Sign of approximation error and remainder (residual)

Question

In the Wikipedia article Taylor series it is said that:

The error incurred in approximating a function by its $n$th-degree Taylor polynomial is called the remainder or residual and is denoted by the function $R_n(x)$.

The article Taylor's theorem clarifies further:

Taylor's theorem describes the asymptotic behavior of the remainder term $$R_k(x) = f(x) - P_k(x),$$ which is the approximation error when approximating $f$ with its Taylor polynomial.

But I would think that the error is of the opposite sign: $P_k(x) - f(x).$ As an analogy, if the true answer is 90 and someone says it is 100 then I think the error incurred is 10, not -10. Which one is correct?

Answer 1

I'm a little late, but here I go.

Discussion of this kind can easily fall in the never-ending rabbit hole of semantic debate.

I agree with your view that the word error should represent, say $E_k$ as in $E_k(x) = P_k(x) - f(x)$, since it can be seen as the error due to the approximation.

However, I believe the word remainder should represent the amount missing from the approximation using Taylor expansion, which then should stay $R_k(x) = f(x) - P_k(x)$, that stems from $f(x) = P_k(x) + R_k(x)$, which reads as: the function f is the Taylor approximation $P_k(x)$ plus some remainder term $R_k(x)$.

At that point, a matter of a minus sign on the definition doesn't haywire the use of Taylor approximation.