Extension of number system

EXTENTION OF NUMBER SYSTEMS: 

EXTENTION OF NUMBER SYSTEMS Created by : S. Gayathri For class ix

Slide 2: 

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 ) ]

Natural numbers on the number line: 

Natural numbers on the number line | | | | | | 1 2 3 4 5 6 . 1 is the smallest natural number. . There exist infinite natural numbers. . Here the natural numbers are represented by blue marks.

Closure property in N: 

Closure property in N + - N is not closed under - &  since all the numbers in the table doesnot belong to N. N is closed under +&x, since all the numbers in the table belongs to N.

Slide 5: 

| | | | | | | | | | | | | -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 INTEGERS ON THE NUMBER LINE All positive numbers , all negative numbers together with zero are called integers. There are infinite Integers. Number line is vacant in between two integers. At any point on the number line ,you find the numbers of greater magnitude as you move to the right and the converse is also true. >>> < < <

Q, the set of all rational numbers: 

Q, the set of all rational numbers In Q, in between any two rational numbers you can find infinitely many rational numbers. Therefore rational numbers are dense on the number line. Q Number line corresponding to Q has lot of gaps .Gaps corresponds to irrational numbers. Therefore rational numbers are not complete on the number line .

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’.

Slide 9: 

Belongs to does not belong to N + x N N Z + - x Z Z Q Q - {0} Q + - x  -   Q is closed under all the four fundamental mathematical operations, of course, Q - {0} is closed under division because division by 0 in not defined in mathematics.

Slide 10: 

REAL NUMBERS ON THE NUMBER LINE Q I Q U =R Numberline corresponding to R has no gaps . Real numbers are dense and complete on the number line. To every point on this number line there corresponds a real number and to every real number there corresponds a point on the number line. I

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

R and mathematical operations: 

R and mathematical operations R R + - X  R Q Q + - X  Q Square roots Non Perfect Square Numbers Square roots Non-Perfect Square Numbers Square roots negative Numbers Belongs to does not belong to

Slide 13: 

N Z Q I R C N  Z  Q  I  R  C

By adding irrational numbers to Q we get a system called ‘ set of all real numbers’ [ R ] Q and I are disjoint sets.: 

By adding irrational numbers to Q we get a system called ‘ set of all real numbers’ [ R ] Q and I are disjoint sets. Q I Q U I = R Q  I = 

In R square root of a negative integer is not possible.: 

In R square root of a negative integer is not possible. By adding new type [Square root -ve numbers] of numbers to R , we can get a bigger number system called the “system of complex numbers ” in which every real number has a square root . C

Slide 16: 

Tell me the smallest number system to which the following numbers belong ? 1. 4 + 4 2. 1 - 3 3. -7 x 2 4. 3  2 5. 4 6.  0. 4 7.  ( - 4 ) 8.  2. 5