# Definition: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 | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x + y \in \R \) | ||||

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

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

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

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

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

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

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

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

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

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

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

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

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

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ 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