Triangle inequality proof

In summary, the Triangle Inequality Proof is a fundamental mathematical concept that states the sum of any two sides of a triangle must be greater than the third side. It is important in various fields and can be proven using the triangle inequality theorem or the Pythagorean Theorem. Its real-world applications include navigation, engineering, and physics. However, there are exceptions to the Triangle Inequality, such as in non-Euclidean geometries or certain cases of degenerate or spherical triangles.
  • #1
pzzldstudent
44
0
My professor said this was the triangle inequality. We're to use mathematical induction to prove it. I've gotten some work done, and after "proving" it, it just seems to easy. :|

http://answerboard.cramster.com/advanced-math-topic-5-321495-0.aspx" [Broken].
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
The work you've done is essentially correct - the idea is to write

[tex]
|a_1 + a_2 + \dots + a_k + a_{k+1}| = |(a_1 + a_2 + \dots + a_k) + a_{k+1}|
[/tex]

then use the hint once, then your induction hypothesis.
 
  • #3
thank you

thanks!
 

What is the Triangle Inequality Proof?

The Triangle Inequality Proof is a mathematical concept that states that the sum of any two sides of a triangle must be greater than the third side. In other words, the shortest distance between two points is a straight line.

Why is the Triangle Inequality Proof important?

The Triangle Inequality Proof is important because it is a fundamental property of triangles and is used in various mathematical and scientific fields. It helps us understand the relationship between the sides of a triangle and is essential in geometry, trigonometry, and calculus.

How do you prove the Triangle Inequality?

The Triangle Inequality can be proven using the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. This can also be demonstrated using the Pythagorean Theorem and the concept of distance between two points.

What are the real-world applications of the Triangle Inequality Proof?

The Triangle Inequality Proof has many real-world applications, such as in navigation and mapmaking, where it is used to calculate the shortest distance between two points. It is also used in engineering, architecture, and physics to determine the strength and stability of structures.

Are there any exceptions to the Triangle Inequality?

Yes, there are exceptions to the Triangle Inequality, such as in non-Euclidean geometries, where the Triangle Inequality does not hold. Additionally, in certain cases, the Triangle Inequality may hold for some triangles but not for all, such as in degenerate triangles or triangles on a sphere.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Precalculus Mathematics Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
6K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
5K
  • Topology and Analysis
Replies
3
Views
1K
Back
Top