prove that root 2 + root 5 is irritational given root 2 and root 5 are irrational. ( prove by contradiction)
Answers
Answer:
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Step-by-step explanation:
Given: √2+√5
We need to prove√2+√5 is an irrational number.
Proof
Let us assume that √2+√5 is a rational number.
A rational number can be written in the form of p/q where p,q are integers and q≠0
√2+√5 = p/q
On squaring both sides we get,
(√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 is incorrect
√2+√5 is an irrational number.
Hence proved.
Step-by-step explanation:
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 supposition is false.
√2+√5 is an irrational number.
Hence proved.