As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. Why not define the gamma function so that $$\Gamma(n) = n!$$ instead? I realize either definition is equally good, but if someone were going to ask me […]

I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, I’m not the only one to do something, other kids will sing and play the piano and such). […]

A quote from Enderton: One might well question whether there is any meaningful sense in which one can say that the continuum hypothesis is either true or false for the “real” sets. Among those set-theorists nowadays who feel that there is a meaningful sense, the majority seems to feel that the continuum hypothesis is false. […]

This is a soft question, but I’ve tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly. It’s easiest to explain with an example, so I’ll give one (it’s fun!) and then pose my questions at the […]

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared for the math I will face in post secondary. How can I get the most out of a […]

I am currently trying to understand why the axiom (schema) of replacement is true in the intuitive hierarchy of sets. The axiom states that if $F$ is a unary operation and if $x$ is a set, then the set $\{F(y):y\in x\}$ exists. I have already checked Shoenfield’s article “Axioms of set theory“. There he describes […]

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I’ve been looking through some books on the topic, and most seem to ‘only’ cover groups, rings and fields. Why is this the case? It seems to me you’d want […]

I’m majoring in mathematics and currently enrolled in Linear Algebra. It’s very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied Mathematics — but I like doing math for the sake of math, not so […]

I see people like Terry Tao and others take extensive notes. But is this really necessary? When I do this I feel like I am rewriting a textbook.

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain […]

Intereting Posts

Prove that $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \leq \frac{3}{2}$
$L^1$ convergence gives a pointwise convergent subsequence
Rearrangements of absolutely convergent series
Proof that the set of irrational numbers is dense in reals
Solving $\lim\limits_{(x,y)\to(0,0)}\;\frac{x^5 + \,y^5}{x^3+\,y^3}$
Good introductory book on Calculus on Manifolds
Prove $\int_{0}^{\infty}{\ln x\ln\left(x\over 1+x\right)\over (1+x)^2}dx$ and $\int_{0}^{\infty}{\ln x\ln\left(x\over 1+x\right)\over 1+x}dx$
A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$
Is $ds$ a differential form?
Matrix Differentiation proof
Prove that if the square of a number $m$ is a multiple of 3, then the number $m$ is also a multiple of 3.
How to calculate: $\sum_{n=1}^{\infty} n a^n$
Quotient ring of Gaussian integers $\mathbb{Z}/(a+bi)$ when $a$ and $b$ are NOT coprime
Proving $\frac{x}{x^2+1}\leq \arctan(x)$ for $x\in .$
Hyperbolic critters studying Euclidean geometry