Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4
Exercise 1.3
Question 1
Prove that is irrational .
Sol :
Let is a rational number.
Therefore, we can find two integers a , b such that
Let a and b have a common factor other than 1 . Then we can divide them by the common factor and assume that a and b are co-prime .
…..eq-(1)
Therefore, a2 is divisible by 5 and it can be said that a is divisible by 5.
Let a = 5k, where k is an integer and putting this value in eq-(1) , we get
(1)
This means that b2 is divisible by 5 and hence, b is divisible by 5.
And also this implies that a and b have 5 as a common factor .
And this is a contradiction to the fact that a and b are co-prime .
Hence, cannot be expressed as
or it can be said that
is irrational .
Question 2
Prove that is irrational
Sol :
Let is rational .
Therefore, we can find two integers a , b such that
Since a and b are integers, will also be rational and therefore
is rational.
This contradicts the fact that is irrational. Hence, our assumption that
is rational is false . Therefore ,
is irrational
Question 3
Prove that the following are irrationals:
(i)
Sol :
Let is rational .
Therefore, we can find two integers a , b such that
is rational as a and b are integers.
will also be rational and this contradicts the fact that
is irrational.
Hence, our assumption that is rational is false . Therefore ,
is irrational
(ii)
Sol :
Let is rational .
Therefore, we can find two integers a , b such that
is rational as a and b are integers.
will also be rational and this contradicts the fact that
is irrational.
Hence, our assumption that is rational is false . Therefore ,
is irrational
(iii)
Sol :
Let is rational .
Therefore, we can find two integers a , b such that
is rational as a and b are integers.
will also be rational and this contradicts the fact that
is irrational.
Hence, our assumption that is rational is false . Therefore ,
is irrational