Proving absolute values theorems

Click For Summary

Discussion Overview

The discussion revolves around proving the theorem related to absolute values, specifically that for all real numbers \(x\) and \(y\), if \(x + y \ge 0\), then \(|x + y| = x + y\). The scope includes mathematical reasoning and proof techniques.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant asks how to prove that \(|x + y| = x + y\) under the condition \(x + y \ge 0\), noting that their textbook assumes this to be true.
  • Another participant suggests letting \(a = x + y\) and questions what \(|a|\) is, implying a need to explore the definition of absolute value.
  • A later reply reiterates the definition of absolute value and states that since \(a = x + y \ge 0\), it follows from the definition that \(|x + y| = x + y\).

Areas of Agreement / Disagreement

Participants generally agree on the definition of absolute value and its application in this context, but the initial question about proving the theorem remains open without a formal consensus on the proof method.

Contextual Notes

The discussion does not resolve the proof method and relies on the definition of absolute value, which may depend on the participants' understanding of mathematical definitions and proof techniques.

tmt1
Messages
230
Reaction score
0
For all real numbers $x$ and $y$ , if $x + y >= 0$ then $|x + y| = x + y$. How would I prove this?

My textbook just assumes this to be true.
 
Physics news on Phys.org
Let $a = x+y$. What is $|a|$?
 
Deveno said:
Let $a = x+y$. What is $|a|$?

$|x + y|$
 
tmt said:
For all real numbers $x$ and $y$ , if $x + y >= 0$ then $|x + y| = x + y$. How would I prove this?

My textbook just assumes this to be true.

The absolute value of a real number is defined as:
$$|a| = \begin{cases}a&\text{if } a\ge 0 \\ -a & \text{if } a<0\end{cases}$$

Since it is already given that $a = x+y \ge 0$, it follows from the definition that $|x+y|=x+y$.
 

Similar threads

Undergrad Absolute difference between increasing sum of squares
  • · Replies 22 ·
Replies
22
Views
3K
High School The infinitesimal in Calculus
  • · Replies 18 ·
Replies
18
Views
3K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad On inverse function theorem in Spivak's CoM
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Determination of error in interpolating polynomial
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Proving that convexity implies second order derivative being positive
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
MHB What is the Absolute Minimum Value of f(x)?
  • · Replies 2 ·
Replies
2
Views
2K
MHB What is the Absolute Maximum Value of f in the Function f(x) = ln(x)/x?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Spivak, Ch. 20: Understanding a step in the proof of lemma
  • · Replies 1 ·
Replies
1
Views
3K