prove that
is an irrational number
Answers
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Question:-
prove that √3 is irrational .
Answer :-
proof :-
we are trying to prove that root 3 cannot be expressed as a fraction if we are trying to prove that something cannot be true it is often useful to assume that it is true and attempt to prove a contradiction .
so let us assume that ;
√3=a/b
where a/b is it fraction in its lowest term .
let us play around with this formula and see what we can come up with ;
√3=a/b
=> √3=a^2/b^2 [ squaring both the sides ]
=> 3b^2=a^2 [ multiplying b square ]
so A square is an even number => A is an odd number
we can therefore Express A as 3c where is the is also an integer .
=> 3b^2=a^2
=> 3b^2 =(3c)^2
=> 3b^2 = 9c^2
=> b^2= 3c^2
we can now see that b square is also even . and B is an odd number .
Assumption :-
but we can assume that a divided by b is a fraction in its lowest term which is clearly not since both A and B are even number so we have a contradiction and have to conclude that our original as a option that root 3 can be expressed as a fraction is false so we can say that root 3 is irrational