Question 238188
assume that sqrt(2) / 2 is a rational number.


then you can set y = sqrt(2)/2 then this means that y is a rational number.


if you multiply both sides of this equation by 2, then you would get 2 * y = sqrt(2).


2 * y would still be a rational number because a rational number times a rational number is a rational number, but sqrt(2) would not because that's how we started.


the equation would be false negating the claim that sqrt(2) / 2 is a rational number.


how do we know that multiplying a rational number by 2 yield a rational number.


first of all 2 is a rational number because it is an integer and any integer can be represented by that number divided by 1 which is a rational number.


take the number y = 1/2


this is clearly a rational number because it's a division of two integers.


now multiply both sides of this equation by 2.


you get 2y = 1 which is also clearly a rational number because an integer is a rational number.


bottom line is the assumption that sqrt(2)/2 is a rational number is a false assumption proven by multiplying both sides of that equation by 2 yielding a false equation.