Number Bases and Polynomials

About a month ago, I was working with our 5th grade math teacher to develop some extension activities for some students in an unleveled class.  The class was exploring place value, and I suggested that some might be ready to explore what happens when you allow the number base to be something other than 10.  A few students had some fun learning to use their basic four algorithms in other number bases, but I made an even deeper connection.

When writing something like 512 in expanded form (5\cdot 10^2+1\cdot 10^1+2\cdot 10^0), I realized that if the 10 was an x, I’d have a polynomial.  I’d recognized this before, but this time I wondered what would happen if I applied basic math algorithms to polynomials if I wrote them in a condensed numerical form, not their standard expanded form.  That is, could I do basic algebra on 5x^2+x+2 if I thought of it as 512_x–a base-x “number”?  (To avoid other confusion later, I read this as “five one two base-x“.)

Following are some examples I played with to convince myself how my new notation would work.  I’m not convinced that this will ever lead to anything, but following my “what ifs” all the way to infinite series was a blast.  Read on!

Level 1–Basic Addition:

If I wanted to add (3x+5)(2x^2+4x+1), I could think of it as 35_x+241_x and add the numbers “normally” to get 276_x or 2x^2+7x+6.  Notice that each power of x identifies a “place value” for its characteristic coefficient.

If I wanted to add 3x-7 to itself, I had to adapt my notation a touch.  The “units digit” is a negative number, but since the number base, x, is unknown (or variable), I ended up saying 3x-7=3(-7)_x.  The parentheses are used to contain multiple characters into a single place value.  Then, (3x-7)+(3x-7) becomes 3(-7)_x+3(-7)_x=6(-14)_x or 6x-14.  Notice the expanding parentheses containing the base-x units digit.

Level 2–Advanced Addition:

The last example also showed me that simple multiplication would work.  Adding 3x-7 to itself is equivalent to multiplying 2\cdot (3x-7).  In base-x, that is 2\cdot 3(-7)_x.  That’s easy!  Arguably, this might be even easier that doubling a number when the number base is known.  Without interactions between the coefficients of different place values, just double each digit to get 6(-14)_x=6x-14, as before.

What about (x^2+7)+(8x-9)?  That’s equivalent to 107_x+8(-9)_x.  While simple, I’ll solve this one by stacking.

Base1

and this is x^2+8x-2.  As with base-10 numbers, the use of 0 is needed to hold place values exactly as I needed a 0 to hold the x^1 place for x^2+7. Again, this could easily be accomplished without the number base conversion, but how much more can we push these boundaries?

Level 3–Multiplication & Powers:

Compute (8x-3)^2.  Stacking again and using a modification of the multiply-and-carry algorithm I learned in grade school, I got

Base2and this is equivalent to 64x^2-48x+9.

All other forms of polynomial multiplication work just fine, too.

From one perspective, all of this shifting to a variable number base could be seen as completely unnecessary.  We already have acceptably working algorithms for addition, subtraction, and multiplication.  But then, I really like how this approach completes the connection between numerical and polynomial arithmetic.  The rules of math don’t change just because you introduce variables.  For some, I’m convinced this might make a big difference in understanding.

I also like how easily this extends polynomial by polynomial multiplication far beyond the bland monomial and binomial products that proliferate in virtually all modern textbooks.  Also banished here is any need at all for banal FOIL techniques.

Level 4–Division:

What about x^2+x-6 divided by x+3? In base-x, that’s 11(-6)_x \div 13_x. Remembering that there is no place value carrying possible, I had to be a little careful when setting up my computation. Focusing only on the lead digits, 1 “goes into” 1 one time.  Multiplying the partial quotient by the divisor, writing the result below and subtracting gives

Base3

Then, 1 “goes into” -2 negative two times.  Multiplying and subtracting gives a remainder of 0.

Base4

thereby confirming that x+3 is a factor of x^2+x-6, and the other factor is the quotient, x-2.

Perhaps this could be used as an alternative to other polynomial division algorithms.  It is somewhat similar to the synthetic division technique, without its  significant limitations:  It is not limited to linear divisors with lead coefficients of one.

For (4x^3-5x^2+7) \div (2x^2-1), think 4(-5)07_x \div 20(-1)_x.  Stacking and dividing gives

Base5

So \displaystyle \frac{4x^3-5x^2+7}{2x^2-1}=2x-2.5+\frac{2x+4.5}{2x^2-1}.

CONCLUSION

From all I’ve been able to tell, converting polynomials to their base-x number equivalents enables you to perform all of the same arithmetic computations.  For division in particular, it seems this method might even be a bit easier.

In my next post, I push the exploration of these base-x numbers into infinite series.

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3 responses to “Number Bases and Polynomials

  1. An interesting post! I’ve also played with this idea and found it useful to show to students to help make connections between polynomial and arithmetic operations. Of course, polynomial arithmetic is easier because there are no carries. Have you seen the recently posted exposition of these ideas at http://gdaymath.com/courses/exploding-dots/ ?

    • It doesn’t surprise me at all that Jim Tanton would have already explored this. I’ve been reading some of his Galley Method ideas lately and might have been unknowingly inspired. I haven’t read the post you provided, but I’m moving there now. Thanks.

  2. Pingback: Base-x Numbers and Infinite Series | CAS Musings

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