Archive for the ‘Theory’ Category

I was trying to get a matrix for a projection of any point to a line given by the following equotations: After some time of research I’ve ended with the following result: Let then Of course the result isn’t new at all, it was just a big efford for me to do that at midnight […]

Today, I noticed an amazing and worth memorizing thing. I turns out that fibbonacci series shows up also in graph theory in one of the simplest graphs: Where’s Fibbonacci hiden here? Well, the graph can be represented as a folowing matrix: By multiplying the matrix by itself and using induction we get: 

Matrixes seem to have a lot of wonderful properties, for example they may help checking if given triangles represented each by three complex numbers (verticles) are simillar. So, two triangles represented by complex numbers and are similar if, and only if: Proof coming soon.

The angle which arms contain chords or tangents of a circle is the half of the sum (if the verticle of the angle lies inside the circle or on it) or difference (if the verticle lies outside) of middle angles based on the curves, which set this angle. The proof can be illustrated by following […]

Problem: If you’ll eliminate every second person from the circle of people going clockwise, which one stays alive? (As first the second person dies) Aswer: if then the person with number stays alive. Equivalently it’s a one-bit-shift-left of number written binary I will not write here the whole solution, thus it can be found in […]

Prove that for every natural number there exists different natural numbers such as sum of any two is divisable by their diffrence. Proof: Because number is divisable by , we can write and of course . Adding those two together: Induction: 1. Sequence 1,2 works 2. Assume that sequence works, then the number is divisable […]