> [!summary] >This proof uses the definition of the absolute value to prove the multiplication rule > **Key Result:** $|a||b| =|ab|$ >[!info]+ Read Time **⏱ 1 min** # 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 $|a|\cdot|b|$ is: > $ |ab| = \begin{cases} ab, & \text{if } ab \geq 0 \\ -ab, & \text{if } ab < 0 \end{cases} > $ Prove that $|a|\cdot |b|=|ab|$ $ \begin{array}{c} \text{Case 1}: a>0, b >0: \\ |a|\cdot|b| = a\cdot b = ab \\ \text{By definition of a absolute value, $a>0$ and $b>0$ so $ab>0$ therefore }\\ ab=|ab| \\ \\ \text{Case 2}: a<0 , b>0: \\ |a| \cdot |b| = -a\cdot b \quad \\ \text{By definition of a absolute value, $a<0$ and $b>0$ so $ab<0$ therefore }\\ -ab = |ab|\\ \\ \text{Case 3}: a<0,b<0:\\ |a|\cdot|b| = -a\cdot -b =ab| \\ \text{By definition of a absolute value, $a<0$ and $b<0$ so $ab>0$ therefore } \\ ab=|ab| \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/pmT4aAtwpfY?si=aZwOwJHZjQZYLCTk" 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> --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. ---