MF57: Polynomials and polynumbers

We begin the important task of defining the fundamental objects of modern algebra. First we review different roles played by polynomials. We are going to base polynomials on something more fundamental called polynumbers, whose arithmetic parallels but is richer than that of the natural numbers and rational numbers.

Category:

Science & Technology

Tags:

• Mathematics
• foundations
• algebra
• polynomials
• polynumbers
• Wildberger
• arithmetic
• field

[jehovajah]

Hey prof, don't knock the warm fuzzy feeling! lol!. This is a good time to introduce Grassmann's Ausdehnungsgroesse, and a bit of Newtonian History. De Moivre, Cotes and Newton were masters of the Multinomial, the name that preceded polynomials. These arose out of the great calculation of the Trigonometric tables, and the logarithmic tables too. These tabular entries were integer in form, and difference formulae were derived to calculate them.This was part of the foundation the calculus.

[njwildberger]

@jehovajah Thanks for the insightful comment(s)!

[Toxie207]

This reminds me of a synthetic nest or an augmented matrix. We are removing the unncecessary powers of x and just concentrating on the ordered set of coefficients which are the important details?

[njwildberger]

Hi Toxie207, Your interpretation is quite right. But not only are we being more concise, we are also being more precise, in that we don't have to wrestle with the nature of that mysterious x.

[Lioobayoyo]

@GR1o6180339887498948

answer's a bit late, but for other :

if you write them hori^zontaly instead of verticaly, from right to left instead of from top to bottom, you'll notice that the multuiplication is VERY usual, it's the same as the one you learned when you were young, for numbers with several figures ;)

[grosucelmic]

Really nice, as usual. This time, with a bracket on top..

[lisp21]

the Polynomials constitute a vector space, plus a convolution operation(the multiplication in your definition).

[GR1o6180339887498948]

I understand how multiplication is done but it's still kind of unusual.