By definition of a fraction, can the numerator or the denominator be a non integer ? For example, can we call the expression $\frac{\pi}{2}$ a fraction ?
The popular answer is no, but since the calculus of fractions is valid even with non integers, shouldn't we call $\frac{\pi}{2}$ a fraction after all ?
On
From Webster's online:
frac·tion | \ ˈfrak-shən \ 1a : a numerical representation (such as ³/₄, ⁵/₈, or 3.234) indicating the quotient of two numbers
Since $\pi$ (roughly 3.14159) is a number, $\frac{\pi}{2}$ is definitely a fraction by this definition (and I checked that Webster's does consider $\pi$ to be a number).
If we didn't call it a fraction, what would we call it? Though on the other hand, there might not be a practical need to call it anything.
As to whether a given fraction can be rewritten so that both the numerator and denominator are integers, well, I think that's a separate question.
Yes, a fraction can have basically any real (or even complex) numbers in the numerator and the denominator. Starting with the real numbers and then allowing fractions does not give you anything new, just other ways to write numbers you already have access too (you don't need fractions to be able to speak about the real number $\frac\pi2$, it's just very convenient). In contrast with the integers, where allowing fractions gives you a whole bunch of new numbers you didn't already have ($\frac12$ is not an integer, for instance).
Note that many people confuse the two concepts (or at least the names) of fractions and of rational numbers. A rational number is a number which may be represented as a fraction with integers in the numerator and denominator. This doesn't limit fractions to only use integers. It just means that if you have non-integers in your fraction, the number you have is not necessarily a rational number. (And in the case of $\frac\pi2$, that's not a rational number. However, $\frac{\sqrt 8}{\sqrt 2}$ is a rational number, because it may also be written as $\frac21$.)