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From Karnataka Open Educational Resources
→Hints for difficult problems
= Hints for difficult problems =
= Hints for difficult problems =
Irrational numbers
Exercise 1.3.1
1.Write four possible irrational numbers between 4 and 5.
Solution:
Consider the squares of 4 and 5
42 = 16 and 52 = 25
We can also wright 4 and 5 as
=4 and =5
Between and there exists
,,,,,,,
2.Write four rational numbers between and
Solution:
root 2 ~1.414 and
root 3 ~ 1.732
Rational numbers of root2 and root 3 is in between 1.4 to 1.7 like 1.45,1.5,1.55, 1.6 ........
3.Prove that + is an irrational number
Solution:
Let us assume on contrary that is a rational number.
Then there exists co-prime positive integers p and q such that
=
-=
(-)2 = ()2
+2- 2.= 5
=
p and q are integers
is rational
But this contradicts the fact that is irrational.
So, our assumption is wrong.
Hence, is irrational
Write an irrational
= Project Ideas =
= Project Ideas =
