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Memo

Properties of Real Numbers

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
$\forall a, b \in R \Rightarrow a + b \in R$	Closed Under Addition
$\forall a, b \in R \Rightarrow a - b \in R$	Closed Under Subtraction
$\forall a, b \in R \Rightarrow a \cdot b \in R$	Closed Under Multiplication
$\forall a, b \in R \Rightarrow a \div b \in / R$	Not Closed Under Division

Exercises

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