# Axiom:Real Number/Axioms

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## Definition

The properties of the field of real numbers $\struct {\R, +, \times, \le}$ are as follows:

\((\R \text A 0)\) | $:$ | Closure under addition | \(\ds \forall x, y \in \R:\) | \(\ds x + y \in \R \) | |||||

\((\R \text A 1)\) | $:$ | Associativity of addition | \(\ds \forall x, y, z \in \R:\) | \(\ds \paren {x + y} + z = x + \paren {y + z} \) | |||||

\((\R \text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall x, y \in \R:\) | \(\ds x + y = y + x \) | |||||

\((\R \text A 3)\) | $:$ | Identity element for addition | \(\ds \exists 0 \in \R: \forall x \in \R:\) | \(\ds x + 0 = x = 0 + x \) | |||||

\((\R \text A 4)\) | $:$ | Inverse elements for addition | \(\ds \forall x: \exists \paren {-x} \in \R:\) | \(\ds x + \paren {-x} = 0 = \paren {-x} + x \) | |||||

\((\R \text M 0)\) | $:$ | Closure under multiplication | \(\ds \forall x, y \in \R:\) | \(\ds x \times y \in \R \) | |||||

\((\R \text M 1)\) | $:$ | Associativity of multiplication | \(\ds \forall x, y, z \in \R:\) | \(\ds \paren {x \times y} \times z = x \times \paren {y \times z} \) | |||||

\((\R \text M 2)\) | $:$ | Commutativity of multiplication | \(\ds \forall x, y \in \R:\) | \(\ds x \times y = y \times x \) | |||||

\((\R \text M 3)\) | $:$ | Identity element for multiplication | \(\ds \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) | \(\ds x \times 1 = x = 1 \times x \) | |||||

\((\R \text M 4)\) | $:$ | Inverse elements for multiplication | \(\ds \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) | \(\ds x \times \frac 1 x = 1 = \frac 1 x \times x \) | |||||

\((\R \text D)\) | $:$ | Multiplication is distributive over addition | \(\ds \forall x, y, z \in \R:\) | \(\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \) | |||||

\((\R \text O 1)\) | $:$ | Usual ordering is compatible with addition | \(\ds \forall x, y, z \in \R:\) | \(\ds x > y \implies x + z > y + z \) | |||||

\((\R \text O 2)\) | $:$ | Usual ordering is compatible with multiplication | \(\ds \forall x, y, z \in \R:\) | \(\ds x > y, z > 0 \implies x \times z > y \times z \) | |||||

\((\R \text O 3)\) | $:$ | $\struct {\R, +, \times, \le}$ is Dedekind complete |

These are called the **real number axioms**.

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Place these into their own pagesIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-3}$ Order properties of real numbers - 1967: Michael Spivak:
*Calculus*... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 6)$ - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.3$: Arithmetic