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:

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

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

$a+b=b+a$

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

$a+0=a$

$a.1=a$

$a.1=a$

$a+(-a)=0$

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

$a.0=0$

$a+b$ is a real number

$a.b$ is a real number

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

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

$a=a$

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

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

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