Why do we need negative one?

In terms of algebraic structures, it is to ensure that the set of integers forms a group: that is, the elements of the set have an additive inverse.

For the same reason we need all the other negative numbers. Negative numbers are regularly used whenever there are two opposites - in which case, one of the opposites is assigned to positive numbers, the other to negative numbers. For example:* Height above (or below) sea level

* Having (or owing) money

* Gaining (or losing) money

* Electrical charges

* Moving forward (or moving backward) - along a line

* Etc.