# Thread: Prove equality

1. ## [RESOLVED]Prove equality

$\displaystyle a$ 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.

2. ## Re: Prove equality

Originally Posted by Fabio010
$\displaystyle a$ and $\displaystyle b$ are real numbers. demonstrate the following equality.
$\displaystyle |ab|=|a||b|$
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 $\displaystyle |a||b|=|ab|$

3. ## Re: Prove equality

Originally Posted by Plato
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 $\displaystyle |a||b|=|ab|$
OK thanks a lot for the help
but

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

4. ## Re: Prove equality

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

5. ## Re: Prove equality

ok

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

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.