# Branch: Number Systems and Arithmetics

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.

### Concepts you will learn in this part of BookofProofs

• 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.

Explanations: 1

Branches: 1
Chapters: 2
Definitions: 3 4
Explanations: 5 6
Motivations: 7 8
Parts: 9 10 11 12
Topics: 13

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition