## Number systems

Leopold Kronecker (1823-1891) famously quipped, “God made the natural numbers; all else is the work of man”. So we start working.

- 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.

תיקונים קלים:

1. הנוסחה (6) צריכה הלהיות

$R_{qn\pm pm, mq}$

2.שלוש שורות מתחת, יש “=” מיותר

Comment by dmitrybat — Monday, November 14, 2011 @ 10:22 |

כולכם מוזמנים לחפש טעויות/שגיאות כתיב/אי-דיוקים וכו’ בכל החומרים שמפורסמים כאן.

דימה

Comment by dmitrybat — Monday, November 14, 2011 @ 10:38 |