Creation of negative numbers

By the same scheme, transform

$$f(x)=x + A$$

(where $A$ is a natural number), if applied to natural numbers, leads to negative integers.

1. Original set: Consider the set of natural numbers
   
   $$1, 2, 3, 4, \ldots$$
   
   

2. Image: Find the image of the set under the transform
   
   $$f(x)=x + 3.$$
   
   
   Set of images is
   
   $$4, 5, 6, 7, \ldots$$
   
   

3. Complement: The set of images is naturally complemented to the set $Z$ of all natural numbers:
   
   $$Z= \{ 1, 2, 3, 4, 5, 6, 7, \ldots \}$$
   
   

4. Inverse Transform: Now apply the inverse transform
   
   $$f^{-1}(x)=x - 3$$
   
   
   to the complemented set $Z$ of images
5. Negative Numbers!

   We arrive at new numbers: These are new negative numbers.
   

   $$f^{-1}( Z ) =\{ -2, -1, 0, 1, 2, 3, 4, 5 \ldots \}$$