• Free Textbooks in Mathematics

    The following is a list of links to useful textbooks in mathematics, available for free on the Internet. All books are legally safe to download, The books are in printable format - Postscript (PS) or Portable Document Format (PDF). You are free to download, read and print them. This is a partial list of "free textbooks" in math, the list is to be updated regularly.
  • Emotions Relationship & Learning

    It's been said that "necessity is the mother of inventions", but necessity is the mother of learning as well. Learning is vital to survival it is a necessity, however necessity determines what we learn. This interaction between necessity and learning is part of "emotional relationship" between the learner his/her topic.
  • Asymptotic Differentiable Function

    Let $f$ be any function asymptotic at zero. Prove/Disprove:
    If $f$ is differentiable everywhere in its domain, then $\lim_{x\to \infty}f\;'(x)$ must be equal to zero?
  • Fibonacci Sequence, A Pattern of Natural Beauty

    The sequence of Fibonacci can be found everywhere in nature. First mentioned by Leonardo Fibonacci who was a medieval mathematician that was trying to solve problem involving the growth of a population of rabbits. It became one of the most famous sequence in all of mathematics. Here is a mathematical description of some of its fundamental properties...
    Fibonnaci as appeared in a snail
  • Steinhaus theorem

    Let $\lambda$ be Borel measure.
    1. If $A$ is mesuarable and $\lambda(A)>0$, then $A-A=\left\{ x-y\; :\; x,y\in A \right\}$ contains a segment $I$ such that $0\in I$
    2. If $A, B$ are measurable sets and $\lambda(A), \lambda(B)>0$, then $A+B$ contains a segment I.
  • Minimum way to move all circles

    Consider a game with three vertical sticks and 5 circles, where each circle has a hole in the middle. The circles arranged in one stick as shown in the picture: The task is to move all the circles to a nearby stick, circles can be move from one stick to another, but a circle cannot be put over a smaller one. What are the minimum number of steps needed to complete this task?
  • List of Computer tools in mathematics

    The following list links to free computer tools for mathematics.
  • Google search tips

    Tips and tricks on Google search techniques.
  • The circle of Apollonius

    By definition, a circle is the locus of all points equidistant from a fixed center. However, the circle of Apollonius is defined differently. In the following post I will define and construct an Apollonian circle.
  • Hahn Decomposition Theorem

    Let $\nu$ be a signed measure over measurable space $(X,\mathcal{M})$. Denote $\tilde{N}=\left\{ u\in \mathcal{M} \; :\; \mbox{$u$ is $\nu$-negative} \right\}$ and $\tilde{P}=\left\{ u\in \mathcal{M} \; :\; \mbox{$u$ is $\nu$-positive} \right\}$
    1.   $\exists N\in \tilde{N}\Big(\exists P \in \tilde{P}\Big( N\cap P=\emptyset \; \mbox{ and }\; N\cup P = X\Big)\Big)$
    2.   $\forall N,N^{'}\in \tilde{N}\Big(\forall P,P^{'}\in \tilde{P} \Big(N\sqcup P =N^{'}\sqcup P^{'}=X \; \Longrightarrow \nu( P\triangle \tilde{P})=\nu( N\triangle \tilde{N})=0\Big)\Big) $
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