Articles of soft question

Why are multiplication and addition associative?

Without workin in a rigorous formal system, how can one intuitively establish that multiplication and addition are associative operations on the real line (including negative numbers)?

How do you define discrete math to a 5 years old kid in a nontechnical way?

How do you define discrete math to a 5 years old kid in a nontechnical way? It seems to me even the formal definition of discrete math is vague for me.

Representation theory of associative algebras applied to Lie Algebras

I’m given to understand that a lot of result on representation theory of Lie Algebras can be obtained by applying known result of representation theory of associative algebras to the enveloping algebras of lie algebras. I understand why it works and that’s not the problem. I’d just like to have a reference of some book […]

Does one necessarily need an MS in Math before taking a PhD in Math?

I finished bachelor’s in mathematical finance and am nearly finished with master’s in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but in pure mathematics particularly stochastic analysis. Is getting into a PhD in pure mathematics possible without a master’s in pure mathematics? If […]

Reference on Infinite Dimensional Manifold

I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold. Now I have two questions or requests: (1) When was infinite dimensional manifold introduced? I guess this may be related to Functional Analysis. But I want more details about its history. (2) I am still […]

Definition of the mathematical proof

How do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or definitions?

Why do we use “if” in the definitions instead of “if and only if”?

This question already has an answer here: Are “if” and “iff” interchangeable in definitions? 14 answers Alternative ways to say “if and only if”? 6 answers Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate] 2 answers

Sine and Cosine Derivatives

The derivatives of: $$\frac{d}{dx}\sin(x)=\cos(x)$$ $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I currently trying to teach a friend of mine calculus, because he does not know it yet. He keeps forgetting how to take the derivatives of $\sin(x)$ and $\cos(x)$. Is there a simple way, or trick to remember the derivatives of $\sin(x)$ and $\cos(x)$?

Diadics and tensors. The motivation for diadics. Nonionic form. Reddy's “Continuum Mechanics.”

I’m taking a course in continuum mechanics. Our book is Continuum Mechanics by Reddy, a Cambridge edition. In the second chapter he introduces tensors and defines them to be polyadics. He is specifically concerned with dyadic which are tensors of rank (2,0). In his presentation of dyadic, the notation is introduced, then some properties are […]

Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?

For recreational purpose, i haven’t seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the Monte Hall problem? Please give a few examples.