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:

P1

Addition is associative: $\forall a,b,c\quad a+(b+c)=(a+b)+c$ (``$\forall$'' means ``for every'').

P2

The number 0 is an additive identity: $\forall a\quad a+0=0+a=a$.

P3

Additive inverses exist: $\forall a \exists (-a){\text{ s.t.{} }} a+(-a)=(-a)+a=0$ (``$\exists$'' means ``there is'' or ``there exists'').

P4

Addition is commutative: $\forall a,b\quad a+b=b+a$.

P5

Multiplication is associative: $\forall a,b,c\quad a\cdot (b\cdot c)=(a\cdot b)\cdot c$.

P6

The number 1 is a multiplicative identity: $\forall a\quad a\cdot 1=1\cdot a=a$.

P7

Multiplicative inverses exist: $\forall a\neq 0 \exists a^{-1}{\text{ s.t.{} }}a\cdot a^{-1}=a^{-1}\cdot a=1$.

P8

Multiplication is commutative: $\forall a,b\quad a\cdot b=b\cdot a$.

P9

The distributive law: $\forall a,b,c\quad a\cdot(b+c)=a\cdot b+a\cdot c$.

P10

The trichotomy for ${\mathbb{P}}$: for every $a$, exactly one of the following holds: $a=0$, $a\in{\mathbb{P}}$ or $(-a)\in{\mathbb{P}}$.

P11

Closure under addition: if $a$ and $b$ are in $P$, then so is $a+b$.

P12

Closure under multiplication: if $a$ and $b$ are in $P$, then so is $a\cdot b$.

P13

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

The generation of this document was assisted by L^ATEX2HTML.