prove that 2√5 is irrational number
Answers
Answer:
If possible, let us assume 2+5 is a rational number.
2+5=qp where p,q∈z,q=0
2−qp=−5
q2q−p=−5
⇒−5 is a rational number
∵q2q−p is a rational number
But −5 is not a rational number.
∴ Our supposition 2+5 is a rational number is wrong.
⇒2+5 is an irrational number.
Answer:
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: