1. TEACHER NAME:SOU.HAJARE S R
Subject :
Algebra

2. Topic :
Real Numbers

3. 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, …… .

4. 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.

5. 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, = 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

6. 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.

7. 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 = { ,-1,0,1, 2, }

8. 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 = { /2,0,1/ }

9. 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

10. *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.”

11. *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= …, 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.

12. 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

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

14. REPRESENTATION OF IRRATIONAL NUMBERS ON THE NUMBER LINE.

15. Example to find the HCF of 240, 128.
Solution:- 240=128 x Here the divisor is 128 and remainder is 112. Apply the division algorithm again on 128 and =112 x Now the divisor is 112 and the remainder is 16. We again apply the division algorithm on 112 and = 16 x 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.

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.

17. 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.

18. 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

19. DETERMIN H.C.F AND L.C.M OF 8, 25
Solution:- 8 = 2 x 4 = 2 x 2 x 2 = = 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.

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