*Number systems and arithmetics* is a branch of mathematics dealing with a formal clarification of what *numbers* are and in which domains numbers allow arithmetical operations like addition, subtraction, multiplication, and division. It also describes how these domains can be extended in a step-by-step manner by certain structural properties to allow such operations.

- You should be acquainted with set theory, especially
- For a deeper understanding, you should know some basic facts about
- algebraic structures, including groups, integral domain, and fields.
- The topological concept of a metric space.

- What are
*natural numbers*, and how the can be defined using the axiomatic method? - What are
*integers*, how they can be defined using natural numbers and how they extend the calculating possibilities of natural numbers? - What are
*rational numbers*, how they can be defined using integers and how they extend the calculating possibilities of integers? - What are
*real numbers*, how they can be defined using rational numbers and how they extend the calculating possibilities of rational numbers? - What are
*complex numbers*, how they can be defined using real numbers and how they extend the calculating possibilities of real numbers? - What are
*quaternions*, how they can be defined using complex numbers and how they extend the calculating possibilities of complex numbers? - What are the differences of the above domains, covering their
*algebraic*,*topological*, and*order*properties.

| | | | created: 2014-02-20 23:39:12 | modified: 2019-01-20 20:34:50 | by: *bookofproofs* | references: [979]

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[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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