Real number

Calculus is based in the system of real numbers and their properties.

Classification

Real numbers are classified as rational numbers (denoted by $Q$), integers ($Z$), whole numbers ($W$), natural numbers, and irrational numbers. In order of inclusion, non-irrational real numbers can be ordered as follows:

$N\subseteq W\subseteq Z\subseteq Q$

Properties

Distributive Property

$a.(b+c)=a.b+a.c$

Commutative property of addition

$a+b=b+a$

Commutative property of multiplication

$a.(b.c)=(a.b).c$

Aditive identity property

$a+0=a$

Multiplicative identiy property

$a.1=a$

Multiplicative identity property

$a.1=a$

Additive inverse property

$a+(-a)=0$

Multiplicative inverse property

$a.{\frac {1}{a}}=1$ where $a\neq 0$

Zero property of multiplication

$a.0=0$

Closure property of addition

$a+b$ is a real number

Closure property of multiplication

$a.b$ is a real number

Addition property of equality

If $a=b$, then $a+c=b+c$

Substitution property

If $a=b$, then $a$ may be substituted for $b$ or conversely

Reflexive (or identity) property of equality

$a=a$

Symmetric property of equality

If $a=b$, then $b=a$

Transitive property of equality

If $a=b$ and $b=c$, then $a=c$

Law of trichotomy

Exactly one of the following holds: $a<b,a=b,a>b$