$\displaystyle a$ and $\displaystyle b$ are real numbers.
demonstrate the following equality.
$\displaystyle |ab|=|a||b|$
I have no idea, how to prove that.
I just know $\displaystyle |a+b|\leq|a|+|b|$ so i dont know how to start.
$\displaystyle a$ and $\displaystyle b$ are real numbers.
demonstrate the following equality.
$\displaystyle |ab|=|a||b|$
I have no idea, how to prove that.
I just know $\displaystyle |a+b|\leq|a|+|b|$ so i dont know how to start.
Just bite the bullet.
There are four cases:
$\displaystyle \begin{gathered} a \geqslant 0\;\& \,b \geqslant 0 \hfill \\ a \geqslant 0\;\& \,b < 0 \hfill \\ a < 0\;\& \,b \geqslant 0 \hfill \\ a < 0\;\& \,b < 0 \hfill \\ \end{gathered} $
Take the last case:
$\displaystyle (-a)(-b)=ab>0$ so $\displaystyle ab=|ab|$.
But $\displaystyle -a=|a|~\&~-b=|b|$ so $\displaystyle |a||b|=|ab|$