Prove that ,√5 is irrational number
Answers
Answer:
√5 is irrational number as it is non-terminating and non-repeating in decimal form
Answer:
hi☺️☺️
___________________________________☺️
Let's prove this by the method of contradiction-
Say, √5 is a rational number.
∴ It can be expressed in the form p/q where p,q are co-prime integers.
⇒√5=p/q
⇒5=p²/q² {Squaring both the sides}
⇒5q²=p² (1)
⇒p² is a multiple of 5. {Euclid's Division Lemma}
⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}
⇒p=5m
⇒p²=25m² (2)
From equations (1) and (2), we get,
5q²=25m²
⇒q²=5m²
⇒q² is a multiple of 5. {Euclid's Division Lemma}
⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}
Hence, p,q have a common factor 5. this contradicts that they are co-primes.
Therefore, p/q is not a rational number. This proves that √5 is an irrational number.
For the second query, as we've proved √5 irrational.
Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.
____________________________________☺️