1. Division by zero is undefined: You cannot divide any number by zero because it does not produce a meaningful result. 2. Dividing zero by any nonzero number results in zero: For any number $a \neq 0$, $\frac{0}{a} = 0$. 3. Dividing a number by 1 leaves it unchanged: For any number $a$, $\frac{a}{1} = a$. 4. Dividing a number by itself equals 1: For any number $a \neq 0$, $\frac{a}{a} = 1$. 5. Division is the inverse operation of multiplication: If $a \times b = c$, then $\frac{c}{b} = a$ and $\frac{c}{a} = b$. 6. When dividing fractions, multiply by the reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$. 7. Division distributes over addition and subtraction only when factoring: $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$, but $\frac{a}{b+c} \neq \frac{a}{b} + \frac{a}{c}$. 8. Dividing negative numbers follows sign rules: $\frac{-a}{b} = -\frac{a}{b}$, $\frac{a}{-b} = -\frac{a}{b}$, and $\frac{-a}{-b} = \frac{a}{b}$. These rules help simplify and understand division operations in algebra and arithmetic.