SOLUTION: Knowing that √2 is an irrational number, argue that √2/2 is also an irrational number. Please help. Even just some ideas would be great. Thank you!

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An Irrational Number is a number that cannot be written as a simple fraction - the decimal goes on forever without repeating. So, since is an irrational number, dividing its result in half is also irrational.

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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.