Definition of an ordered field: An ordered field is a field containing a subset of elements closed under addition and multiplication and having the property that every element in the field is either 0, in the subset, or has its additive inverse in the subset.
Examples of ordered fields We will begin with the ones for addition:
A1. For all x, y ∈ R, x + y ∈ R and if x = q and y = z, then x + y = w + z
A2. For all x, y ∈ R, x+y=y+x
A3. For all x,y,z ∈ R, x+(y+z) = (x+y)+z
A4. There is a unique real number 0 such that x+0=x for all x ∈ R
A5. For each x ∈ R, there i a unique real number -x such that x+(-x) =0
Now here are the ones for multiplication
M1. For all x,y ∈ R, x⋅y ∈ R and if x=w and y=z, then x⋅y=w⋅z
M2. For all x,y ∈ R, x⋅y=y⋅x
M3. For all x,y,z ∈ R, x⋅(y⋅z)=(x⋅y)⋅z
M4. There is a unique real number 1 such that 1 =/= 0 and x⋅1 = x for all x ∈ R
M5. For each x ∈ R with x =/= 0 there is a unique real number 1/x such that x⋅(1/x)=1.
DL. For all x,y,z ∈ R, x⋅(y+z) x⋅y + x⋅z
These 11 axioms are called "field axioms" because they describe something called a "field" in algebra. Things like A2 and M2 are called commutative laws while A3 and M3 are called associative laws. DL stands for Distributive Law. Thanks to A1 and M1, we can think of addition and multiplication as functions that maps RxR into R.
The other basic operations are as follows: x-y is x+(-y), x/y is defined as x⋅1/y, 1+1=2 and x⋅x=x²
