TITLE: When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$?
QUESTION [10 upvotes]: $$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$
In other words, a rational number is a number that can be written as one integer over another.
For an integer, the denominator is $1$ in that case. For example, $5$ can be written as $\dfrac 51$.

Is $5$ a rational number? Or is $\dfrac 51$  a rational number? I'm not able to figure out what the definition is actually saying. What are the numbers that cannot be written as one integer over another?

Irrational numbers are the numbers that cannot be written as one integer over another. Roots of numbers that are not perfect squares are examples of irrational numbers.

However, what is this then: $\dfrac {\sqrt 7} {2}$?

REPLY [3 votes]: Although there are many answers here already, unless if I have missed something none of them actually proves that $\sqrt{7}/2$ is irrational.  As already noted in the comments to amWhy's answer, it is not valid to say that if $a$ is not an integer then $a/b$ is irrational.  However, it is valid to say that if $a$ is irrational and $b$ is rational, then $a/b$ is irrational.  (This is because if $a/b$ and $b$ were both rational, then their product $a$ would be rational because the product of rational numbers is always rational.)  Because $\sqrt{7}$ is irrational (as shown in wece's answer) and 2 is rational, the ratio $\sqrt{7}/2$ is irrational.