- Construct non-positive integers by adjoining “formal solutions” to the equations for
- Embed into , identifying the above solution with the difference for .
- Define arithmetic operations on these “new numbers” via manipulations with the corresponding equations.
- Prove that with the “new numbers” the addition operation is always invertible, and is always solvable with any .
- The construction can be essentially reproduced to define fractions as “formal solutions” to the equations of the form with . To avoid an obvious non-uniqueness, consider only case where do not vanish simultaneously.
- Derive the formulas for addition/subtraction and multiplication/division. Note that these formulas sometimes give the forbidden combination .
- Two ways to solve the problem:
- keep the addition/subtraction always defined, but exclude the root of , or
- keep the “ideal element” and have a nice picture and lots of simplification in geometry, but live with arithmetic prohibitions.
- The ring of rationals and the “circle” of the “rational projective line” :
Download the pdf file for the complete exposition.