Question

prove that root 5 an irrational number.

Answer 1

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→Let us assume that √5 is an irrational number.

→Therefore, √5=p/q (where p and q are integers and q in not eual to 0 and they are co prime numbers)

now,

→ √5q=p. ( cross multiply)
→on squaring
→5q^2=p^2...........(1)
→ i.e p square is divisible by 5
Thus, p is also divisible by 5 ( theorem )
now let p= 5a
→5q^2=(5a)^2. ......( by 1)
→ q^2=5a^2
I.e q square is also divisible by 5
→ hence, q is also divisible by 5

Therefore , p and q have atleast 5 as their common factor .

→ but this contradicts the fact that p and q are co prime numbers.

→ thus our assumption is wrong

→ hence √5 is an irrational number
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Hope it helps.

Answer 2

Let us assume that √5 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, √5 = p/q

    p = √5q

we know that 'p' is a rational number. so √5 q must be rational since it equals to p

but it doesnt occurs with √5 since its not an intezer

therefore, p =/= √5q

this contradicts the fact that √5 is an irrational number

hence our assumption is wrong and √5 is an irrational number.

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