Field

Things like real numbers and complex numbers come from Fields.

A field is a set F with two binary operations on F called addition (+) and multiplication (*).

A binary operation on F takes the form

$F x F = F$

The operation can take any two elements of F and the result is an element of F. Note: this can mean taking the same element twice.

The Axioms below are true for a field:

Associativity

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

$(a*b)*c=a*(b*c)$

Commutativity

$a+b=b+a$

$(a*b=b*a$

Additive Identity

$a+0=0+a=a$

Multiplicative Identity

$a*1=1*a=a$

Additive Inverse

$a+(-a)=0$

Multiplicative Inverse

$a*\dfrac{1}{a}=1$

Distributivity

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