Prove that 3+ root 5 is irrational
Answers
Here,
Let us assume that 3 + √5 is a rational number.
Then we can find co prime numbers a and b such that 3 + √5 = a/b
⇒ √5 = a/b - 3
⇒ √5 = a-3b/b
∴ a and b are integers , a-3b/b is a rational.
But we know that √5 is an irrational number.
Our assumption is wrong as irrational number ≠ rational number. This contradiction is due to our assumption that 3 + √5 is a rational number. rational number.
So, 3 + √5 is an irrational number.
Answer:
Step-by-step explanation:
Let
3 + √5 is a rational number.
we can find co prime numbers p and q of 3+ root 5
3 + root 5 = p/q [ where p and q are integer , q ≠ 0 and q and p are co - prime number ]
root 5 = p/q -3
root 5 = p - 3q/p
we know that p/q is a rational number.
.°. √5 is also a irrational number.
This contradicts our assumption.
.°. 3 +√5 is an irrational number.
∴ p and q are integer