Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If , and be the three sides of a triangle, then neither can be greater than , nor can be greater than , nor can be than .
Consider the triangle in the image, side shall be equal to the sum of other two sides and , only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, represents the side if represents side and represents side . A modulus is nothing, but the distance of a point on the number line from point zero.
For example, the distance of and from on the initial line is . So we may write that .
Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately.
Triangle Inequality for Real Numbers
For arbitrary real numbers and , we have
This expression is same as the length of any side of a triangle is less than or equal to (i.e., not greater than) the sum of the lenghts of the other two sides. The proof of this inequality is very easy and requires only the understandings of difference between ‘the values’ and ‘the lengths’. Values (like ) can be either negative or positive but the lenghts are always positive. Before we proceed for the proof of this inequality, we will prove a lemma.
Lemma: If , then if and only if .
Proof: ‘if and only if’ means that there are two things to proven: first if then , and conversely if then .
Proof: Suppose . Then . But since, can only be either or , hence . This implies that, .
Or, . (Proved!)
And conversely, assume . Then if , we have and from assumption, . Or . And also, if , . In either cases we have . (Proved!)
This is the proof of given lemma.
Now as we know and . Then on adding them we get
Hence by the lemma, . (Proved!)
Generalization of triangle inequality for real numbers can be done be increasing the number of real-variables.
or, in sigma summation:
Triangle Inequality for Vectors
Notations used in this theorem are such that represents the length (or norm) of vector in a vector space.
The length of a vector is defined as the square-root of scalar product of the vector to itself. i.e., .
Now, we can write
( and so for )
Comparing (1) and (2), we get that
Triangle Inequality for complex numbers
Theorem: If and be two complex numbers, represents the absolute value of a complex number , then
The proof is similar to that for vectors, because complex numbers behave like vector quantities with respect to elementary operations. You need only to replace and by and respectively.
Triangle Inequality in Eucledian Space
Before introducing the inequality, I will define the set of n-tuples of real numbers , distance in and the Euclidean space .
1. The Set
The set of all ordered n-tuples or real numbers is denoted by the symbol .
Thus the n-tuples
where are real numbers and are members of . Each of the members is called a Co-ordinate or Component of the n-tuple.
We shall denote the elements of by lowercase symbols , , etc. or simply , , ; so that each stands for an ordered n-tuple of real numbers.
and for any real number .
Also we write
2. Distance in
We define a quantity
or, that is
and we describe as the distance between the points and .
If , we write
so that is a non-negative real number. The number which denotes the distance between point and origin is called the Norm of . The norm is just like the absolute value of a real number. And also,
4. The Euclidean Space
The set equipped with all the properties mentioned above is called the Euclidean space of dimension .
Some major properties of the Euclidean Space are:
Properties A, B and C are immediate consequences of the definition of ). We shall now prove, property D, which is actually Triangle inequality.
Theorem:Prove that .
From the definition of norm,
(Since from Cauchy Schwartz Inequality)
Replacing and by and repectively, we obtain:
(from the definition of norm).