prove that √2 - √5 is irrational.
Answers
hey there :
Let √2+√5 be a rational number.
A rational number can be written in the form of p/q where p,q are integers.
√2+√5 = p/q
Squaring on both sides,
(√2+√5)² = (p/q)²
√2²+√5²+2(√5)(√2) = p²/q²
2+5+2√10 = p²/q²
7+2√10 = p²/q²
2√10 = p²/q² - 7
√10 = (p²-7q²)/2q
p,q are integers then (p²-7q²)/2q is a rational number.
Then √10 is also a rational number.
But this contradicts the fact that √10 is an irrational number.
.°. Our assumption was incorrect
√2+√5 is an irrational number.
Answer:
Explanation:
Let us think 2+√5 is rational. So it can be written in the form of p/q where p and q are co-prime and q not equal to 0
2+√5 = p/q
√5 = p/q -2
√5 = p-2q/p
But p and q are integer, so p-2q/p is rational but √5 is irrational. This contradicts happen due to our wrong assumption that 2+√5 is rational. Hence, 2+√5 is Irrational.