Can the sum of two sides of a triangle be less than the third side I was studying elementary vectors and complex numbers together, when I thought: is it possible for the sum of two sides of a triangle be less than the third side.
Can a triangle with complex sides (or any type of numbers invented by man) ever be such that: sum of its two sides is less than the third?
 A: The triangle inequality
$$
||x+y||\le ||x||+||y||
$$
is one of the defining properties of a norm in a vector space, so in  any normed vector space this inequality is verified.
But, if we relax this request, we can have a vector space that is equipped with a symmetric bilineaar form that is not positive defined, and, from this form, we can define a ''distance'' that does not necessarlily satisfies the triangle inequality.
We can construct a vector space of this kind using immaginary coordinates as in the case of the relativity theory, that can be formulated in a space with three real coordinates (the ''space'' coordinates) and one imaginary coordinates ( the ''time'' coordinate). This space is called  a Minkowski space-time and  in it we can define a ''Minkowski distance''  that is not positive defined. This means that the square of the distance between two points (events) of this space can be negative. 
In this space there are points such that the trangle inequality is reversed, in the sense that we have:
$$
||x+y||\ge ||x||+||y||
$$
