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 at 0:03

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! $\endgroup$ – Gnumbertester Jul 15 at 1:10

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 at 8:42
  • $\begingroup$ Standard terminology in textbooks I have used. $\endgroup$ – mlchristians Jul 12 at 13:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.