Sergei Yakovenko's blog: on Math and Teaching

Sunday, November 13, 2011

Lecture 3, Nov 10, 2011 (Thu)

Number systems

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

  1. Construct non-positive integers by adjoining “formal solutions” to the equations x+n=m for n\ge m
  2. Embed \mathbb N into \mathbb Z, identifying the above solution with the difference m-n for m>n.
  3. Define arithmetic operations on these “new numbers” via manipulations with the corresponding equations.
  4. Prove that with the “new numbers” the addition operation is always invertible, and x+n=m is always solvable with any n,m\in\mathbb Z.
  5. The construction can be essentially reproduced to define fractions as “formal solutions” to the equations of the form qx=p with p,q\in\mathbb Z. To avoid an obvious non-uniqueness, consider only case where p,q do not vanish simultaneously.
  6. Derive the formulas for addition/subtraction and multiplication/division. Note that these formulas sometimes give the forbidden combination 0\cdot x=0.
  7. Two ways to solve the problem:
    • keep the addition/subtraction always defined, but exclude the root of 0\cdot x=1, or
    • keep the “ideal element” and have a nice picture and lots of simplification in geometry, but live with arithmetic prohibitions.
  8. The ring \mathbb Q of rationals and the “circle” of the “rational projective line” \mathbb Q P^1:

Download the pdf file for the complete exposition.


  1. 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 | Reply

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


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

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