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

$a.(frac1a)=1$ where $a\ne 0$

Zero property of multiplication

$a.0=0$

Closure property of addition

$a+bisarealnumber===Closurepropertyofmultiplication===<math>a.bisarealnumber===Additionpropertyofequality===If<math>a=b$, then $a+c=b+c$

Substitution property

If $a=b,thenamaybesubstitutedforborconversely===Reflexive(oridentity)propertyofequality===<math>a=a===Symmetricpropertyofequality===If<math>a=b$, 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: $a<b,a=b,a>b$