1. ## [RESOLVED]Prove equality

$a$ and $b$ are real numbers.

demonstrate the following equality.
$|ab|=|a||b|$

I have no idea, how to prove that.

I just know $|a+b|\leq|a|+|b|$ so i dont know how to start.

2. ## Re: Prove equality

Originally Posted by Fabio010
$a$ and $b$ are real numbers. demonstrate the following equality.
$|ab|=|a||b|$
Just bite the bullet.
There are four cases:
$\begin{gathered} a \geqslant 0\;\& \,b \geqslant 0 \hfill \\ a \geqslant 0\;\& \,b < 0 \hfill \\ a < 0\;\& \,b \geqslant 0 \hfill \\ a < 0\;\& \,b < 0 \hfill \\ \end{gathered}$

Take the last case:
$(-a)(-b)=ab>0$ so $ab=|ab|$.
But $-a=|a|~\&~-b=|b|$ so $|a||b|=|ab|$

3. ## Re: Prove equality

Originally Posted by Plato
Just bite the bullet.
There are four cases:
$\begin{gathered} a \geqslant 0\;\& \,b \geqslant 0 \hfill \\ a \geqslant 0\;\& \,b < 0 \hfill \\ a < 0\;\& \,b \geqslant 0 \hfill \\ a < 0\;\& \,b < 0 \hfill \\ \end{gathered}$

Take the last case:
$(-a)(-b)=ab>0$ so $ab=|ab|$.
But $-a=|a|~\&~-b=|b|$ so $|a||b|=|ab|$
OK thanks a lot for the help
but

even if $ab<0$ isnt $ab= |ab|$ ??

4. ## Re: Prove equality

Originally Posted by Fabio010
OK thanks a lot for the help
but even if $ab<0$ isnt $ab= |ab|$ ??
No one said it was. Nowhere was that ever even mentioned.
This is true: If $ab<0$ then $-ab=|ab|$

5. ## Re: Prove equality

ok

so i can prove with all cases :/???
because if $-ab = |ab|$
$-a = |a|~\&~b = |b|~so |a||b| = |ab|$

sorry if i am being confusing.

6. ## Re: Prove equality

??can i prove with all cases?

7. ## Re: Prove equality

Originally Posted by Fabio010
??can i prove with all cases?
What does that mean?

8. ## Re: Prove equality

Sorry i was confusing.
Now i understood it completely.

Thanks for the help.