# Inequality - Proof that √(a^2)<√(b^2) does not imply a<b

1. Aug 22, 2014

### Akitirija

Hi everyone!

First of all thank you for a great forum! I downloaded the app and find it ingenious!

The problem stated above is from "3000 Solved Problems in Calculus".

The book solves this problem simply by stating: "No. Let a=1 and b=-2".

However, I am curious to know if it is possible to provide a more algebraic proof, or of this is the only way to prove it. As I really cannot provide any attempts of my own, I will just ask if anyone know what topic I have to study in order to find the answer. If I fail after that, I will return to you again.

(Maybe it should be mentioned that I am interested in doing Calculus 1,but I want the fundamentals in order first.)

Yours sincerely,
Aki

2. Aug 22, 2014

### HallsofIvy

Staff Emeritus
I am puzzled by this. Are you saying that this proof is too simple and you want a harder proof?

The problem is to show that "if $\sqrt{a^2}< \sqrt{b^2}$ then a< b" is NOT true. A standard way to show that an "if-then" statement is not true is to give a "counter example". While no number of "examples" will show that such a statement is true one example in which it does not work is enough to show that it is NOT true.

3. Aug 22, 2014

### Akitirija

Ah, cool! I see your point. I'm not really used to proofs at all. Thank you very much for the quick reply!

4. Aug 22, 2014

### Ray Vickson

Just to be clear: what the book gave was not a "proof", it was a "disproof". Even if you do not know how to do proofs you may be able to do disproofs, as they are very different concepts.

5. Aug 22, 2014

### Akitirija

I see. Thank you for the clarification!