> [!summary] This proof uses the definition of an absolute value to prove the quotient rule > **Key Result:** $\frac{|a|}{|b|}= |\frac{a}{b}|$ assuming that $b\neq 0$ # Mathematical Proof > [!warning] Assumptions To prove the absolute value multiplication rule by [[Proof by Cases|proof by cases]] assume the following: > - The definition of the absolute value of a $\frac{|a|}{|b|}$ is: > $ > | \frac{a}{b}| = > \begin{cases} > \frac{a}{b}, & \text{if } ab \geq 0 \\ > -\frac{a}{b}, & \text{if } ab < 0 > \end{cases} > $ > - $b \neq 0$ Prove that $\frac{|a|}{|b|}= |\frac{a}{b}|$ $ \begin{array}{c} \\ \text{Case 1:} \quad a>0 , b>0: \\ \frac{a}{b} = |\frac{a}{b}| \quad\text{By definition } \\ \\ \text{Case 2:} \quad a>0, b<0: \\ -\frac{a}{b}= |\frac{a}{b}|\quad \text{By definition} \\ \\ \text{Case 3:} \quad a<0 , b<0: \\ \frac{-a}{-b}=\frac{a}{b}= |\frac{a}{b}| \quad \text{By definition} \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/0ozGYp6MXWY?si=qcY4fA4XRtKXiSeU" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>