Numbers as Polynomials

144 = b² + 4b + 4 = (b + 2)² = 12²

The idea that numbers are polynomials is an old one that goes back to March and April 2004, when I created the category Numbers. At the time, it didn't seem like a big idea, so I just stuck a few paragraphs about it onto the end of Fractions in Base 2. But, since then, I've referred to it in a lot of places, and now it bugs me that it doesn't have its own essay. That's why I'm making one!

Here's the plan.

• This essay will refer to all past essays that talk about numbers as polynomials.
• Future essays that talk about numbers as polynomials will refer to this essay. That will make them appear in the backlinks (at right).
• I'm also adding a note to Fractions in Base 2 just to put in a couple of forward links.

From the beginning, the idea of numbers as polynomials has been connected to the idea of negative digits, so the history of negative digits that I wrote (History and Other Stuff, about halfway down) is also a pretty good history of numbers as polynomials. Let's just make a copy of the whole thing.

• At the end of Fractions in Base 2, I talked about numbers as polynomials over the base, and joked that there ought to be a digit with value −1.
• Almost immediately I started taking the joke seriously. A few months later, in an essay about divisibility tests in other bases, I officially started using the hash mark as a negative digit. Here are the main facts translated to the new notation.

  101	= 11 × 11
  1001	= 11 × 111
  1001	= 11 × 111

• A year and a half after that, in Repunits, I put the hash mark to good use once more, and the letter X had a cameo as Digit with Value −2 in this crazy number.

  1110011211001111110010101010100111111001121100111

• I used the hash mark one more time in The Multiplication Table even though I was already planning to write this essay.

Here are all the other places that I've talked about numbers as polynomials.

• History and Other Stuff contains not only the history but also other stuff. In particular, it ends with some nice examples of numbers as polynomials.
• Powers of N points out that powers of 11 are interesting.
• In Reflection Symmetry I stumbled on a peculiar identity.

  1011 × 111 = 110001

With that, the survey of past essays is complete.

I don't have any new information to add at this time, but there is one bit of old information that I'd like to emphasize. Just as integers can be factored into primes, polynomials can be factored into irreducible polynomials. However, irreducible polynomials are much coarser grains than primes. A number regarded as a polynomial sometimes breaks into more than one irreducible polynomial, but an irreducible polynomial regarded as a number often breaks into more than one prime. You can see the whole process play out in the second and third tables in Repunits, just keep in mind that repunits are more reducible than most polynomials.

Here's one more complication. With negative digits, a number can be written in many different ways, and the different ways correspond to different polynomials. The different polynomials must ultimately yield the same set (multiset) of prime factors, but apart from that, they don't have much in common. For example, one might be irreducible and another reducible.

104 = b² + 4
116 = b² + b − 6 = (b − 2)(b + 3) = 12 × 13