# Search results

1. ### All natural numbers

An example is \binom{n+m}{k}=\displaystyle\sum_{i=0}^{k}\binom{n}{i}\binom{m}{k-i}. It is asked to prove it for all naturals n,m,k. They say it is enough to prove it for example just for m. The reason, which was brought forward, is that in the induction step we assume the truth of the...
2. ### All natural numbers

Hello! Question: if it is asked to prove a statement A(n_1,...,n_k) for all natural numbers n_1,...,n_k, is it actually enough to check its truth by induction on just one of the counters, say n_1?
3. ### Empty family of sets

Do you mean that "for all" incorporates "for no"?
4. ### Empty family of sets

Hi! I'd like to ask the following question. Does it make sense to take unions and intersections over an empty set? For instance I came across a definition of a topological space which uses just two axioms: A topology on a set X is a subset T of the power set of X, which satisfies: 1...
5. ### Symmetry of Polynomials

Ok the combination is clear: -f(b-x)+a=f(b+x)-a Thank you! :-) Oh sorry I've realised that everything is ok at the very moment you've sent your post...and deleted the thing!
6. ### Symmetry of Polynomials

Ok the combination is clear: -f(b-x)+a=f(b+x)-a Thank you! :-)
7. ### Symmetry of Polynomials

Hi! Brief question: I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane. Are there such conditions?
8. ### How much math does a math professor remember?

How can you be sure that those theorems you leave out as somewhat irrelevant will not provide your area of interest with unexpected insights and applications in the future? Nowadays number theory(!) finds unexpected applications in physics. Now, number theory is traditionally recognized as...
9. ### Square Root of Complex Numbers

Hi! I've got a question. There is a nice formula for finding square roots of arbitrary complex numbers z=a+bi: \frac{1}{\sqrt{2}}(\epsilon\sqrt{|z|+a}+i\sqrt{|z|-a}) where epsilon:=sing(b) if b≠0 or epsilon:=1 if b=0. I've just looked it up and it's nice to use it to find complex roots of...