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.

Theoretical minimum (in a nutshell)

You should be acquainted with set theory, especially
- set operations,
- functions,
- order relations, and
- countability.
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.

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.