Absolute Value Proof: Showing |a|=sqrt(a^2) and |a/b|=|a|/|b|

[kittykat52688]

Homework Statement

If a,b are real numbers and b does not equal zero show that |a|=sqrt(a^2) and |a/b|=|a|/|b|.

Homework Equations

I know that |ab|=|a||b| and a^2 = |a|^2

The Attempt at a Solution

Attempt at showing that |a|=sqrt(a^2):
|a|=sqrt(a^2)
|a|^2=(sqrt(a^2))^2
|a|^2=a^2
a^2=a^2

Not sure how to do the second part.

 

[Mark44]

Homework Statement

If a,b are real numbers and b does not equal zero show that |a|=sqrt(a^2) and |a/b|=|a|/|b|.

Homework Equations

I know that |ab|=|a||b| and a^2 = |a|^2

So |a| = |a/b * b| = |a/b||b|. What can you conclude about |a/b|?

The Attempt at a Solution

Attempt at showing that |a|=sqrt(a^2):
|a|=sqrt(a^2)
|a|^2=(sqrt(a^2))^2
|a|^2=a^2
a^2=a^2

It looks like you are assuming that |a| = sqrt(a^2) (which is what you need to prove), and concluding that a^2 equals itself.

Instead of going about it this way, I would suggest using two cases: one with a >= 0 and the other with a < 0. Can you show for each case that sqrt(a^2) = |a|?

Not sure how to do the second part.

 

[The Chaz]

It's reasonable to expect that the definition of the absolute value function would surface at some point.

Your first attempt is invalid because you assumed the statement that you were trying to prove. If you reversed the steps, you would have to use the square root property, and it wouldn't work.

Here's a sample of what you would do to prove that |ab| = |a|*|b|.
Case: a > 0; b > 0.
Then ab > 0 and |ab| = ab = |a|*|b|.

Case : a > 0; b < 0 (or vice versa)...
Case : a < 0; b < 0.

Give that a shot.