NUMBER THEORY

NUMBER:

$Unique\; sequence\; of\; elements\; which\; used\; to\; count\; the\; collection\; of\; individuals\; OR\;$

$Mathematical\; object\; used\; to\; count\;, \;measure\; and\; label .$

TYPES OF NUMBER

$NATURAL\; NUMBER:$          $\mathbb{N}=\{1,2,3,...\}$

$WHOLE\; NUMBER:$             $\mathbb{W}=\{0,1,2,3,...\}$

$INTEGER:$                               $\mathbb{Z}=\{0,±1,±2,…\}$

$RATIONAL\; NUMBER:$        $\mathbb{Q}=\{x|x =p/q\;  and\; p,q\in \mathbb{Z} ,q\neq 0\}$

$IRRATIONAL\; NUMBER:$   $\mathbb{Q}^{'}=\{x|x\neq p/q\;  and\; p,q\in\mathbb{Z},q\neq 0\}$

$REAL\; NUMBER:$                    $\mathbb{R}=\mathbb{Q}\cup\mathbb{Q}^{'}$

$COMPLEX\; NUMBER$           $\mathbb{C}=\{x+\iota y|x,y\in R\}\; where\; \iota\; is\; imaginary\; unit.$

NOTE:

$\star\;\mathbb{N}\subset \mathbb{W}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset \mathbb{C}$

NUMBER THEORY (HIGH ARITHMETIC):

$Mathematical\; theory\; that\; studies\; the\; properties\; and\; relation\; of\; integers.$

POLAR FORM OF COMPLEX NUMBER

$Z=x+iy=r(\cos\theta+\iota\sin\theta)=rcis\theta$

$Where\; r=|Z|=\sqrt{(x^2+y^2 )}\;and$

$Principle\; Value\; \theta=\tan^{-1}(\frac{y}{x})=Arg(Z)        \;\;0\leq\theta\leq2\pi$

$Normal \;Value\; arg(Z)=Arg(Z)+2n\pi \;\;-\pi\leq\theta\leq\pi$

$\circ\;This\; is\; polar\; form\; of\; (x,y)\; in\; polar \;co-ordinates\; (r,\theta)$