## 0Preface Branches

The non-commercial open textbook **BookOfProofs** (**BoP**) was launched in February 2014. It is open to both – students and educators who want to participate as co-authors.

The site uses the axiomatic method to systematically derive the foundations of different branches of mathematics, computer sciences, and physics. In the meanwhile, the project offers thousands of definitions, theorems, proofs, and examples at undergraduate and university level. But there is still a lot to do… **We warmly encourage you to become a co-author!** Share with your readers, pupils or students your knowledge about different concepts and theorems in mathematics, physics or computer sciences. You can also post your examples, exercises, and solutions related to the specific topics you are interested in.

### Typesetting Possibilities

Typesetting of maths and diagramming is enabled using MathJax and XyJax, for instance

$$\int_0^\infty\frac{x^2 + 2x + 3}{x}dx$$

$$ \begin{xy} \xymatrix@R=1pc{ \zeta \ar@{|->} [dd] \ar@{.>}_\theta [rd] \ar@/^/^\psi [rrd] \\ & \xi \ar@{|->} [dd] \ar_\phi [r] & \eta \ar@{|->} [dd] \\ P_{F}\zeta \ar_t [rd] \ar@/^/ [rrd]|!{[ru];[rd]}\hole \\ & P_{F}\xi \ar [r] & P_{F}\eta } \end{xy} \begin{xy} \xymatrix @W=3pc @H=1pc @R=0pc @*[F-] { : \save+<-4pc,1pc>*{\it root} \ar[] \restore \\ {\bullet} \save*{} \ar `r[dd]+/r4pc/ `[dd] [dd] \restore \\ {\bullet} \save*{} \ar `r[d]+/r3pc/ `[d]+/d2pc/ `[uu]+/l3pc/ `[uu] [uu] \restore \\ 1 } \end{xy} $$### Interactivity

**BookofProofs** places value on interactive learning mathematics. When you contribute to its open book, you can embed great external tools into your work. For instance, using JSXGraph, you can create interactive visualizations of mathematical concepts which are not apparent when just describing the dry theory. As an example, below you can find an interactive figure demonstrating graphically the multiplication of two complex numbers $x$ and $y$ depending on their values you can change by dragging the points $x$ or $y$. In particular, if both lie on the x-axis, the multiplication corresponds to the multiplication of real numbers – try out $(-1)\cdot( -2)$:

Thus, surprisingly, a graphical demonstration of complex numbers can also provide an insight to the question, why two negative real numbers, when multiplied together, give a positive real number.

Sagecell is a web service providing an interface to SageMath – a free open-source mathematics software system and one of the worldâ€™s largest collections of open source computational algorithms at your fingerprints.

You can even run R code inside your contributions. R is a free software environment for statistical computing and graphics with thousands of powerful libraries for different computational purposes. rdrr.io is a web service providing an interface to it:

All contents of the project are licensed under CC BY-SA 3.0 and are provided “AS IS”, without any warranty of any kind, and on a volunteer basis by our community.

- Please use the buttons to the left in order to expand and collapse the chapters of the book.

| | Contributors: *bookofproofs*

## 78491.User Manual - Quick Reference

## 70782.Introduction

## 273.Set Theory

## 254.Logic

## 475.Analysis

## 596.Algebra

## 577.Topology

## 758.Geometry

## 829.Combinatorics

## 7310.Probability Theory and Statistics

## 6211.Number Theory

## 6812.Graph Theory

## 10813.Knot Theory

## 11914.Game Theory

## 10515.Theoretical Physics

## 7016.Theoretical Computer Science

## 8717.Solving Strategies and Sample Solutions to Mathematical Problems

## 10918.Riddles, Puzzles and Brain-Teasers

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