Is $\frac{x}{2}$ an algebraic fraction? If yes, isn't it improper? Then how to turn it proper?

Question

According to Wikipedia on algebraic fractions, $\frac{x}{2}$ seems an algebraic fraction. Then by definition, as the degree of the numerator $1$ ($x^1$) is larger than the degree of the denominator $0$ ($x^0$), it is an improper algebraic fraction. Now how can I turn it to a proper one? Long division gives either $x - \frac{x}{2}$, $x - x + \frac{x}{2}$, $x - x + x - \frac{x}{2}$, $x - x + x - x + \frac{x}{2}$, ..., which suggests impossibility. Should we exclude $\frac{x}{2}$ as an algebraic fraction?

Answer 1

This is an interesting question. I didn't see any explicit mention of whether a rational function with constant denominator should be considered improper. If you apply the strict definition then it would seem to be.

But I think there is a way out. Quoting from the Wikipedia article you linked to:

Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction.

In this case we have $$ \frac{x}{2} = \frac{1}{2}x $$ So the “polynomial part” of $\frac{x}{2}$ is $\frac{1}{2}x$, and the “proper rational fraction part” is $0$.