Why $f(x)=\pi$ is a rational function? Is a constant function a polynomial even though the constant is a transcendental?

Question

Wikipedia said

A constant function such as $f(x) = π$ is a rational function since constants are polynomials. The function itself is rational, even though the value of $f(x)$ is irrational for all x.

But the definition stated: A function ${\displaystyle f(x)}$ is called a rational function if and only if it can be written in the form $${\displaystyle f(x) = {\frac {P(x)}{Q(x)}}}$$ where ${\displaystyle P\,}$ and ${\displaystyle Q\,}$ are polynomial functions of ${\displaystyle x\,}$ and ${\displaystyle Q\,}$ is not the zero function. The domain of ${\displaystyle f\,}$ is the set of all values of ${\displaystyle x\,}$ for which the denominator ${\displaystyle Q(x)\,}$ is not zero.

We can rewrite $f(x)=\pi$ as $f(x)=\frac{\pi}{1}$

I could agree if $1$ is a polynomial since it's a non-zero constant function that is a polynomial of degree 0. But $\pi$ is a trancendental number here. I mean, i never saw a polynomial with a constant of $\pi$. Can you explain it to me?

Thanks.

Answer 1

$\pi$ is a polynomial of degree $0$ of $\mathbb R[x]$. Therefore, the constant real map equal to $\pi$ is a rational function. Which doesn't mean that $\pi$ is a rational number.

The difficulty is that we are (abusively) naming two different objects as $\pi$:

• The real number.
• A real map.

Answer 2

$\pi$ is just a number here, we could calculate in a $\pi$-based number system, it would be a constant still.

Rational function means that it is a fraction of of two polynomials, now whether you take $\mathbb{R}$, $\mathbb{C}$ or even $\mathbb{Q}$ it doesn't matter because the terms rational number and rational function address different issues.

For example let's say you have the function $\pi f(x)$, how do you know this is rational or not?

f(x) could be a constant function or a $\dfrac{x}{\pi}$, the important part is that it satisfies the form $$a_{n} x^n+a_{n-1} x^{n-1}+...+a_0 \hspace{6mm} \text{where } a_i \in \text{arbitrary } \mathbb{F} \text{ (field)}$$, then it's a polynomial. From which you can constuct rational functions.

Answer 3

$f(x) = \pi$ is a polynomial function. The coeficients in it do not have to be integers for it to be a polynomial. Yet, the number $\pi$ is irrational.

There are differences between a rational function and a rational number.