Real numbers have certain properties when it comes to operations, identities, and equalities. When performing operations on real numbers five different properties can be used.

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	$(a\cdot b)\cdot c=a\cdot (b\cdot c)$

These properties can be used when solving equations or simplifying expressions to easier find the correct solution. Real numbers have two important identities (equations that always hold true).

Additive Identity	$a+0=0$
Multiplicative Identity	$a\cdot 1=a$

The equalities for real numbers indicates that different operations can be performed on equations and still yield the same equation.

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 $\frac{a}{c}=\frac{b}{c}$

Finally, there are two special equalities; Symmetric Property of Equality and Transitive Property of Equality. Real numbers can be written in different ways. For example $0.75=\frac{3}{4}$ and $2.5=2\frac{1}{2}.$ The Symmetric Property of Equality then implies that the order does not matter. Given three real numbers, $a,b,$ and $c,$ the transitive property of equality refers to: Real numbers are said to be closed under addition and multiplication. That is, adding or multiplying two real numbers results in a real number.