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?

  • 16
    $\begingroup$ Ratio of two polynomials $\endgroup$
    – saulspatz
    Jul 12 '19 at 0:03
  • $\begingroup$ @saulspatz Why is term "function" in there when the numerator and denominator can only be polynomials? For example $\cos(x)/\sin(x)$ is a ratio of functions, but not a rational function as per the definition. $\endgroup$ Mar 1 '20 at 18:57
  • $\begingroup$ @AlJebr It's hopeless to ask for rational explanations of language. It's that way because that's the way it is. $\endgroup$
    – saulspatz
    Mar 1 '20 at 19:05

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 $q \neq 0$.

  • $\begingroup$ Is this really the reason, do you have any source? "Rational" just means "pertaining to ratio", and the fact that the same word is used in another case doesn't mean the other case is reason for using it for this case. $\endgroup$
    – JiK
    Jul 12 '19 at 8:42
  • $\begingroup$ Standard terminology in textbooks I have used. $\endgroup$ Jul 12 '19 at 13:58

a rational < insert thing > is pertaining to a ratio. A rational number is a ratio of integers, a rational expression is a ratio of expressions,
and a rational function is a function whose result comes from evaluating a rational expression.

  • $\begingroup$ rational root is another use. a root of a polynomial that is a rational number. $\endgroup$
    – user645636
    Jul 12 '19 at 0:31
  • 1
    $\begingroup$ JW stop editing my posts. I don't care if you have editor duty. $\endgroup$
    – user645636
    Jul 12 '19 at 1:12
  • 2
    $\begingroup$ You don't own your answers in StackExchange, and JW made just a few typo corrections to this post. $\endgroup$
    – JiK
    Jul 12 '19 at 8:41
  • $\begingroup$ It's why I left my previous math community. $\endgroup$
    – user645636
    Jul 13 '19 at 17:10

As the other answers suggest, "rational" here means "being a ratio". However, of course you could write everything, say $x$, as a ratio, namely as $\dfrac{x}{1}$.

(OK my very precise math friends, not "everything", but everything that's contained in some monoid with $1$; certainly everything that would usually be called a "number" or a "function".)

So the actual question is: What are the "more elementary elements" (numbers or functions) of which the "rational" numbers, or functions, are ratios?

It turns out that we call

rational numbers = ratios (quotients) of integers

rational functions = ratios (quotients) of polynomials.

So there seems to be a deeper analogy hidden here, that is

Polynomials are among functions what integers are among numbers; polynomials are "the integers among the functions";

and that is true to a surprisingly large extent.

  • $\begingroup$ I never considered that analogy, very interesting! $\endgroup$ Jul 15 '19 at 1:10

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