2 is an irrational number.[2 is not a rational number.]
Step-by-step explanation:
Let ✓2 is rational number
there exist positive integers a and b such that
✓2=a/b where, a and b, are co-prime i.e.
⇒(✓2)^2=(a/b)^2
=>2=a^2/b^2
=>2b^2=a^2
=> b^2=a^2/2
=>hence 2is factor of a^2
so 2 is factor of a --------( i)
Hence we can write a =2c
⇒(2c)^2/2=b^2
=> 4c^2/2=b^2
=> 2c^2=b^2
=> c^2=b^2/2
=> hence 2is factor of b^2
so 2 is factor of b -------- (ii)
From (i) and (ii), we obtain that 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. This means that our supposition is wrong.
Answers & Comments
Answer:
2 is an irrational number.[2 is not a rational number.]
Step-by-step explanation:
Let ✓2 is rational number
there exist positive integers a and b such that
✓2=a/b where, a and b, are co-prime i.e.
⇒(✓2)^2=(a/b)^2
=>2=a^2/b^2
=>2b^2=a^2
=> b^2=a^2/2
=>hence 2is factor of a^2
so 2 is factor of a --------( i)
Hence we can write a =2c
⇒(2c)^2/2=b^2
=> 4c^2/2=b^2
=> 2c^2=b^2
=> c^2=b^2/2
=> hence 2is factor of b^2
so 2 is factor of b -------- (ii)
From (i) and (ii), we obtain that 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. This means that our supposition is wrong.
Hence,
2 is an irrational number.