02. Axioms for arithmetic in ℝ

Axioms for Arithmetic in $\mathbb{R}$

Sum: Unique real number $a+b$ for every $a,b\in \mathbb{R}$
Product: Unique real number $a\cdot b$ for every $a,b\in \mathbb{R}$
Additive Inverse / Negative: Unique real number $-a$ for $a\in \mathbb{R}$
Multiplicative Inverse / Reciprocal: Unique real number $\frac{1}{a}$ for $a\in \mathbb{R}$ with $a\ne 0$
Zero / Additive Inverse: Special element $0\in \mathbb{R}$
One / Multiplicative Identity: Special element $1\in \mathbb{R}$

For $a,b,c\in \mathbb{R}$ we also have
$+$ is commutative:

$$a+b=b+a$$

$+$ is associative:

$$a+(b+c)=(a+b)+c$$

Additive Identity:

$$a+0=a$$

Additive Inverse:

$$a+(-a)=0$$

$\cdot$ is commutative:

$$a\cdot b=b\cdot a$$

$\cdot$ is associative:

$$a\cdot (b\cdot c)=(a\cdot b)\cdot c$$

Multiplicative Identity:

$$a\cdot 1=a$$

Multiplicative Inverse:

$$\text{ if }a\ne 0\text{ then}a\cdot \frac{1}{a}=1$$

$\cdot$ distributes over $+$:

$$a\cdot (b+c)=a\cdot b+a\cdot c$$

The first 4 are addition and the rest are for multiplication

To Avoid Total Collapse

$$0\ne 1$$

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Field $\mathbf{F}$

Let $\mathbf{F}$ be a set with operations $+$ and $\cdot$ that satisfy the axioms
Then

$$\mathbf{F}\text{ is a field}$$

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Example of Fields

1. $\mathbb{R}$ - Real numbers
2. $\mathbb{Q}$ - Rational numbers
3. $\mathbb{C}$ - Complex numbers

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