how it was derived ...
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211 views

Recovering the spatial Fourier transform from the space-time Fourier transform

This CW question is aimed at developing some intuition (grokking) about a certain formula of Fourier analysis. Any kind of explanation (physical, geometrical, analytical ...) is welcome. If we ...
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Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
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Convolution intuition: clarifying Terence Tao's “blurring”/“fuzz” interpretation

On this math.MO post, "What is convolution intuitively?", Terence Tao's answer (in the case where one function is a bump function) involves "blurring" and "fuzz." Could someone clarify his ...
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Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics, he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I ...
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Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
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573 views

Can anyone explain intuitively why increasing the circumference by 1 meter always increases the radius by 15.9cm?

I found the mathematical proof, and it is obviously correct. But how can the increase in radius be constant regardless of the starting circumference? With a very small circle, the increase should be ...
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189 views

Intuition for Multiple Summation becoming One Summation - Nothing too formal/rigorous please

Source. I grok addition is associative and commutative, and a term can be moved into other summations iff these other summations aren't summing this term. Hence I grok $$\sum_{i,j} g_{ij} \sum_r ...
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What really are determinants? [closed]

As a layman, it is not clear to me what does a determinant stand for and how it could be computed. So, what really is a determinant, meaning, why should we care about them and how is the best way to ...
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Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = ...
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Finding a planar graph satisfying these properties

I need to construct a 3-regular connected planar graph with a planar embedding where each face has degree 4 and 6. In addition, each vertex must be incident with exactly one face of degree 4. seems ...
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782 views

What is contour integration

So I recently took a course that involves contour integration. I understood how to perform the integrals out, but I never got a hold of what the physical meaning was. I understand the introductory ...
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Confused with Eigenvalues and Eigenvectors and Vector transformations

Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here: I am actually performing automatic operations on finding them, but ...
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Stirling numbers practice question 1

This is a practice question and not HW Q. Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{v,w,x,y,z\}$. Determine the number of functions $f:A \longrightarrow B$ where a) $f(A)=\{v,x\}$ Ans. $2!*S(7,2)$ I ...
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55 views

derangements basic practice question 2

This is not HW just a practice question from the text. Q.. (NOTE: I find that both part a and b say the same thing but the answers are different) Ten women attend a business luncheon. Each women ...
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1answer
166 views

Derangements basic practice question

practice questions not Homework I have problems with this questions that I have answers for but cant understand how the answer was derived. Q.1. In how many ways can the integer $1,2,3,...10$ be ...
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138 views

Intuition for orthogonal vectors in $\Bbb R^n$

Two vectors in $\Bbb R^n$ are orthogonal iff their dot product is $0$. I'm aware that the dot product can be defined in other spaces, but to keep things simple let's restrict ourselves to $\Bbb R^n$. ...
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what does $|x-2| < 1$ mean?

I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ ...
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$ \{x : P(x)\} $ vs. $ \{P(x) : x\} $ -— When are these set-builder notations the same and different?

I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. ...
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Operations on negative integers

I was trying to teach my younger sister some math, and it drifted on to integers, and operations on negative integers. So questions like: a) $-3+2 = ?$ b) $2- (-3)= ?$ c)$-3 -2 = ?$ had to be ...
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Associativity of $\iff$

In this answer, user18921 wrote that the $\iff$ operation is associative, in the sense that $(A\iff B)\iff C$ $A\iff (B\iff C)$ are equivalent statements. One can brute-force a proof fairly ...
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Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$

The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar ...
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mathematical maturity

So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel ...
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motivation of additive inverse of a Dedekind cut set

My understanding behind motivation of additive inverse of a cut set is as follows : For example, for the rational number 2 the inverse is -2. Now 2 is represented by the set of rational numbers less ...
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Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
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Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D ...
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Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
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The orthogonal projection onto a plane - explanation

Could somebody explain, why orthogonal projection onto a plane with equation $x_1+x_2+x_3=0$ is given by $$y=(x_1,x_2,x_3)-\bigg( \frac{x_1+x_2+x_3}{3}\bigg)(1,1,1)$$ I don't understand, why we sum ...
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74 views

Understanding Continuity of Functions

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$ I also know that by definition, a function $f(x)$ ...
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225 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
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2answers
167 views

Integral and series convergence intuition

I have this problem I ran into during my studies to the upcoming exam: I don't feel I have the intuition of whether a series or an integral converges or not. What are the things I should look for when ...
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Yet Another Monty Hall Question - Please advise if alternative scenario proves the same principle

Okay, I'm very embarrassed that there are already 71 questions (based on search of "monty hall") and I'm going to post another one. I read the first 5 before succumbing to choice-overload. I'll try to ...
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169 views

Real number construction : additive inverse of a cutset

For a given cut set $\alpha$, what is the intuition behind considering the set of all such $p$'s such that some number less than $-p$ does not belong $\alpha$ as the inverse of $\alpha$ ? i.e. ...
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Intuitive interpretation of the Fourier transform of the derivatives of a function

Let $f$ be a smooth function, $f(t)\in C^\infty(\mathbb R)$, and $F$ be its Fourier transform $$F(\omega):=\mathcal F f\,(\omega)\,=\,\int_{-\infty}^\infty \mathrm e^{-\mathrm i\omega ...
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Why should $|2^\mathbb{N}|>|\mathbb{N}^2|$ be true?

I've been thinking a bit about infinite things lately, and this question I had wondered about came back to me. One of the classic expository demonstrations of Cantor's work is the two equally ...
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Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...