Question

Prove that $|ab|=|a||b|$ for any numbers $a$ and $b$.

Answer 1

Prove of $|ab|=|a||b|$:

We can prove this by definition of modulus,

Case-1: When $a>0$ and $b>0$,

Then the equation opens modulus as,

$|ab|=|a||b|$

$\Rightarrow \left(a\right)\left(b\right)=\left(a\right)\left(b\right)$

LHS=RHS

Case-2: When $a>0$ and $b<0$,

Then the equation opens modulus as,

$|ab|=|a||b|$

$$\begin{gathered} \Rightarrow \left|\left(a\right)\left(b\right)\right|=\left|\left(a\right)\right|\left|\left(b\right)\right| \\ \Rightarrow \left|ab\right|=\left(a\right)(-b) \\ \Rightarrow -\left(ab\right)=-ab \\ \Rightarrow ab=ab \end{gathered}$$

LHS=RHS

Case-3: When $a<0$ and $b<0$,

$|ab|=|a||b|$

$$\begin{gathered} \Rightarrow \left|(-a)(-b)\right|=\left|\left(a\right)\right|\left|\left(b\right)\right| \\ \Rightarrow \left|ab\right|=-\left(a\right)\times -\left(b\right) \\ \Rightarrow \left(ab\right)=ab \end{gathered}$$

LHS=RHS

For all the four cases the equation $|ab|=|a||b|$ is satisfied.

Hence,it is proved that $|ab|=|a||b|$.