Let F denote either the set of real numbers or the set of complex numbers:

1. Addition is commutative: \(x+y = y+x, \forall x,y \in F\)
2. Addition is associative: \(x+(y+z) = (x+y)+z, \forall x, y, z \in F\)
3. There is a unique element \(0 \in F: x+0=x, \forall x \in F\)
4. Multiplication is commutative: \(xy = yx, \forall x,y \in F\)
5. To each \(x \in F\) there corresponds a unique element \((-x) \in F: x+(-x)=0\)
6. Multiplication is associative: \(x(yz) = (xy)z, \forall x,y,z \in F\)
7. There is an unique non-zero element \(1 \in F\) such that \(x.1 = x, \forall x \in F\)
8. To each non-zero \(x \in F\) there corresponds a unique element \(x^{-1} \in F\) such that \(xx^{-1}=1, \forall x \in F\)
9. Mulitiplication distributes over addition; that is, \(x(y+z) = xy+xz, \forall x,y,z \in F\)
10. Addition associates with each pair of elements \(x,y \in F\) an element \((x+y) \in F, \forall x,y \in F\)
11. Multiplication associates with each pair \(x, y\) an element \(x.y \in F, \forall x,y \in F\)

To summarize:

Property	Addition	Multiplication
Closure	\(x+y \in F\)	\(x.y \in F\)
Commutativity	\(x+y=y+x\)	\(x.y=y.x\)
Identity	\(x+0=x\)	\(x.1=x\)
Associativity	\(x+(y+z)=(x+y)+z\)	\(x(yz)=(xy)z\)
Inverse	\(x+(-x)=0\)	\(xx^{-1}=1\)
	Multiplication distributes over addition \(x(y+z)=xy+xz, \forall x,y,z \in F\)

• The set of rational numbers Q is a subfield of complex numbers C.
• The set of all complex numbers of the form \(x+y\sqrt{2}\) where x and y are rational, is a subfield of C.