Polynomials are just numbers in base x
Here’s a neat idea: A polynomial is just a number in (an unknown) base
Bases
When we use the number
So what does
What about if we change the base from base-10 (decimal) to base-2 (binary)? What does
More generally, what does
In this way, we can use a number to represent a polynomial expression in much the same way that we already use numbers to represent arithmetic expressions.
But… 427 is not a binary number!
Some careful readers may have objected at the thought of
This is normally true. Normally, we do not allow a single digit to be equal to or larger than the base. For base-10, this means no digits larger than 9. For base-2, this means no digits larger than 1. However, let us suspend our normal conventions for a moment. Is there really anything wrong with using a digit that is larger than the base?
So, although we may be on somewhat unfamiliar ground, let’s continue exploring a world in which we allow digits to be arbitrarily large and see where this takes us.
Addition
First, we re-learn addition, without carryover. Consider the example of adding
Notice two things. First, when we add 7 and 4, instead of writing 11 down, we only write a 1 and we carry the 1. Second, our answer is the number
If we now try this addition again, but we allow digits of arbitrary size and forget the carryover rule, it would look something like
This answer is hard to write down in standard form;
Adding polynomials
If the example above felt pointless, then hopefully this will help. What is
- a)
(aka ) or - b)
(aka )
It’s clearly b, but why? Why does carrying over make intuitive sense when we’re dealing with numbers in base-10, but not when we’re dealing with polynomials?
At the risk of being overly pedantic1, in base-10, we can convert a 10 in one place into a 1 in the next place because
When we’re dealing with polynomials in base-x, we can’t carry over because we don’t know how many
Neat. But… is it useful?
By treating polynomials as numbers (but in some unknown base
We just saw how adding the polynomials
How about multiplying polynomials? What if instead of adding
By treating polynomials as numbers, we can use the standard method for multiplying numbers to multiply polynomials. I think that’s quite a bit more compelling, but we can do even better.
What is
Going farther
Solving polynomial equations
This framing gives us a new way to think about solving polynomial equations. When solving an equation for
It’s quite easy to see that, in this case,
Again, it’s obvious that
Negative and non-integer bases
We know that when solving an equation for
Solving the equation results in
Nothing revolutionary, but an interesting perspective.
Prime polynomials?
Now that we’re treating polynomials as numbers, let’s try to extend a property normally reserved for numbers: the property of being “prime”. We define a prime number to be one that is not a product of two smaller numbers.
Try to find two smaller numbers (in base-x, i.e. polynomials) that can multiply to
I’ve only just thought of this idea (primeness for polynomials), and so I haven’t yet thought of any useful applications for it, but math has a long history of finding useful applications for bizarre and esoteric ideas.
-
We may have crossed that bridge long ago. ↩