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## The axiomatic construction of natural arithmetic

Natural arithmetic is the science of natural numbers: it is based on addition, multiplication, natural order and divisibility. Now, all these operations and relations are defined on the basis of the single successor function, whose properties are brought together in...

## Russell’s paradox and the emergence of class theory

Russell’s paradox or antinomy is a very simple paradox in naive set theory, which arises when one tries to define a “set of all sets”. Its resolution relies on the introduction of the notion of class and the distinction of sets among classes. Thanks...

## The natural scalar (or dot) product: a numerical combination of vectors

The scalar or dot product of two vectors in real space is a real number that takes into account the direction, sense and magnitude of both vectors. 1.The natural scalar product in the Euclidean plane 1.1.From the distance between two points to the scalar product In...

## Gaussian integers: an imaginary arithmetic

Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular, we can determine which are the usual prime numbers that...

## What is a complex number? A simple geometric approach

There are various ways of defining complex numbers. The most direct way is to look at them as points or vectors of the Euclidean plane. Addition and multiplication are then defined using the coordinates. 1. The set $$\mathbb C$$ of complex numbers 1.1. A complex...