Question

prove that √3 is a irrational number

Answer 1

Step-by-step explanation:

let us assume that √3 is rational

so , rational no. are of the form p\q where p & q are co prime

so √3=p\q

squaring both sides,

we have 3= p square \ q square

cross multiplying

so 3q square=p square

which means 3divides p square 3 divides p

now , let us take.

p= 3c. so we get,

3q square = (3c) square

3q square =9c square

q square =3c

which means 3 divides q square 3 divides q

it therefore contradicts our assumption wrong

therefore √3 is irrational

Answer 2

Answer:

if possible √3 be a rational no. then its simplest form be p/q , where p&q are integers having no common factor other than 1

Step-by-step explanation:

now,

√3=p/q

squaring both side

(√3)²= (p/q)²

3 = p²/q²

cross multiplication

3q² = p² .......(I)

3 divides p²

3 divides p

let,

p= 3b

put the value of p in eqñ. 1

3q²= p²

3q²= (3b)²

3q²= 9b²

q²= 3b²

3 divides q²

3 divides q

thus, 3 is a common factor of p and q

but, this contradiction that p and q have no common factor other than 1

the contradiction arises bu assuming that √3 ia rational.

Hence, √3 is irrational no.