The Axioms of the Field of Real Numbers

# The Field of Real Numbers

We will now look at some axioms regarding the set of real numbers $\mathbb{R}$. We will note that an "axiom" is a statement that isn't meant to necessarily be proven and instead, they're statements that are given. These axioms are rather straightforward and may seem trivial, however, we will subsequently use them in order to prove many simple theorems and build a foundation for the set of real numbers.

# The Axioms of the Field of Real Numbers

Let $\mathbb{R}$ denote the set of real numbers and let $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ denote the binary operation of addition and let $\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ denote the binary operation of multiplication. Then for all $a, b, c \in \mathbb{R}$, the following axioms hold:

• Axiom A1: $a + b = b + a$ (Commutativity of Addition).
• Axiom A2: $a + (b + c) = (a + b) + c$ (Associativity of Addition).
• Axiom A3: There exists an element $0 \in \mathbb{R}$ such that $a + 0 = 0 + a = a$ (Existence of an Additive Identity).
• Axiom A4: There exists an element $-a \in \mathbb{R}$ such that $a + (-a) = (-a) + a = 0$ (Existence of Additive Inverses).
• Axiom M1: $a \cdot b = b \cdot a$ (Commutativity of Multiplication).
• Axiom M2: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ (Associativity of Multiplication).
• Axiom M3: There exists an element $1 \in \mathbb{R}$ such that $a \cdot 1 = 1 \cdot a = a$ (Existence of a Multiplicative Identity).
• Axiom M4: There exists an element $a^{-1} \in \mathbb{R}$ such that $a \cdot a^{-1} = a^{-1} \cdot a = 1$ (Existence of Multiplicative Inverses).
• Axiom D: $a \cdot (b + c) = a \cdot b + a \cdot c$ (Distributivity of Multiplication over Addition).

We note that these axioms define a special algebraic structure known as a field, so we say that $\mathbb{R}$ is a field under the operations of $+$ addition and $\cdot$ multiplication.