Rational Number In Decimal Representation
Number system of Class 9
In this a finite number of digit occurs after decimal i.e. 1/2= 0.5, 0.6875, 0.15 etc.
NON-TERMINATING AND REPEATING (RECURRING DECIMAL):
In this a set of digits or a digit is repeated continuously.
PROPERTIES OF RATIONAL NUMBER
If a,b,c are three rational numbers.
(i) Commutative property of addition. a + b = b + a
(ii) Associative property of addition (a+b)+c = a+(b+c)
(iii) Additive inverse a + (-a) = 0
0 is identity element, -a is called additive inverse of a.
(iv) Commutative property of multiplications a.b. = b.a.
(v) Associative property of multiplication (a.b).c = a.(b.c)
(vi) Multiplicative inverse
1 is called multiplicative identity and 1/a is called multiplicative inverse of a or reciprocal of a.
(vii) Distributive property a.(b+c) = a.b + a.c
question Prove that is an irrational number.
Solution: Let where r be a rational number
Squaring both sides
⇒ 3 + 2 – 2 =r2
⇒ 5 – 2 =r2
Here, 5 –2 is an irrational number but r2is a rational number.
∴ L.H.S. ≠ R.H.S.
Hence it contradicts our assumption that is a rational number.
(b) Irrational Number in Decimal Form :
= 1.414213 ...... i.e. it is not-recurring as well as non-terminating.
= 1.732050807 ...... i.e. it is non-recurring as well as non-terminating.
question Insert an irrational number between 2 and 3.
question Find two irrational number between 2 and 2.5.
Solution: 1st Method :
Since there is no rational number whose square is 5. So is irrational..
Also is a irrational number.
2nd Method: 2.101001000100001.... is between 2 and and it is non-recurring as well as non-terminating.
Also, 2.201001000100001......... and so on.
question Find two irrational number between and .
Solution: 1st Method :
Irrational number betweenand
2nd Method : As = 1.414213562 ...... and = 1.732050808......
As , and has 4 in the 1st place of decimal while has 7 is the 1st place of decimal.
∴ 1.501001000100001......., 1.601001000100001...... etc. are in between and
question Find two irrational number between 0.12 and 0.13.
Solution: 0.1201001000100001......., 0.12101001000100001 .......etc.
question Find two irrational number between 0.3030030003..... and 0.3010010001 .......
Solution: 0.302020020002...... 0.302030030003.... etc.
question Find two rational number between 0.2323323332..... and 0.252552555255552.......
Solution: 1st place is same 2.
2nd place is 3 & 5.
3rd place is 2 in both.
4th place is 3 & 5.
Let a number = 0.25, it falls between the two irrational number.
Also a number = 0.2525 an so on.