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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Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
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47 views

Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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19 views

Understanding Inequalities.

If $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a\lt\frac{2(a-1)}{2+\ln a})$ equals $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1})$. Does that mean that $-1 + \sqrt{2a-1}$ = ...
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24 views

Show that $ \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $ \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$

Show that $\displaystyle \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $\displaystyle \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$ The closest I can get is $$ ...
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42 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
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39 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
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55 views

Using De-moivres to solve the following problem:

Part (i) I can solve and understand that the solutions are $Z=e^\frac{2ki\pi}{5}$ for $k = 0,1,2,3,4$ Its the part (ii) I cannot understand. Could someone kindly give me a ...
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1answer
50 views

Using De Moivre's theorem with relation to the argument of a complex number

Given that $Z^4 = 64(\cos\pi+ i\sin\pi)= 64(-1+0i) = -64$ I understand that the argument [$arg(Z^4)$] is $\pi$, now if instead given the form $Z^4 =64(-1+0i)$ and I desired to find the argument ...
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Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
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64 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
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The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
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21 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
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180 views

Why is a raised to the power of Zero is 1? [duplicate]

Why is $a^0=1$ $\forall a \in Z, a\neq0$. I understand $2^4=2\cdot2\cdot2\cdot2$ How can I express $a^0$. I am serious about the practical proof of this
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43 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
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Why is it called a primitive root?

I am looking for a paper or reference that explains why primitive roots are called primitive roots. I know what they are but was wondering if there was a reason?
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175 views

What if we removed all the irrational numbers from the real number line? [closed]

(1) Imagine drawing the real number line and tippexing out all the irrational numbers. What would the resulting shape look like- would there even be a line? (2) And what about if we ...
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Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
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1answer
39 views

Gambling interpretation of conditional probability

In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : $$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in \mathscr{G} $$ where at ...
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36 views

Single variable justification for the multivariate chain rule.

I $\def\d{\mathrm d}\def\p{\partial}$am going to ask everyone to switch their paradigms to that of the real line. I am looking for a "lowbrow" explanation of the following phenomena. I am talking ...
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Why isn't the Cantor Set contradictory?

So you start with a 1-dimensional stick, remove the middle third of it, leaving 2 pieces. From each of these 2 pieces, remove the middle third. Etc. Whatever is left at the end of infinitely many ...
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3answers
230 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
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Getting an intuitive feel for induced representations

I'm reading about induced representations for research. Particularly, I'm trying to get a firm grasp on the finite group case before venturing on to the locally compact case. I've been looking at ...
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67 views

Differentiable curves that are not smooth

We call a curve admitting a parameterization $t\to z(t)$, $t\in[0,1]$ differentiable if the vector function $z$ is differentiable. We call the curve smooth if it is differentiable and its derivative ...
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Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
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1answer
27 views

Understanding arguments to functions in $\mathbb R^n$

Example of two theorems I have problems with: Mean value theorem: $U\subseteq\mathbb R^n$ open, $f:U\to\mathbb R^m$continuously differentiable, $x\in U$, $\xi\in\mathbb R^n$ such that $x+t\xi\in U$ ...
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$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$ How can I explain this to a school student who doesn't know what a limit is?
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110 views

Why do we denote $S^1$ for the the unit circle and $S^2$ for unit sphere?

Maybe a quite easy question. Why is $S^1$ the unit circle and $S^2$ is the unit sphere? Also why is $S^1\times S^1$ a torus? It does not seem that they have anything in common, do they?
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68 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
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91 views

Can a statement in FOL be equivalent to two separate independent statements?

This may seem like a dumb question, and it certainly seems dumb to me asking it, but I'm running into a contradiction. I'm looking at the problem of finding a statement $\phi$ such that $\psi$ and ...
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29 views

Meaning of “Identify a set with another set” in group theory

There is a exercise problem that asks "Identify a set with another set ". I don't understand what I should do. Do I need to establish a bijection between them? Thanks EDIT-I: Actual question: G is a ...
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1answer
91 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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The physical meaning of ${n \choose k} = {n \choose n-k}$.

They say that $${n \choose k}={n \choose n-k}.$$ Can someone explain its physical meaning? Among many problems that use this proof, here is an example: The english alphabet has $26$ letters ...
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What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
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1answer
60 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...
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101 views

Understand the $\operatorname{Hom}$ Functor.

Via Wikipedia I see that $\operatorname{Hom}_C(A,-): C \rightarrow \textbf{Set}$ a covariant functor which maps each object $X$ in $C$ to the set of morphisms $\operatorname{Hom}_C(A,X)$. I am trying ...
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1answer
48 views

Method of Variation of Parameters - Assigning zero works?

I have yet to find a decent answer on this, and so I don't think this question is inappropriate. Also, this question is mainly meant for people that are very familiar with this method. In the method ...
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1answer
97 views

“Geometric” proof of Rouche's theorem on the number of zeros?

I understand the analytic proof of Rouche's theorem as presented in Stein and Shakarchi's complex analysis - $|f(z)| > |g(z)|$ on the boundary circle C ensures that the argument principle can be ...
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393 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
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How to imagine “tensoring with Serre's twisted sheaf”

What has an algebraic geometer in mind when (s)he sees $\otimes \mathcal{O}(1)$? I think it has something to do with an intersection of a hypersurface...? Thanks, Adrian
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Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
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131 views

What application is there for a non-Hausdorff topological space?

I'm learning basic topology and as I understand it, a good way to intuit what an open set is, is that it determines which elements are near each other. However, in a non-Hausdorff space, it would be ...
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What is the intuition behind the Poisson distribution's function?

I'm trying to intuitively understand the Poisson distribution's probability mass function. When $X \sim \mathrm{Pois}(\lambda)$, then $P(X=k)=\frac{\lambda^k e^{-k}}{k!}$, but I don't see the ...
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70 views

convert continuous random variable to a discrete one for the given exponential distribution

I understand that the following question requires converting continuous r.v. to discrete r.v. But How can we get a PMF from the CDF of continuous distribution? It involves dividing continuous values ...
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80 views

Inuition regarding Lowenheim-Skolem applied to models of set theory

According to wikipedia, ...the Löwenheim–Skolem Theorem states that for every signature $σ$, every infinite $σ$-structure $M$ and every infinite cardinal number $κ ≥ |σ|$, there is a ...
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In regards to lagrange multipliers, Confusion about derivation.

In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers. "Since the gradient vector for a given ...
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1answer
25 views

Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$. Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$. ...
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2answers
58 views

Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function y=1/x, x>0 (say x and y both measure meters) about the x axes by 2pi you get a shape which has infinite surface area and finite ...
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“Algebraic indistinguishability” [duplicate]

When people talk about Galois theory they often say that the basic idea behind it is that certain numbers are "algebraicaly indistinguishable". I never really understood what this means in a way that ...
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2answers
104 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
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58 views

How to illustrate that

there is $a, n \in \mathbb Z^{+}$ and prime number $p$, with relationship: $$p|a^{n}$$ It's straight forward that $p|a$, but I can't find a proper illustration of it.