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*********IRRATIONAL NUMBER*********

Before defining irrational number let's know about other sets of numbers.
Natural numbers:Set of numbers starting from one.

Whole numbers: Set of numbers starting from zero is called whole number.

Integers:Set of numbers less than zero is represented by - negative sign. The set of whole numbers
and negative numbers is called integers.

Rational numbers:Set of numbers which can be expressed in the form p/q ,where p and q are both integers
and q is not equal to zero,p and q have no common factors other than one.
Irrational numbers:Set of numbers which can't be expressed in the form p/q
where p and q are both integers.
and q is not equal to zero,pand q have no common factors other than one.e.g:

Properties of irrational number:

*Sum of irrational no. and rational no. is irrational number.

*Product of non zero rational number and irrational number is irrational number

*Negative of a irrational number is irrational number

********CHECK OUT SOLVED EXAMPLES********

1)Insert five rational numbers between 1/2 and 1/3
Ans: (1/2)+(1/3) =5/6,mean value of 1/2 and 1/3=(5/6)/2=5/12
mean value of 1/2 and 5/12= 11/12= (11/12)/2 =11/24 = 11/24
mean value of 1/2 and 11/24 =(12+11)/24 = (23/24)/2 = 23/48
mean value of 1/3 and 3/8 = (1/3)+(3/8)= (17/24)/2=17/48
mean value of 1/3 and 5/12 = (9/12)/2 = 3/8
5/12,11/24,23/48,17/48,3/8 are five mean values.

check the answer!!!
 2) Prove that 7 1/2 is irrational number.
Ans: Let 7 1/2 be rational number.
(7)1/2 = p/q , such that it has no common factor other than 1...(i)
(7 1/2) 2 = (p/q) 2
7 = (p/q) 2 , (p) 2 =7(q)2
which shows 7 is the common factor which condradicts rational number defination or eq(1)
so, 7 is irrational number.

check the answer!!!
3) Arrange the following in Ascending order
7 1/2 ,3*(6) 1/2 , 5*(2)1/2,2*(3)1/2,181/2

7 1/2 = 7 1/2
3*(6) 1/2=(9*6) 1/2=(54) 1/2
5*(2)1/2= (25*2)1/2= (50)1/2
2*(3)1/2 = (4*3)1/2 = (12)1/2
181/2 = 181/2 = 181/2
Now in ascending order:

check the answer!!!
 4) Rationalise the denomenator of 3/((18)1/2-2)

9*(2)1/2+6)= 3*((18)1/2+2) /((18)1/2+2)*((18)1/2-2))

using formulae(a2-b2)=(a+b)(a-b)

= 9*(2)1/2+6)/(18-4) = 9*(2)1/2+6)/14

check the answer!!!
Smplify :3*((18)1/2+2) +((18)1/2+2)-((18)1/2-2)
=3*(18)1/2+(18)1/2-(18)1/2+ 6-2-2
= (3+1-1)*(18)1/2+2
= 3*(18)1/2+2
= 9*(2)1/2 + 2
check the answer!!!
 Find p+q,p-q ,p=((2)1/2+2) /((2)1/2-2)),q = ((2)1/2-2) /((2)1/2+2))

Ans:p+q = {((2)1/2+2) /((2)1/2-2))} + {((2)1/2-2) /((2)1/2+2))}

={((2)1/2+2) *((2)1/2+2))+ ((2)1/2-2)*((2)1/2-2)}/((2)1/2+2) *((2)1/2-2))

= ( 2+4 + 2*2*(2)1/2 ) +( 2+4 -2*2*(2)1/2 )/(2-4)

={6+4*(2)1/2 ) +( 6 -4*(2)1/2 )}/(-2)

= (12)/-2 = -6

p-q = {((2)1/2+2) /((2)1/2-2))} - {((2)1/2-2) /((2)1/2+2))}

={((2)1/2+2) *((2)1/2+2))- ((2)1/2-2)*((2)1/2-2)}/((2)1/2+2) *((2)1/2-2))

= ( 2+4 + 2*2*(2)1/2 ) - ( 2+4 -2*2*(2)1/2 )/(2-4)

={6+4*(2)1/2 ) -( 6 -4*(2)1/2 )}/(-2)

= 8*21/2/-2 = -4*21/2


check the answer!!!

********TRY BY YOURSELF********

1)Find four rational numbers between 1 and 2.

2) Find five rational no.between 5/6 and 6/5.

3)Insert five rational numbers between 1 and -1 excluding zero.

4)Find the mean values of following:
a) 3/4 and 4/6
b)4/3 and 7/3
c)5.66 and 6.77

5)Check this rational number whether it is terminating or non termenating decimal

 a) 13/6   b) 22/7  c) 45/9  d) 63/9  e) 123/4   f) 45/8


6) Check whether it is terminating , non terminating or non terminating repeating decimal.

 a) 453/3   b)567/7   c) 234/34    d)33333/11     c) 23444/45     f)23456/131      g) 23456/234
7) Prove that following are irrational no.

  a)4*(2)1/2      b)(10)1/2       c)7*(7)1/2   d)4*(12)1/2  e)(27)1/2


8)Check whether following are irrational or rational number:

 a) 4*(4)1/2    b)14*(64)1/2   c)274*(2)1/2   d)(29)1/2   e)(121)1/2


9) Which of the following are surds

 a)4*(2)1/2  b) 4*(2)1/2  c)4*(2)1/2  d)4*(2)1/2  e) 4*(2)1/2


 10)Find the square and cube of following surds.
    a)(32)1/2  b)a)41*(12)1/2 c)14*(27)1/3  d)4*(256)1/4  e)(36)1/3

ABOUT RATIONALISATION

1.Rationalise the denomenator of following irrational number:

   a)1/( 13(7)1/2)  b)3/ (3(5)1/2) c)5/(3(2)1/2) d)(3(12)1/2)/(3(21)1/2)

2.Rationalise the numerator of following irrational number:

  a) 23/(3)1/3 b) 4*(144)1/2/12 c)4*(12)1/2 d)4*(81)1/4

3. Find the square of following irrational number:

  a) 2*(2)1/2  b)14*(2)1/2 c)12*(12)1/2 d)(81)3/4 e)43*(2)5/6

4.Find the cube of following irraional number:

  a)3*(12)1/3 b)4*(9)1/2 c)4*(27)2/3d)4*(2)1/2

SIMPLIFICATION

1)3*(12)1/3 + 4*(9)1/2

2)3*(12)1/3 - 4*(9)1/2

3)3*(81)1/4 - (49)1/2

4)(3*(121)1/2)) /((4*(729)1/3)

5)(3*(256)1/3)*(2500)1/2

6)(6120)1/3/(8000)1/3

7) p = (3)1/2 +(2)1/2 ,q=(24)1/2 - (32)1/2 ,Find the value of following:

 a) p+q  b)p-q  c) p/q  d)p*q  e)(p+q)/(p-q) f)p-



                                                                                                                                                                                                                           
   

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