Proving that a > b iff a^2 > b^2 for a, b > 0

[d1zz]

I am having problems with proving alot of the questions asked in the text book. I don't know what to do or how to start it.

For example:

a > b <=> a^2 > b^2

(Let a > 0 and b > 0)

How would I prove that question? Gr.12 calculas did not really delve into the concept of proving mathematical equations. Is there a method, book (I have to read or you recommend) that would help in learning how to prove different types of questions?

How did you learn how to prove everything? Hints, explanations, and help is very much appreciated.

Thanks alot in advance.

 

[pka]

First it is important to note that we are given that \(\displaystyle a > 0\quad \& \quad b > 0\) otherwise the statement is false.
We are to prove an ‘if and only if’.
First assume that \(\displaystyle a > b\). Because both are positive, we can multiply by both
\(\displaystyle \begin{array}{rcl}
a > b & \Rightarrow & \;a^2 > ab \\
& \Rightarrow & \;ab > b^2 \\
& \Rightarrow & \;a^2 > b^2 \\
\end{array}.\)

Now for the only if part:
\(\displaystyle \begin{array}{rcl}
a^2 > b^2 & \Rightarrow & \;a^2 - b^2 > 0 \\
& \Rightarrow & \;\left( {a - b} \right)\left( {a + b} \right) > 0 \\
\end{array}.\) We know that \(\displaystyle \left( {a + b} \right) > 0\) so that means that \(\displaystyle \left( {a - b} \right) > 0\) or \(\displaystyle a > b.\)