class 9 maths REAL NUMBERS We have learnt about various types of numbers in our earlier classes. Let us review them and know more about numbers. Natural numbers: Counting numbers are known as natural numbers. We always start counting anything from 1 always. So natural numbers starts from 1. Thus, 1, 2, 3, 4, 5, 6, etc., are all natural numbers. Whole numbers: All natural numbers together with 0 form the collection of all whole numbers. Thus, 0, 1, 2, 3, 4, 5, 6, etc., are all whole numbers. ØEvery natural number is a whole number. Ø0 is a whole number which is not a natural number Integers: All natural numbers, 0 and negative of natural numbers form the collection of all integers. Rational numbers whose denominator is 1 is called Integer. Thus, –4, –3, –2, –1, 0, 1, 2, 3, 4, etc. are all integers. ØEvery natural number is an integer. ØEvery whole number is an integer. p q
Rational numbers: Any number of the form , where p and q are integers and q¹ 0, are known as rational numbers. Operations on rational numbers: For any two rational numbers p/q and m/n, we define Addition + = Subtraction
- =
Multiplication
× =
Division
÷ = × =
,
where m≠0, n≠0
SIMPLEST FORM OF A RATIONAL NUMBER: A rational number
p is said to be in simplest form, if p and q are integers having no common factor q
other than 1 (i.e. both p and q are relatively co-prime). Here q¹ 0 2 3 4 5 6 1 , , , etc. is . 4 6 8 10 12 2
Thus, the simplest form of each of , Similarly, the simplest form of
6 2 76 4 is and that of is . 9 3 133 7
TO FIND RATIONAL NUMBER(S) BETWEEN TWO GIVEN RATIONAL NUMBERS: Method 1.
Let x and y be two rational numbers such that x<y. Then,
Method 2.
1 ( x + y) is a rational number between x and y. 2
Let x and y be two rational numbers such that x<y. Suppose we want to find n rational numbers between x and y. Let d =
y-x . n +1
Then, n rational numbers between x and yare: ( x + d), ( x + 2d), ( x + 3d), ..., ( x + nd)
Ex.
Find three rational numbers between -
Sol.
Let x = - , y = -
3 7
d=
y-x = n +1
-
3 1 and - . 7 7
1 and n = 3. 7
1 æ 3ö - ç- ÷ - 1 + 3 7 è 7ø = 7 7 3 +1 3 +1
-1+ 3 7 = 2´1 = 1 = 4 7 4 14
So, three rational numbers between -
3 1 and - are 7 7
( x + d), ( x + 2d), ( x + 3d) æ 3 1öæ 3 2öæ 3 3ö ç - + ÷, ç - + ÷, ç - + ÷ è 7 14 ø è 7 14 ø è 7 14 ø
Þ-
- 6 +1 - 6 + 2 - 6 + 3 , , 14 14 14
5 4 3 3 1 , - , - are the three rational numbers between - and 14 14 14 7 7
DECIMAL REPRESENTATION OF RATIONAL NUMBERS: Every rational number can be represented either as non-terminating but repeating (recurring) decimal. For example:
a
terminating
2 1 8 4 9 5 = 0.80, = 1.8, = 0.625 and = 0.666..., = 0.1666..., = 1.142857142857... 5 !!!5! 8 !!! 3 !!!!! 6 !!!!#!7!!!!!!!!" $ !#!! " $ Ter min ating
Thus
Non - ter min ating but repeating
–4, –3, –2, –1, 0, 1, 2, 3, 4, etc. are all integers.
decimal
or
a