Posted on July 12, 2007 @ 05:28:12 PM by Paul Meagher
Why should you care what a polynomial is? There are lots of reasons, but one basic one is because decimal numbers can be formally described as polynomial sums:
454 = 4*10^{2} + 5*10^{1} + 4*10^{0}
Here is the PHP equivalent:
<?php
// Decimal number as polynomial sum.
$hundreds = 4; $tens = 5; $ones = 4;
echo $hundreds * pow(10, 2) + $tens * pow(10, 1) + $ones * pow(10, 0);
?>
Which suprisingly outputs:
454
The number 454 can be viewed as a quadratic number because it can be represented with a second degree polynomial:
P_{2}(a, x) = a_{2}x^{2} + a_{1}x^{1} + a_{0}x^{0}
Which is equivalent to:
P_{2}([4, 5, 4], 10) = 4* 10^{2} + 5*10^{1} + 4*10^{0}
You can represent other number systems as particular types of polynomial sums where the x term is either 2, 8, 16, or 60 if your number system is binary, octal, hexidecimal, or sexigesimal.
So one reason to know something about polynomials is because of the insight it offers into our own numbering system and why it works.
Recreational Exercise
Is there a problem with viewing polynomials as fundamental building blocks for numbers when the generalized polynomial expression itself uses a number system to represent the idea of decreasing series of exponents in its terms? Are number systems a consequence of polynomial theory or are polynomials accurate and useful rationalizations of number systems whose foundation is elsewhere (e.g., set theory, logic, embodied metaphor)?
