The theorem that $\binom {n} {k} = \frac {n!} {k! (n-k)!}$ already assumes $0!$ is defined to be $1$. Otherwise this would be restricted to $0 <k < n$. A reason that we do define $0!$ to be $1$ is so that …

Sep 6, 2011 · Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used …

Mar 13, 2011 · So what IS the Holy Bible / The Great Standardization Document of All Definitions for Mathematics? Because people are often fighting over different definitions of mathematical entities, 0 …

6 days ago · Find all $x$ and $y$ such that $\prod_ {d\mid x} (d\pm1) =y!$ Note: This question appeared in our school contest that happened last year. In simple means, does there ...

Oct 26, 2012 · What's the difference between Fourier transformations and Fourier Series? Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?

Hint: prove inductively that a product is $> 1$ if each factor is $>1$. Apply that to the product $$\frac {n!} {2^n}\: =\: \frac {4!} {2^4} \frac {5}2 \frac {6}2 \frac {7}2\: \cdots\:\frac {n}2$$ This is a prototypical …

What I would say is that you can multiply any non-zero number by infinity and get either infinity or negative infinity as long as it isn't used in any mathematical proof. Because multiplying by infinity is …

6 days ago · In the 7th letter, Plato mentions that inhabitants of Sicily follow a lifestyle incompatible with philosophy, and such sharp criticism arguably led to an invasion. Socrates also emphasizes lifestyl...

Apr 1, 2026 · As the title says, what is exactly the battle between something you like, something you hate and something you 'like to like' ? Let's just say, Martin is a very bright student, in 5th grade, he …

Nov 27, 2019 · Your answer is already solved, but I would like to add a trick. If the rank of an nxn matrix is smaller than n, the determinant will be zero.