What is the definition of distance mean in math?

Answer 1

There are many different ways to define distance, and these definitions lead to different metric spaces.

Here are three examples that may help illustrate different ways of defining distances:

The Euclidean distance between two points is the length of the straight line joining the two points.

However, that definition does not work in the context of distant locations on the surface of a sphere - such as the earth. The straight line would usually require tunnelling through the earth. To find the shortest distance form London to Sidney, for example, you need to first establish the "great circle" which is a circle centred at the centre of the earth and passing through the two cities. The distance between the two cities in is the length of the minor arc of this great circle.

Another metric was developed by Minkowski and known as the Manhattan or taxicab distance. This defines the distance between two points as the sum of the distances between them in the North-South direction and East West direction. The concept is based on the grid-like layout of downtown Manhattan and the metric represents the road distance between two locations (although it seems to ignore 1-way streets!).

Each one of these is a different definition of distance - appropriate in some circumstances. There are many more possible definitions. In fact many graduates in pure mathematics will have done a course in metric spaces which examines different ways of defining "distance" and their implications on "measure".