Calculus by Spivak
Ch 1 Basic Properites of Numbers
(P1) Associative law for addition
If$ a, b \text{ and } care any numbers, then
$ \qquad a + (b + c) = (a + b) + c.
(P2) Existance of an additive identity
If$ ais any number, then
$ \qquad a+0=0+a=a.
(P3) Existance of additive inverses
For every number$ a, there is a number$ -asuch that
$ \qquad a+(-a)=(-a)+a=0.
(P4) Commutative law for addition
If$ a \text{ and } bare any numbers, then
$ \qquad a+b=b+a.
(P5) Associative law for multiplication
If$ a, b, \text{ and } care any numbers, then
$ \qquad a\cdot (b\cdot c)=(a\cdot b)\cdot c.
(P6) Existance of a multiplicative identity
If$ ais any number, then
$ \qquad a\cdot 1 = 1\cdot a = a.Moreover$ 1 \neq 0.
(P7) Existance of multiplicative inverses
For every number$ a\neq 0,there is a number$ a^{-1}such that
$ \qquad a\cdot a^{-1} = a^{-1}\cdot a = 1.
(P8) Commutative law for multiplication
If$ a \text{ and } bare any numbers, then
$ \qquad a\cdot b = b\cdot a.
(P9) Distributive law
If$ a, b, \text{ and }care any numbers, then
$ \qquad a\cdot (b+c) = a\cdot b + a\cdot c.
Let$ Pbe the collection of all positive numbers.
(P10) Trichotomy law
For every number$ a,one and only one of the following holds:
(i) $ a=0
(ii) $ ais in the collection$ P.
(iii) $ -ais in the collection$ P.
(P11) Closure under addition
If$ a \text{ and } bare in$ P,then$ a+bis in$ P.
(P12) Closure under multiplication
If$ a \text{ and } bare in$ P,then$ a\cdot bis in$ P.
Theorem 1
For all number $ aand$ b, we have
$ |a+b| \le |a|+|b|.