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

  • Thread starter Thread starter Akitirija
  • Start date Start date
  • Tags Tags
    Inequality Proof
Click For Summary

Homework Help Overview

The discussion revolves around the logical implications of the inequality involving square roots, specifically whether the statement √(a^2) < √(b^2) necessarily implies a < b. The original poster references a problem from a calculus book and expresses interest in understanding the proof or disproof of this statement.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster seeks a more algebraic proof beyond the counterexample provided in the book. Participants discuss the nature of proofs and disproofs, with some emphasizing the role of counterexamples in demonstrating the falsity of an implication.

Discussion Status

The conversation is ongoing, with participants clarifying the difference between proofs and disproofs. The original poster has expressed gratitude for the insights shared, indicating a productive exchange of ideas, though no consensus on a formal proof has been reached.

Contextual Notes

The original poster mentions a desire to build foundational knowledge in calculus before tackling the problem, suggesting a potential gap in prerequisite understanding that may influence the discussion.

Akitirija
Messages
17
Reaction score
0
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
 
Physics news on Phys.org
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}&lt; \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.
 
Ah, cool! I see your point. I'm not really used to proofs at all. Thank you very much for the quick reply!
 
Akitirija said:
Ah, cool! I see your point. I'm not really used to proofs at all. Thank you very much for the quick reply!

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.
 
I see. Thank you for the clarification!
 

Similar threads

Need help in manipulating rational absolute value inequalities
  • · Replies 11 ·
Replies
11
Views
3K
Can You Solve This Proof Involving Inequalities and Negations?
  • · Replies 10 ·
Replies
10
Views
3K
An inequality involving ##x## on both sides: ##\sqrt{x+2}\ge x##
  • · Replies 3 ·
Replies
3
Views
3K
Solving an inequality for a change of variables
  • · Replies 15 ·
Replies
15
Views
2K
Matrices: if AB=A and BA=B, then B^2 is equal to?
  • · Replies 18 ·
Replies
18
Views
4K
Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##
  • · Replies 18 ·
Replies
18
Views
2K
Solve Inequality: Algebraic Proof of a<b<c<1/12
  • · Replies 3 ·
Replies
3
Views
2K
Manipulating an inequality in the bisection method
  • · Replies 2 ·
Replies
2
Views
2K
Solving Simultaneous Inequalities
  • · Replies 7 ·
Replies
7
Views
2K
Spivak's "Calculus": AM-GM inequality problem.
  • · Replies 6 ·
Replies
6
Views
4K