DescriptionCounting is doubtless one of the earliest skills of humankind. The German mathematician Leopold Kronecker is quoted to have said ''God created the natural numbers and all the rest is the work of men.''Whilst numbers seem to be the most basic concepts in mathematics, there are numerous problems concerning just the natural numbers which challenged curious minds for centuries, for example, the distribution of prime numbers. Many problems here are still widely open and often considered to be amongst the hardest in Mathematics. In this project we will focus on various aspects and questions about numbers and number systems. Already at school, we get used to work with real numbers and take them for granted. But how to prove that they exist? Are they really well defined? The school of Pythagoras denied the existence of irrationals and it is said that they punished Hippasus of Metapontum to death for his discovery that the diagonal in a square of side length one is irrational. What is the motivation for the introduction of complex numbers and why were so many flaws in early attempts to prove the Fundamental Theorem of Algebra? What are transzendental numbers and how to prove that pi or e are transzendental? Then there is the strange world of p-adic numbers where infinite sums converge if and only if their terms converge to zero -- in view of the harmonic series, this statement is fundamentally wrong in the world of real numbers... What are quaternions and Cayley numbers and division algebras and what are they good for? We may have a glimpse at nonstandard numbers and the introduction of real numbers via Conway games. We will also look at interesting concepts like continuous fractions and approximations of irrationals by rationals as well as Diophantine Problems and Minkowski's Geometry of Numbers. These examples should provide enough evidence that there are many directions for students to branch out in their discoveries about numbers and number systems. Our sources will be the excellent monograph ''Numbers'' by Ebbinghaus, Hermes, Hirzebruch, Koecher, Mainzer, Neukirch, Prestel and Remmert, where each chapter is devoted to a special number system. We will consult the classic ''An Introduction to the Theory of Numbers'' by Hardy and Wright as well as Burger's book ''Exploring the Number Jungle''. The latter is a very good introduction into problems in Number Theory and Diophantine Approximation. This book has plenty of challenging problems and questions while it gives first insights into many interesting special topics in this area. The book is an ideal platform to start and then to branch out into many different specialised topics like, e.g., ''History of irrational and transcendental numbers'', ''Connections between diophantine approximation and hyperbolic geometry'', ''Minkowski's and Pick's theorem and applications'', etc... All three books are available in the Library of Durham University. CorequisiteThere is no explicit corequisite. Useful courses to be taken simultaneously are here, however, Geometry or Number Theory.ResourcesSome recommendable Books covering the general theory are
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