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Question

# prove that √5 is not a rational number. Hence, prove that 2 - √5 is also irrational.

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Solution

## Let us suppose that √5 is a rational number. Hence, √5 can be written as ab where both a and b are co-primes.√5=ab⇒√5b=a⇒5b2=a2∴ a25=b2 ......... 1We know that, if a number p divides q2, it will divide q as well.Here, 5 divides a2. Hence, it must divide a as well. ∴ a5=c (c= any integer)a=5c ..... 2From 1 and 2, we get:25c2=5b2b2=5c2b25=c2Again, 5 divides b2. Hence, it will also divide b. Hence, 5 is a factor of both a and b. Therefore a and b are not co-primes. So, what we assumed is not true.Hence, √5 is irrational.Now, according to the properties of irrational numbers, the sum or difference of a rational number and an irrational number is always an irrational number.Hence, (2−√5) is also irrational.

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