Real numbers have certain properties when it comes to operations, identities, and equalities.

Operations
Name	Property
Distributive Property	$a\cdot (a+c)=a\cdot b+a\cdot c$
Commutative Property of Addition	$a+b=b+a$
Commutative Property of Multiplication	$a\cdot b=b\cdot a$
Associative Property of Addition	$(a+b)+c=a+(b+c)$
Associative Property of Multiplication	$(a\cdot b)\cdot c=a\cdot (b\cdot c)$

Identities
Name	Property
Additive Identity	$a+0=0$
Multiplicative Identity	$a\cdot 1=a$

Equalities
Name	Property
Addition Property of Equality	If $a=b,$ then $a+c=b+c$
Subtraction Property of Equality	If $a=b,$ then $a-c=b-c$
Multiplication Property of Equality	If $a=b,$ then $a\cdot c=b\cdot c$
Division Property of Equality	If $a=b,$ then $\frac{a}{c}=\frac{b}{c}$
Reflexive Property of Equality	Any real number is equal to itself
Symmetric Property of Equality	If $a=b,$ then $b=a$
Transitive Property of Equality	If $a=b$ and $b=c,$ then $a=c$

Closure Property of Real Numbers
$\forall a,b\in \mathbb{R}\Rightarrow a+b\in \mathbb{R}$	Closed Under Addition
$\forall a,b\in \mathbb{R}\Rightarrow a-b\in \mathbb{R}$	Closed Under Subtraction
$\forall a,b\in \mathbb{R}\Rightarrow a\cdot b\in \mathbb{R}$	Closed Under Multiplication
$\forall a,b\in \mathbb{R}\Rightarrow a÷b\notin \mathbb{R}$	Not Closed Under Division