Question

prove that root 3 is irrational ​

Answer 1

Answer:

we can prove it

Step-by-step explanation:

let us assume √3 is rational number

expressed in p/q form

p and q are co-primes and q not equal to 0

√3 =p/q

3 = p^2/q^2

3q^2=p^2 this is equation 1

it means 3 divides p^2 . this means 3 divides p because each factor should appear two times for square to exist.

so we have p=3r

r is some integer.

p^2= 9r^2 this is equation 2

from 1 and 2 equations

3q^2= 9r^2

q^2= 3r^2

it demonstrate√3 is irrational number.

Answer 2

Answer:

Let us assume that √3 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√3 = p/q { where p and q are co- prime}

√3q = p

Now, by squaring both the side

we get,

(√3q)² = p²

3q² = p² ........ ( i )

So,

if 3 is the factor of p²

then, 3 is also a factor of p ..... ( ii )

=> Let p = 3m { where m is any integer }

squaring both sides

p² = (3m)²

p² = 9m²

putting the value of p² in equation ( i )

3q² = p²

3q² = 9m²

q² = 3m²

So,

if 3 is factor of q²

then, 3 is also factor of q

Since

3 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

hence,. √3 is an irrational number

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