Extension of Number systems

Slide 1: 

N = set of all natural numbers = { 1 ,2 ,3, 4 ,…..} Z = set of all integers = {…-3, -2, -1 , 0 , 1 , 2 , 3 , ..} Q = set of all rational numbers = { p / q : p & q are integers and q = 0} I = set of irrational numbers [ numbers which are not rational ] R = set of all real numbers[ rational numbers and irrational numbers ] C = set of all complex numbers [ set of all real numbers and including roots of negative numbers eg :  (-4 ) ]

Slide 2: 

N Z Q I R C N  Z  Q  I  R  C

Closure property in N : 

Closure property in N + -

An addition table for integers : 

An addition table for integers + As all the numbers in this addition table are integers, Z is closed under the operation addition . Same is the case with subtraction {addition of negative integer} and multiplication {repeated addition}.

Fundamental operation in Z & Q : 

Fundamental operation in Z & Q  Q - { 0 } is closed under the operation division.  n . d stands for not defined. + ½ does not belong to Z. Therefore Z is not Closed under the operation ‘division’.

Square roots in Q & R : 

Square roots in Q & R number Square root in Q 2 3 5 6 7 8 Square root in R number 2 Q I I Q I I I I Q number Square root in R n.d n.d n.d 0 1 2 3 Q U I = R