Commutative Properties
- Commutative Property of Addition:
  - If $a, b$ are real numbers, then $a + b = b + a$
- Commutative Property of Multiplication:
  - If $a, b$ are real numbers, then $a \cdot b = b \cdot a$
Associative Properties
- Associative Property of Addition:
  - If $a, b, c$ are real numbers then $(a + b) + c = a + (b + c)$
- Associative Property of Multiplication:
  - If $a, b, c$ are real numbers then $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

Distributive Property:
- If a,b,ca,b,c are real numbers then
  - $a (b + c) = a b + a c$
  - $(b + c) a = b a + c a$
  - $a (b - c) = a b - a c$

Identity Properties
- Identity Property of Addition: For any real number a: $a + 0 = a 0 + a = a$ 0 is the additive identity
- Identity Property of Multiplication: For any real number a: $a \cdot 1 = a 1 \cdot a = a$ 1 is the multiplicative identity
Inverse Properties
- Inverse Property of Addition: For any real number a: $a + (- a) = 0 - a$ is the additive inverse of a
- Inverse Property of Multiplication: For any real number a: $(a \neq 0) a \cdot \frac{1}{a} = 1 \frac{1}{a}$ is the multiplicative inverse of a
Properties of Zero
- Multiplication by Zero: For any real number a, $\begin{matrix} a \cdot 0 = 0 & 0 \cdot a = 0 & The product of any number and 0 is 0. \end{matrix}$
- Division of Zero: For any real number a, $\begin{matrix} \frac{0}{a} = 0 & Zero divided by any real number, except itself, is zero. \end{matrix}$
- Division by Zero: For any real number a, $\frac{a}{0}$ is undefined and $a \div 0$ is undefined. Division by zero is undefined.