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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What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
7
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3answers
92 views

How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$

A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure). The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = ...
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1answer
21 views

Understanding a complex set description

I am trying to understand the following math expression, describing the possible sets of a 15 Puzzle: The expressions for tile and ...
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1answer
42 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
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3answers
50 views

Is $(-1)^{1/8} + (-1)^{7/8}$ ever a value whose real component is $0$?

Is $$(-1)^{1/8} + (-1)^{7/8}$$ ever a value whose real component is $0$? Is this ever true in modular arithmetic, hypercomplexes, and/or both?
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30 views

explanation for a combinatorial identity involving the binomial coefficient

I am looking for an intuitive explanation for the identity: $$\binom{n}{h}\binom{n-h}{k} = \binom{n}{k}\binom{n-k}{h}$$ Thanks!
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106 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
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2answers
39 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
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0answers
23 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [duplicate]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
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40 views

Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [closed]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection $h:A\rightarrow ...
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2answers
37 views

Intuitive significance of harmonicity

I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it ...
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0answers
295 views

How to present calculus? Asking for some excellent intuitive referance.

$\text{Dear}$ mathematicians, amateurs, learners, students et al; I learned calculus when I was 13 years old, I was at the time able to evaluate some easy derivatives, integrals, some tricky limits ...
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2answers
44 views

Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
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3answers
76 views

Intuition behind Descartes' Rule of Signs

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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0answers
27 views

Descartes' Rule of Signs

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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3answers
24 views

Intuition - Divisibility results - If c divides some integers, then c divides any linear combination of them.

Not querying about proofs here. I don't want to memorize, thence are there intuitions or illustrations for them? As a student, how else can I remember these results? Origin - Elementary Number ...
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1answer
30 views

Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
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1answer
36 views

Intuition - If a = qb + r, then gcd(a,b) = gcd(b, r)

Origin - Elementary Number Theory, Jones, p5, Lemma 1.5 Are there any illustrations? I tried en.wikipedia.org/wiki/Euclidean_algorithm and the first picture to the right - ...
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0answers
13 views

Intuition about the where the beta distribution has its maximum

I've stumped myself trying to develop an intuition about why the beta distribution should have its maximum where it does. I can differentiate easily enough, and I can manage a simple argument based on ...
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1answer
42 views

Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
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1answer
68 views

Existence of a real uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
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62 views

what problem do quadratics solve?

Areas of Math were developed to deal with problems which couldn't be tackled with preexisting methods. So, for example, 2D geometry was used to solve measuring problems with land, basic algebra for ...
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1answer
39 views

What values or form of values can we get for these multiplications modulo a prime?

If we have four complex values, all of the form $a + b i$, for integers $a$ and $b$, we can label them $c$, $d$, $e$ and $f$. Now if we want to find $g$ and $h$ such that $$g \equiv ce \equiv df ...
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2answers
99 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
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1answer
37 views

Mentally visualizing functions of complex numbers

I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head. When I think about real functions, I tend to mentally visualize them as ...
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4answers
107 views

Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
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1answer
22 views

Seeing complex roots on the graph of a polynomial

When I sketch the graph for a general second degree polynomial $y = ax^2 + bx + c$ it is easy to "see" its roots by looking at the points where $y=0$. This is true also for any $n$-degree polynomial. ...
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1answer
34 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
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8answers
196 views

Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

I believe my trouble is that the identity, $e^{i \pi} = -1$, comes down to the definition of the exponentiation of $i$, which seems rather obscure to me. This is my current understanding of ...
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1answer
25 views

How can we prove that we can get any $f$ modulo a prime $p$, that satisfies these equations?

We can suppose that we have four naturals not equal to zero: $a, b, c, d$. Further, we're working modulo a prime $p$. Now if we find $a, b, c, d$ that satisfy: $$f \equiv a \cdot c \equiv b \cdot d ...
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2answers
73 views

Is there an intuition for why power sets come in powers of $2?$

My title is a bit sloppy. First let me say that I perfectly understand the proof that $|P(S)|=2^{|S|}$, I am not asking for an easy to understand proof. My question is more whether there is an ...
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2answers
186 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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2answers
34 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
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70 views

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?
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4answers
65 views

Intuitive idea of axiom of choice

I'm currently reading a book on set theory and it gives the following formulation of the axiom of choice: Let $X$ be a non-empty set. Then there is a function $g: ...
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5answers
104 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
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1answer
66 views

“Poissonization” and intuition

In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization". "Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with ...
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1answer
46 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
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1answer
47 views

What does it mean for a function to “quickly” approach $0$?

We can talk about how "quickly" an infinite series approaches $0$ by talking about an asymptotic bound on its terms - a series that is $O(1/x)$ converges more slowly than one that is $O(1/x^2)$, etc. ...
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1answer
34 views

Degree of a divisor for algebraically closed fields vs not closed ones

Suppose we have an algebraically closed field $F$ and we consider the projective space $\mathbb{P}^1$ over $F$. If we consider some divisor $D = n_P P + n_Q Q +n_s S$, then we say the degree of $D$ ...
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3answers
94 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
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1answer
44 views

Seeking intuitive explanation of Clifford Algebra

Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical ...
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0answers
31 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
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1answer
41 views

Which math discipline should i learn to become familiar with rewriting equations?

In my self study of calculus, I've found that there are examples in the books i read where the author rewrites an equation or expression either as part of a logical step in a proof, or to simplify it ...
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1answer
75 views

Axiom of Choice and Zorn's Lemma Equivalence: some intuition

$$ \text{Axiom of Choice $\Rightarrow$ Zorn's Lemma } $$ $$\text{Axiom of Choice $\Leftarrow$ Zorn's Lemma } $$ I feel mathmatically immature to go through these proofs now. My quesiton therefore is: ...
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1answer
93 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
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0answers
41 views

what does the sine function tell you about an input

Intuitively, I'm trying to understand the significance of the input in a sine function. I'm currently, trying to develop intuition behind sinusoids and what the input tells you about the output and ...
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3answers
56 views

What is an irrotational vector field intuitively?

I understand that, by definition, a vector field is irrotational if the rotation is zero, but what does this intuitively mean? I have an idea of what it could physically be, which I've concluded by ...
0
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1answer
32 views

Is it possible to always get the optimal formula regardless of the derivation method?

Today I've tried to solve a geometric problem (collision point between two circles in a specific situation). I found a working solution but I'm not sure if it was optimal (maybe my solution took more ...
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0answers
65 views

How to explain the significance of $\pi$ to a child? [closed]

In honor of $\pi$ Day, I thought I would pose this question. How would you explain the significance of $\pi$ to a child of, say, 9 years of age? While that's certainly an age that is old enough to ...