# Triangle Inequality

Posted on the 20 January 2012 by Gaurav Tiwari @wpgaurav

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 .

Triangle

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.

Visual representation of Triangle inequality

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 lengths 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 lengths 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.
As,

or, in sigma summation:
.

## Triangle Inequality for Vectors

Theorem: If

and
are vectors in
(vector space in n-tuples or simply n-space), we have
.

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

or,

(
and so for
)
Similarly,
.
Comparing (1) and (2), we get that

Since,
.

## 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.
i.e.,

etc.

We define,

and
for any real number
.
Also we write

and
.

## 2. Distance in

If

and
.
We define a quantity
as

or, that is

and we describe
as the distance between the points
and
.

## 3. Norm

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:
A.
.
B.
.
C.
.
D.
.
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

.

Image via Wikipedia

From the definition of norm,

.
(Since
from Cauchy Schwartz Inequality)
We have

or,
.
Or,
.

Replacing

and
by
and
repectively, we obtain:

(from the definition of norm).
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