# Triangle Inequality

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 $a$ , $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$ , nor$b$ can be greater than $c+a$ and so $c$ can not be greater than $a+b$ .

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$ , 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, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$ . A modulus is nothing, but the distance of a point on the number line from point zero.

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$ . So we may write that $|5|=|-5|=5$ .

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 $x$ and $y$ , we have
$|x+y| \le |x|+|y|$ .
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 $4.318, 3, -7, x$ ) 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.