Polynomials/Numbers

Polynomial Representation of Numbers

All of the real numbers of a given radix can also be expressed as a polynomial, where the variable is the radix.

For example, in base 10, we can split a number up into its ones, tens, hundreds, and so on. Suppose we have a number n,

$n=125=1\times 100+2\times 10+5\times 1$

This is equivalent to a polynomial representation in terms of the radix (10):

$n=125=1\times 10^{2}+2\times 10^{1}+5\times 10^{0}$

and if we replace the 10 with an x, we get a polynomial:

$n=125=x^{2}+2x+5$

This generalizes to larger numbers:

$n=3,654,987=3\times 10^{6}+6\times 10^{5}+5\times 10^{4}+4\times 10^{3}+9\times 10^{2}+8\times 10^{1}+7\times 10^{0}$

That's just a polynomial with as many terms as digits, and the unknown $x$ replacing the base:

$n=3x^{6}+6x^{5}+5x^{4}+4x^{3}+9x^{2}+8x+7$

It also generalizes to binary numbers or hex numbers:

$n_{2}=101010=1\times 2^{5}+1\times 2^{3}+1\times 2^{1}=42$

This number becomes the polynomial:

$n_{2}=x^{5}+x^{3}+x$

(where, for binary numbers, the set of possible coefficients is just the set ${0,1}$.)

Here is a hex number, converted to the equivalent polynomial:

$n_{16}=BEEF=B\times 16^{3}+E\times 16^{2}+E\times 16^{1}+F\times 16^{0}$

Replacing the letters with their decimal equivalents, we get the base 10 equivalent number:

$11\times 16^{3}+14\times 16^{2}+14\times 16^{1}+15\times 16^{0}=48,879$

Python for Radix Conversions

Note that the radix conversions above (some non-decimal base into decimal base) are straightforward to do with Python - when you create an integer object, you can pass a string containing the number, then pass a second argument that specifies the radix:

$ python

>>> int('101010',2)
42

>>> int('BEEF',16)
48879

>>> int('DEADBEEF',16)
3735928559