# Thread: triangle inequality

1. ## triangle inequality

Sir / Madam,

In geometry, whether the triangle inequality is a + b > c or a + b >= c, where a, b and c are the sides of the triangle.

If we put a + b = c. Then there is no triangle.

However, we say triangle inequality is a + b >= c in vectors and when study complex numbers.

Which one is correct?

kindly explain the difference, if any.

Thanks

with warm regards,

Aranga

2. ## Re: triangle inequality

if a, b, and c are the lengths of the three sides of a triangle then the "triangle inequality" says:
a+ b> c
a+ c> b and
b+ c> a.

It makes no difference at all if you think of the three lengths as lengths of vectors or absolute values of complex numbers so I am not at all clear what you are asking.

3. ## Re: triangle inequality

sorry for not making myself clear.

Which one of the following correctly explains the "Triangle inequality"

a + b > c [strictly greater than] in all cases as you explained above

or

a + b >= c ? [allowing the case of equality also, where no triangle is formed]

4. ## Re: triangle inequality Originally Posted by arangu1508 sorry for not making myself clear.

Which one of the following correctly explains the "Triangle inequality"

a + b > c [strictly greater than] in all cases as you explained above

or

a + b >= c ? [allowing the case of equality also, where no triangle is formed]
When dealing with triangles we must have a + b > c. But we can easily have the case of vectors where a + b = c. This is quite possible for vectors, just not triangles.

-Dan

5. ## Re: triangle inequality

sum of 2 shorter sides > longer side

6. ## Re: triangle inequality

thank you. it is very useful.

7. ## Re: triangle inequality

The term "triangle inequality" is generally applied to any inequality of the form $|a+b| \le |a| + |b|$. The $a$ and $b$ can be anything from real numbers to elements of an abstract normed vector space, where the elements don't have anything to do with triangles. For example you might be thinking of the space $C[0,1]$ of continuous real valued functions on $[0,1]$ with the norm $\|f\| = max|f(x)|$ on $[a,b]$. In that case you would see the triangle inequality written as $\|f+g\| \le \|f\| + \|g\|$. In almost all cases, you can't use strict inequality without making exceptions, so you basically never see it written with a strict inequality. The case of lengths of sides of a (non-degenerate) triangle is the only one that comes to mind where you get the strict inequality, and even then, you have to list the "exception" of non-degenerate.

8. ## Re: triangle inequality Originally Posted by DenisB sum of 2 shorter sides > longer side
In a triangle the sum of any two sides is greater than the third.

-Dan

9. ## Re: triangle inequality

why the name triangle inequality? It is misleading.

10. ## Re: triangle inequality Originally Posted by arangu1508 why the name triangle inequality? It is misleading.
You know the difference between the vector method and the triangle method by inspection.

And in any event the only time the two are different is if all the vectors are along one line.

-Dan

11. ## Re: triangle inequality Originally Posted by arangu1508 why the name triangle inequality? It is misleading.
Historical accidents are often misleading. This history begins with a simple statement:
The sum of the lengths to any two sides of a triangle is greater that the length of the third side. Someone called that the triangle inequality.
Then as number theory developed relations such as $|a-b|\le|a-c|+|c-b|$ which looks so much like the above quote it was natural to use the name triangle inequality.