I recently revisited rational functions, which I had studied a few months back, and came up with the question, why are rational functions called "rational"?
I tried googling the question, however, I found no answer. A neat math reference text I keep also had no explanation. The ranges of rational functions can include irrational numbers, so I ask, what is the meaning behind the name, rational function? Does anyone have a reference about this nomenclature?
In fact, according to Wikipedia:
The coefficients of the polynomials need not be rational numbers; they may be taken in any field K.
This confirms my suspicions that this is not the reason. What am I missing?
Best Answer
Because of their similarity to rational numbers; i.e.---
A function $f(x)$ is called a rational function provided that
$$f(x) = \frac{P(x)}{Q(x)},$$
where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial; just like a real number $n$ is called a rational number, provided that
$$ n = \frac{p}{q},$$
where $p, q$ are integers and $q \neq 0$.