Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: (7) Next: Homework Assignment 2
Previous: Class Notes for Thursday September 11, 2003

The 13 Postulates

this document in PDF: Postulates.pdf

Everything you ever wanted to know about the real numbers is summarized as follows. There is a set $ {\mathbb{R}}$ ``of real numbers'' with two binary operations defined on it, $ +$ and $ \cdot$ (``addition'' and ``multiplication''), two different distinct elements 0 and 1 and a subset $ {\mathbb{P}}$ ``of positive numbers'' so that the following 13 postulates hold:

Addition is associative: $ \forall a,b,c\quad a+(b+c)=(a+b)+c$ (``$ \forall$'' means ``for every'').
The number 0 is an additive identity: $ \forall a\quad a+0=0+a=a$.
Additive inverses exist: $ \forall a \exists (-a){\text{ s.t.{} }}
a+(-a)=(-a)+a=0$ (``$ \exists$'' means ``there is'' or ``there exists'').
Addition is commutative: $ \forall a,b\quad a+b=b+a$.
Multiplication is associative: $ \forall a,b,c\quad a\cdot (b\cdot
c)=(a\cdot b)\cdot c$.
The number 1 is a multiplicative identity: $ \forall a\quad
a\cdot 1=1\cdot a=a$.
Multiplicative inverses exist: $ \forall a\neq 0 \exists
a^{-1}{\text{ s.t.{} }}a\cdot a^{-1}=a^{-1}\cdot a=1$.
Multiplication is commutative: $ \forall a,b\quad a\cdot b=b\cdot a$.
The distributive law: $ \forall a,b,c\quad a\cdot(b+c)=a\cdot
b+a\cdot c$.
The trichotomy for $ {\mathbb{P}}$: for every $ a$, exactly one of the following holds: $ a=0$, $ a\in{\mathbb{P}}$ or $ (-a)\in{\mathbb{P}}$.
Closure under addition: if $ a$ and $ b$ are in $ P$, then so is $ a+b$.
Closure under multiplication: if $ a$ and $ b$ are in $ P$, then so is $ a\cdot b$.
The thirteenth postulate is the most subtle and interesting of all. It will await a few weeks.

Here are a few corollaries and extra points:

  1. Sums such as $ a_1+a_2+a_3+\dots+a_n$ are well defined.
  2. The additive identity is unique. (Also multiplicative).
  3. Additive inverses are unique. (Also multiplicative).
  4. Subtraction can be defined.
  5. $ a\cdot b=a\cdot c$ iff (if and only if) $ a=0$ or $ b=c$.
  6. $ a\cdot b=0$ iff $ a=0$ or $ b=0$.
  7. $ x^2-3x+2=0$ iff $ x=1$ or $ x=2$.
  8. $ a-b=b-a$ iff $ a=b$.
  9. A ``well behaved'' order relation can be defined (i.e., the booloean operations $ <$, $ \leq$, $ >$ and $ <$ can be defined and they have some expected properties).
  10. The ``absolute value'' function $ a\mapsto \vert a\vert$ can be defined and for all numbers $ a$ and $ b$ we have

    $\displaystyle \vert a+b\vert\leq \vert a\vert+\vert b\vert. $

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Dror Bar-Natan 2003-08-14