Triangle Inequality

1. Jun 20, 2009

razored

http://math.ucsd.edu/~wgarner/math4c/derivations/other/triangleinequal_files/eq0007S.gif [Broken]

Why did they introduce the <= sign?

I cannot think of any numbers that would violote the =. So why introduce the <?

Last edited by a moderator: May 4, 2017
2. Jun 20, 2009

bucher

Because if "a" or "b" (not both) is negative, then the answer would be less than the formula with the absolute value of "a" or "b".

3. Jun 20, 2009

razored

That's not true, plugin a=-3 and b=2. In fact, try any set of numbers and you will see.

4. Jun 20, 2009

HallsofIvy

Okay, I will: $|-3+2|^2= 1^2= 1. |-3|+ |2|= 3+2= 5$ and $5^2= 25$. 1 is definitely less than 25!

Now, what do YOU get? (Or did you do |a+b|2 and (a+b)2 rather than |a+b|2 and (|a|+ |b|)^2?)

5. Jun 20, 2009

Fredrik

Staff Emeritus
I think (or at least I hope) that the OP is referring to the last line (not the one before). The last one should start with a =.

6. Jun 20, 2009

HallsofIvy

That is not my interpretation of that particular way of writing mathematics.

If it were $|a+b|^2= (a+ b)(a+ b)= a^2+ 2ab+ b^2\le |a|^2+ 2|a||b|+ |b|^2= (|a|+ |b|)^2$, in one line, then, yes, the last two are equal. But my understanding of
$$\begin{array}{cc}|a+ b|^2&= (a+b)(a+b)\\ &= a^2+ 2ab+ b^2\\ &\le |a|^2+ 2|a||b|+ |b|^2 \\ &\le (|a|+ |b|)^2$$
is that the left side, here $|a+ b|^2$, is "copied" down the left. That is, it is
$$\begin{array}{cc}|a+ b|^2&= (a+b)(a+b)\\|a+ b|^2&= a^2+ 2ab+ b^2\\|a+ b|^2&\le |a|^2+ 2|a||b|+ |b|^2 \\|a+ b|^2&\le (|a|+ |b|)^2$$

7. Jun 20, 2009

Dunkle

Your interpretation is correct (but I know you don't need me to tell you that). Most authors write this way, and it even saves ink!

8. Jun 20, 2009

razored

The last one should be with <=. I understand why it is there; I was just pissed because I would have never thought to put the <= after putting up absolute values around 2ab.

Thanks for the help.

HallsOfIvy, I do like your interpretation of the math better, more lucid. The GIF I posted is hotlinked from some website I found. Unfortunately, the book I got this problem (spivak) from uses the same notation as the GIF image.