Question

Prove that √5 is irrational.​

Answer 1

Answer:

answe

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Answer 2

we have to prove √5 is irrational

let us assume the opposite ,

i.e ,,,√5 is irrationala

Hence , √5 can be written in the form of a/b

where a and b (b not equal 0) are coprimes(non common factor other than 1)

Hence √5=a/b

√5b=a

squaring both sides

(√5b)square =a square

a square /5 = b square

hence 5is divided a square

(By theorem :-if p is a prime number , and p is divided a square , where is a positive number )

so, 5 shall divided a also ....(I)

hence, we can say

a/5=c where some c is integer

so,a=5c

Now we know that

5b square =a square

putting a=5c

[5bsquare] =[5csquare ]

5bsquare =25c square

bsquare =1/5x25c square

bsquare =5c square

b square /5=c square

Hence 5 divided b also ......ii

(by theorem:- if p is a prime number and p divide a square then p divides 0, where a is positive number )

so, 5 divided b also ......(2)

by (1)and( 2)

5 divides both a and b

so a, and b are not co primes

Hence ,our assumption is wrong

√5 is irrational

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