The Set R of Rational Numbers - Jim Zenn

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Properties: Properties of Fields, in Particular $\Q$

Given the definition of addition and multiplication over a field, where the sum $a + b$ and the product $ab$ also represent rational numbers, the following properties hold in the field:

A1 (Associative Law of Addition). $a + (b+ c) = (a + b) + c, \forall a, b, c $

A2 (Commutative Law of Addition). $a+ b = b + a, \forall a, b$

A3. $a + 0 = a, \forall a $

A4. $\forall a, \exists -a $ such that $a + (-a) = 0 $

M1 (Associative Law of Multiplication). $a(bc) = (ab)c, \forall a, b, c $

M2 (Commutative Law of Multiplication). $ab = ba, \forall a, b $

M3. $a \cdot 1 = a, \forall a $

M4. $\forall a \neq 0, \exists \inv{ a } such that a \inv{ a } = 1 $

DL (Distributive Law). $a(b+c) = ab + ac, \forall a, b, c $

In particular, the set $\Q$ is a field and these properties hold in $\Q$.

Properties: The Ordering Structure of $\Q$

O1. Given $a $ and $b $, either $a \leq b $ or $b \leq a $.

O2. If $a \leq b $ and $b \leq a $, then $a = b $.

O3 (Transitive Law). If $a \leq b $ and $b \leq c $, then $a \leq c $.

O4. If $a \leq b $, then $a +c \leq b + c$.

O5. If $a \leq b $ and $0 \leq c $, then $ac \leq bc $.

Fields with these properties are called ordered fields.

Theorem : Consequences of Field Properties

$a + c = b + c \implies a = b$
$a \cdot 0 = 0, \forall a$
$(-a)b = -ab, \forall a,b$
$(-a)(-b) = ab, \forall a, b$
$ac = bc$ and $c \neq 0 \implies a = b$
$ab = 0 \implies a = 0$ or $b = 0 , \forall a, b, c \implies \R$

Proof

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Proof

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Theorem : Consequences of the Ordered Field Properties

Define $a < b $ as $a \leq b $ and $a \neq b $.

If $a \leq b $, then $-b \leq -a $.
if $a \leq b $ and $c \leq 0 $, then $bc \leq ac $.
If $0 \leq a $ and $0 \leq b $, then $0 \leq ab $.
If $0 \leq a^2, \forall a $.
$0 < 1 $.
If $0 < a$, then $0 < \inv{ a } $.
If $0 < a < b $, then $0 < \inv{ b } < \inv{ a }$, for $a, b, c \in \R $.

Definition: Absolute Value

$\abs{ a } = a $ if $ a \geq 0 $ and $\abs{ a } = -a $ if $ a \leq 0 $.

$\abs{ a } $ is called the absolute value of $a$.

Definition: Distance

For numbers $ a $ and $ b $, we define the distance between $ a $ and $ b $ $\dist{a, b} = \abs{ a - b }$.

Properties: Properties of the Absolute Value

$\abs{ a } \geq 0, \forall a \in \R $
$\abs{ ab } = \abs{ a } \cdot \abs{ b }, \forall a, b \in \R $
$\abs{ a + b } \leq \abs{ a } + \abs{ b }, \forall a, b \in \R $

Corollary : Triangle Inequality

$\dist{ a, c } \leq \dist{ a, b } + \dist{ b, c }, \forall a, b, c \in \R$