Set of Real Numbers

Section: Simple Consequences From the Axioms of Multiplication

Introduction

This section describes rules for multiplying numbers and solving equations involving the multiplication of their terms. Although these rules are well-known for anybody who visited, say, a primary school, it is usually not well-known outside the academic word (!) that they are consequences, which follow from the axioms of multiplication, valid for real numbers \(\mathbb R\), because they form an algebraic structure called a field.

What is remarkable is the bigger context, in which these calculation rules are also valid. There are many other algebraic structures, known as abelian groups, in which one can “multiply” and “divide”. Surprisingly, the “things” one can manipulate in these ways have not necessarily to be numbers (!). For details, please refer to the respective section dealing with groups.

As expected, all real numbers (except 0), together with the multiplication as a way of manipulating numbers, form a special case of an abelian group, called the multiplicative abelian group of real numbers and denoted by \((\mathbb R\setminus \{0\},\cdot)\). The word multiplicative is used to distinguish \((\mathbb R\setminus \{0\},\cdot)\) from another abelian group in this field - its additive abelian group \((\mathbb R, + )\), for which a separate section is dedicated.

The main difference between the two groups \((\mathbb R, + )\) and \((\mathbb R\setminus \{0\},\cdot)\) is that the number \(0\) has an additive inverse (see footnote¹), while it has no multiplicative inverse (see footnote²). Therefore it is included in the first and excluded from the second group. In other respects, both groups can be regarded as having a very similar structure.

¹ i.e there exist an \(x\in\mathbb R\) with \(0+x=0\).

² i.e there is no \(x\in\mathbb R\) with \(0\cdot x=1\), see also Why is it impossible to divide by \(0\)?

Subordinated Structure:

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