Question

Show that 1/√3 is irrational.

Answer 1

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-----------------------HERE'S YOUR SOLUTION------------------------------


let  us rationalise this term.


 then rationalization answer :-√3/3


now see here you know that  '3' is rational for sure???

okk so let us now prove that the term root 3 is irrations.

let us assume that √3 is a rational number(contradiction).

now.

we can write it in the form p/q.

so some conditions:-  here p and q are integers and q is not equal to 0 and also their hcf is 1.

now .
=> p/q=√3
(squarring)
=> p²/q²=3
=>p²=3q² ;;;;_ >>>> as here p is the factor and q is the multiple of p and respectively..

thus alltering...do the same by keeping a variable in place of q.

so.. now..

3 will be a factor of both  p and q which contradicts are  statement. that they bioth have a hcf of 1.

so we make a conclusion that although 2 is rational but √3 is irrational.

thanks :))

hope it helps you..

Answer 2

let us assume to the contrary that 1/√3 is rational number .

1/√3= P/Q { where p and Q are co-prime and Q not equal to 0}

√3 P =Q .1

√3 = Q/P

√3 = Irrational number

Q/P =Rational

Irrational not equal to rational.

This is a contradiction has arisen by the wrong assumption because of our incorrect assumption that 1/√3 is rational.

hence, 1/ √3 is irrational .{proved}