IRRATIONAL NUMBERS
All real numbers which are not rational are called irrational numbers.
There are decimals which are non-terminating and non-recurring decimal.
Example: 0.303003000300003…
Hence, we can conclude that
An irrational number is a non-terminating and non-recurring decimal and cannot be put in the form
Solved Examples Based on Irrational Numbers
Question:
Prove that
Solution:
Let
\
Squaring both sides
Þ
Þ
Let p = 2r
Þ
Þ
Þ
Hence, p and q have 2 as a common factor or p and q are not co-prime.
So, our supposition is wrong.
\
Question:
Prove that
Solution:
Suppose
Þ
Squaring both sides
Þ
Þ
LHS is
So, our supposition is wrong.
Hence,
Question:
Show that there is no positive integer n for which
Solution:
Let there be a positive integer n for which
Which means
or
Rationalizing LHS, we get
Þ
Þ
Þ
or
Adding and subtracting (i) and (ii) we get
Þ
Þ RHS of both are rational.
\
Þ
This is impossible as any two perfect squares differ at least by 3.
Hence, there is no positive integer n for which (