# Tagged Questions

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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### How to see that $\text{gcd}(a,b) = \text{gcd}(a-b,b)$?

I'm trying to understand why $\text{gcd}(a,b) = \text{gcd}(a-b,b)$. What is clear to me is that the $\text{gcd}$ divides $a,b$ and also $a-b$ (let's assume $a\ge b$). But then it seems to me we ...
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### What does it mean for a set to be countably infinite?

Why distinguish between countable and uncountable? What advantages does this property have? I haven't studied much set theory but I am writing about the set of algebraic vs transcendental numbers and ...
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### Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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The reason I ask is because according to this source: the $\fbox{$\color{blue}{\mathrm{PDF}}$}$ for the sum of two Exponential Density Functions is $$\rho(x_1,x_2)\mathrm{d}x_1 \mathrm{d}... 2answers 131 views ### Which general physical transformation to the number space does exponentiation represent? Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while 2 + 4 is ... 7answers 3k views ### Why does the fundamental theorem of calculus work? I've known for some time that one of the fundamental theorems of calculus states:$$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$Despite using this formula, I've yet to see a proof or even a ... 2answers 80 views ### What is the difference between a Poisson and an Exponential distribution? For a Poisson distribution:$$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$where \mu is the mean number of occurrences. For an Exponential distribution:$$f(x;\lambda) = \begin{cases} \...
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Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? What is the true nature of this paradox ? I don't really understand this ?
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### How to see symbol manipulation from an intuitive perspective in math?

I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon ...
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### Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...
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### Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. ...
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### Vector Integration - Intuition

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a ...
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### What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?

I'm trying to understand the origin of a certain formula used in the solution to the following question: This question relates to the position probability density for a classical particle ...