$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$
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: $\dfrac {\sqrt 7} {2}$?
Best Answer
Any number for which it is possible to express as the ratio or quotient of integers is a rational number. So yes, $5$ is rational, because it is possible to express this as $\frac 51, \frac {10}{2}...$.
$5$ is also an integer. Every integer is a rational number, but not all rational numbers, e.g. $\frac 12$, is an integer. We know $\frac 12$ is rational, because it is the quotient of two integers.
However, $\dfrac {\sqrt 7}{2}$ is not a ratio of integers. It is a ratio of a non-integer, namely $\sqrt 7$, over an integer. So $\dfrac{\sqrt 7}{2}$ is not rational.