Sergei Yakovenko's blog: on Math and Teaching

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” $\mathbb Q P^1$:

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

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

דימה

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

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