Triangle Inequality <= Sign Explained

In summary, the introduction of the <= sign saved writers from having to use parentheses around absolute values when they are less than two.
  • #1
razored
173
0
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 <?
 
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  • #2
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
bucher said:
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".

That's not true, plugin a=-3 and b=2. In fact, try any set of numbers and you will see.
 
  • #4
Okay, I will: [itex]|-3+2|^2= 1^2= 1. |-3|+ |2|= 3+2= 5[/itex] and [itex]5^2= 25[/itex]. 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
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
That is not my interpretation of that particular way of writing mathematics.

If it were [itex]|a+b|^2= (a+ b)(a+ b)= a^2+ 2ab+ b^2\le |a|^2+ 2|a||b|+ |b|^2= (|a|+ |b|)^2[/itex], in one line, then, yes, the last two are equal. But my understanding of
[tex]\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[/tex]
is that the left side, here [itex]|a+ b|^2[/itex], is "copied" down the left. That is, it is
[tex]\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[/tex]
 
  • #7
HallsofIvy said:
That is not my interpretation of that particular way of writing mathematics.

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
Fredrik said:
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 =.

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.
 

What is the triangle inequality theorem?

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

What is the symbol for the triangle inequality theorem?

The symbol for the triangle inequality theorem is the less than or equal to sign (≤).

How is the triangle inequality theorem used in geometry?

The triangle inequality theorem is used to determine whether three given lengths can form a valid triangle. It is also used to prove other theorems and properties in geometry.

What happens if the sum of two sides of a triangle is equal to the third side?

If the sum of two sides of a triangle is equal to the third side, the resulting shape is not a triangle but a straight line. This is known as a degenerate triangle.

Can the triangle inequality theorem be applied to any polygon?

No, the triangle inequality theorem only applies to triangles. It cannot be applied to any other polygon because it specifically deals with the relationship between three sides.

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