# Order of a polynomial

I'm attempting to read a book from 1978 called A Practical Guide to Splines by Carl de Boor. In it he says: A polynomial of order n or of degree < n is a function of the form $$p(x) = a_1 + a_2x + \ldots + a_nx^{n-1}$$

I'm getting a bit confused as Wolfram and Wikipedia both say order and degree are the same. Has the meaning of the word order changed in recent decades?

• The quote from de Boor's book can be found here on page 1: amazon.com/Practical-Splines-Applied-Mathematical-Sciences/dp/… – JavaMan Jun 7 '11 at 17:49
• It seems that de Boor has defined a polynomial of order $n$ to be a polynomial of degree $< n$. Is your question about how he is using the word "order" in this particular book, or is your question concerning the history of the word "order"? – JavaMan Jun 7 '11 at 17:51
• @DJC: My question is mainly about the word "order" in this book. I'm also comparing it to other definitions I know (where order = degree) and trying see if the definitions reconcile. In your opinion, are order and degree interchangeable as some respondents have suggested below? Is de Boor's definition of order different to that on Wolfram? (wolframalpha.com/input/?i=polynomial+order). My confidence in my math ability is often low, and this can throw up blocks in understanding the simplest things! – PeteUK Jun 7 '11 at 19:40
• The word "order" has many meanings (for example, multiplicative order). I would imagine in de Boor's eyes, the words "order" and "degree" refer to the same thing, although most people refer to the degree of the polynomial. However, I do not think that understanding the history of the word is important to understand what de Boor is trying to convey. Good luck with your studies! – JavaMan Jun 7 '11 at 20:46
• @DJC I think order in de Boor's eyes is always degree + 1. Thanks for your kind words. – PeteUK Jun 9 '11 at 9:23

At least in the study of splines, order still has the same meaning as the one used by de Boor. Makes sense, order counts the number of control points. And if the best chunk determined by $5$ control points happens to have degree $2$, it is still determined by $5$ control points.
The meaning has not changed, here $\deg p = n - 1$ which is as he says less than $n$. Order and Degree are interchangeable.