They are closed under addition: $a\+b$ is a real number

They are closed under multiplication: $a\cdot b$ is a real number

Addition is associative: $a\+\left(b\+c\right)\=\left(a\+b\right)\+c$ 

Multiplication is associative: $a\cdot \left(b\cdot c\right)\=\left(a\cdot b\right)\cdot c$

Addition is commutative: $a\+b\=b\+a$ 

Multiplication is commutative: $a\cdot b\=b\cdot a$

They are distributive: $a\cdot \left(b\+c\right)\=a\cdot b\+a\cdot c$

0 is the additive identity: $a\+0\=a$

1 is the multiplicative identity: $a\cdot 1\=a$

Each number has an additive inverse: $a\+\left(-a\right)\=0$

Each number except for 0 has a multiplicative inverse: $a\cdot a^{-1}\=1$