Why is this an irrational number

Hi:
I need to know why is 5√2/2 an irrational number? My teacher said that it is because √2 is already an irrational number, but I don't understand that. If I can write it as a fraction then why is it irrational?

Only those fractions that contain integers in both the numerator and denominator are rational numbers. Here, the numerator contains √2, which is not an integer.

It's not enough to write the number as a fraction. To be rational, the number must be written as a fraction in which the numerator and denominator are integers. 5√2/2 does not represent a rational number because 5√2 is not an integer. So, the question becomes can √2 be written as a fraction with an integer numerator and integer denominator. It can't. Therefore, √2 is irrational and 5√2/2 is irrational.

we can think of 5sqrt(2)/2 as (5/2) * sqrt(2).

We know sqrt(2) is irrational [here are some proofs: http://en.wikipedia.org/wiki/Square_root… ]

As it turns out, if R is rational and J is irrational, then R*J must be rational [here is a proof: http://www.math.upenn.edu/~nate/teaching… Exercise 1]

Since we know 5/2 is obviously rational (we can write it as a fraction of integers) then we know that 5/2 * sqrt(2) must be irrational, and so 5*sqrt(2)/2 must be irrational.

Although √2 may seem like a rational number, it is not. It can be expressed simply, but since it cannot be made into a fraction, it is irrational.

because it is asking something of you that is totally and completely unwarranted, what an arrogant and odious number
*Goes off in a huff*

Because it cannot be expressed with fraction containing only integers, i.e. 2/6, 34/26, etc.

that can be written as a ratio of integer no. that is rational. here 5root2 is not integer.