TEACHER NAME:SOU.HAJARE S R Subject : Algebra

Topic : Real Numbers

REAL NUMBERS *Classify the following numbers into five groups natural numbers, whole numbers, rational numbers and Irrational numbers . 2/3, 7, -4/5, 0, -10, 0.3, √2, √3, 0.10111213…… .

Let us compare these answers : Classification Natural numbers Whole numbers Integers Rational numbers Irrational numbers Ameya 7 -10 0.3 √2 Bismilla 0,7 0,7,-10 2/3,-0.3 √2, √3 Chaitanya 0,7,-10,2/3,-4/5,0.6 0.1011… Let us compare these answers : Here we observe that thinking process of Chaitanya about number systems is more clear than others.

Natural numbers:-The counting numbers 1,2,3,4, … are called natural numbers. We write them as a set N = {1,2,3,4, …} Is the sum of two natural numbers a natural number ? Dolly says, "yes.” Since 4 is N, 7 is N, 4 + 7= 11 is N . Is the product of two natural numbers a natural number ? Elizabeth says, ”yes.” Since 5 is N and 8 is N, 5 x 8 = 40 is N

Is the subtraction of two natural numbers a natural number ? Fakir says, “No.” Since 3 is N , 7 is N, 3-7 = -4 N What do we conclude from above three examples? Teacher says that the set of natural numbers is adequate for addition and multiplication but not for subtraction.

Whole Numbers : The union of set of natural numbers and zero is a set of WHOLE NUMBERS & the set is denoted by ‘W’. W = {0,1,2,3,4,. . . . . . . .} Integer : The set of natural numbers, zero and opposite of all natural numbers is called as set of integers and denoted by I. I = {. . . . . -2,-1,0,1, 2, . . . . .}

Is the division of two integers a integer? Govind says, “No.” Since 2 is I, 5 is I, but 2/5 is I. Rational Number : If p & q are integers (q‡0) then the number p/q is called a rational number and the set of rational number is denoted by Q. Q = {. . . . . -1/2,0,1/2 . . . . .}

Are all the natural numbers a rational number ? Hemant says, “yes.” Since 2 is N and 2= 2/1 is I Are all the whole numbers a rational number ? Indu says, “yes.” Since 0 is W and 0= 0/5 is I Are all the integers a rational number ? Jaheer says, ”yes.” Since –3 is I and -3= -3/1 is I Rational numbers include natural numbers, whole numbers and integers

*Natural numbers:- The smallest natural number is 1, and the largest number cannot be defined. *Whole numbers:- The smallest whole number is 0. *Integers:-The German word “Zahlen” means to count and “zahl” means number. *Rational number:- The word rational is derived from the word “Ratio.”

*Equivalent numbers:- 1/3 = 2/6 = 3/9…… These are equivalent numbers. *1/2=0.5, 1/3=0.333…, 1/4=0.25, 1/5=0.2, 1/6=0.161616…, 1/10=0.1. NOTE:-Rational whose denominators factors are 2 ,5 or 2 and 5 are terminating decimals. *Terminating decimals and non terminating recurring decimals are both rational numbers. *Rational numbers include natural numbers, Whole numbers and integers. Note : Terminating decimals and non-terminating recurring decimals are both rational numbers.

TO FIND RATIONAL NUMBERS BETWEEN TWO RATIONALS If a & b are two rational numbers then (a+b)/2 is between a and b. For ex. Between 3 and 4 there is a number (3+4)/2, i.e., 7/2. Between 3 and 7/2 there is a number (3+7/2)/2=13/4. Also, between 3/2 and 4 there is a number (3/2+4)/2=11/4. . CONCLUSION : There are infinite rational numbers between two rational numbers

*Archimedes was a Greek genius who first calculated the digits in decimal expansion of π. He found the value of π lies between 3.140845 and 3.142857. 3.140845 < π < 3.142857. *Aryabhatta was the greatest Indian mathematician who found the value of π correct to four decimal places. π =3.1416 (approximately)

REPRESENTATION OF IRRATIONAL NUMBERS ON THE NUMBER LINE.

Example to find the HCF of 240, 128. Solution:- 240=128 x 1 +112 Here the divisor is 128 and remainder is 112. Apply the division algorithm again on 128 and 112. 128=112 x 1 + 16. Now the divisor is 112 and the remainder is 16. We again apply the division algorithm on 112 and 16. 112 = 16 x 7 + 0 Now remainder is 0 and HCF of 112 and 16 is 16. Therefore we can say that HCF of 240 and 128 is 16. Here HCF(240,128)= HCF of (128,112)= HCF of (112,16)=16.

We know that any natural number can be expressed as a product of primes and this is unique apart from the order in which the prime factors occur. For ex. , 3 = 3 x 1, 4 = 2 x 2 = 22 , 6 = 2 x 3, 8 = 2 x 2 x 2 = 23, 36 = 2 x 2 x 3 x 3 = 22 x 32, 192 = 8 x 24 = 4 x 2 x 4 x 6 = 2 x 2 x 2 x 2 x 2 x 2 x 3 = 26 x 3 In general when a composite number decomposes into primes and are written in ascending order.

FUNDAMENTALTHEOREM Every composite number can be expressed as a product of prime numbers and this factorization is unique apart from the order in which the prime factor occur . This fundamental theorem of arithmetic helps us to find H.C.F and L.C.M of numbers.

Determine H.C.F. and L.C.M. of 120, 90 Solution:- 120 = 8 x 15 = 2 x 2 x 2 x 3 x 5 = 23 x 3 x 5 90 = 6 x 15 = 2 x 3 x 3 x 5 = 2 x 32 x 5 H.C.F. = 21 x 31 x 51 = 30

DETERMIN H.C.F AND L.C.M OF 8, 25 Solution:- 8 = 2 x 4 = 2 x 2 x 2 = 23 25 = 5 x 5 = 52 H.C.F of (8, 25) = 1 **No common factor between 8 and 25 L.C.M of (8, 25) = 8 x 25 = 200 NOTE: H.C.F of any two or more prime numbers is always 1. L.C.M of any two or more prime numbers is equal to their products.

Hajare S. R. Creator : üÖ Std - 9th Subject- Algebra B.Sc.B.Ed Std - 9th Subject- Algebra Topic- Real Numbers