Set of Real Numbers

Section: Simple Consequences From the Axioms of Addition

Introduction

There are some simple consequences following from the axioms of addition, valid in the field of real numbers \(\mathbb R\). These consequences establish the rules taught in the high school for adding numbers and solving equations involving the addition of their terms.

Please note that the rules described and proven in this section are valid not only for the real numbers, but for many other algebraic structures, known as an abelian groups. This is because the real numbers together with the addition form a special case of such an abelian group, called additive abelian group and denoted by \((\mathbb R, + )\). The word additive is used to distinguish \((\mathbb R, + )\) from another abelian group in this field, the multiplicative abelian group \((\mathbb R\setminus \{0\},\cdot)\), for which analogous rules of calculation and equation solving are described in a separate section.

Subordinated Structure:

Contribute to BoP:

add a new Subsection  N

add a new Motivation  N

add a new Example  N

add a new Application  N

add a new Explanation  N

add a new Interpretation  N

add a new Axiom  N

add a new Definition  N

add a new Proposition  N

add a new Lemma  N

add a new Theorem  N

add a new Algorithm  N

add a new Open Problem  N

add a new Bibliography (Branch)  N

add a new Comment (Branch)  N