By Bobby Neal Winters
Long division is hard. I want to state that up front. I remember when I first encountered it in grade school that it hurt my head. There were a lot of rules and you had to be able to know your multiplication tables and guess and try and guess again. I learned how to act pitiful to get my mother to do it for me.
More than forty years later, I will admit that it’s a useful technique. Let’s talk about long division long enough to get into trouble. Consider the easiest case: dividing a number by a smaller number. For example, 7 divided by 2.
Two time three is six is the largest multiple of two that is less than seven, so we put down a 3 on the top line and subtract 6 from this leaves 1. When you are a little kid who doesn’t know about decimals, you stop there and say there is a remainder of 1, but when you know about decimals, you put a decimal point after the 3 on the top line and you put a 0 down after the one to make it ten. Two times five is ten exactly, so you put down a 5 after the decimal and stop. This yields that 7/2=3.5, which, of course, is the correct answer. You can always check your answer--but our students never do--by multiplying 3.5 by 2 to get 7.
This is the easy case not because we are dividing a bigger number by a smaller number, but because the process terminates after a finite number of steps. Consider a case that doesn’t like two divided by seven. This is illustrated below:
Note that since seven is larger than two, we have to put down a 0 followed by a decimal on the top line. We then put zero times seven on the line below 2 getting--surprise, surprise--0. We then take zero from two and put down 2 on the next line. We put a 0 by it to make it twenty. We then note that seven times two is fourteen so we put 14 below the 20. Taking fourteen from twenty leave six, so we put down 6 on the next line and follow it by a 0 to make sixty. Now seven times eight is fifty-six, so we put 56 below the 60.
You know the drill. This will literally go on forever. But before too long this will begin to repeat so that we get 2/7 is equal to 0.285714285714285714285714... .
This works nicely because we have a place-value system to represent numbers.
Mathematicians are never satisfied with just numbers, however. We like to use letters too. We work with entities (I almost wrote things there. Things is sloppy writing, so I used entities there instead. It means things, by the way.) called polynomials. If you’ve had an algebra class you’ve seen something like 2+x or 2-x-x2. You can divide on of these by the other two. For example, let’s divide 2-x-x2 by 2+x:
I was tempted to explain this division. Indeed I wrote a paragraph of it, but I lost consciousness in the process. If you’ve had the course, you can do it, but if not, I am not going to abuse your good nature by teaching it here. Let it be sufficient to say that we go through the same motions as we do when we do the long division of numbers.
Just as in the case of long division of numbers, we can have non terminating cases here too. Consider 2+x divided 2-x-x2:
Here the final quotient is the infinite series 1+x+x2+x3+x4+... . This might ring a bell for some of you because not only is it 2+x divided by 2-x-x2 it is also 1/(1-x). As PeeWee Herman used to say, “I meant that to happen.”
The equation 1/(1-x)=1+x+x2+x3+x4+... is quite famous among math geeks. It is called the Geometric Series. I can literally talk for hours about this. My students will testify to this, as will the piles of legs in the classroom which my students have gnawed off like coyotes in futile attempts at escape.
You can have a lot of fun with this equation. Let x=0.5. The left hand side, i.e. 1/(1-x), becomes 2 while the right hand side becomes 1+0.5+ (0.5)2+ (0.5)3+... . If you add enough terms of that right hand side, you get as close to the sum of 2 as you desire.
If you let x=1, then the left hand side becomes 1/0 by zero while the right hand side becomes 1+1+1+.... which is infinity. If you make yourself believe that 1/0 is equal to infinity, then this is not a problem. Let’s push it further, though. Let x=-1. Then the left hand side is equal to 1/2. The right hand side, however, becomes 1-1+1-1+1-1+.... . Let’s refer to this right hand side as S. It is possible to argue that S is equal to zero; it is possible to argue that S is equal to one. All of the arguments make the assumption that you can treat arithmetic operations done an infinite number of times the same way you can those done a finite number of times.
What happened to resolve this is a good example of how mathematicians operate. We have a formula here that gives good answers in certain circumstances and nonsense in others; we determine exactly what those conditions are. In this case, we get good answers exactly when the absolute value of x is less than one and nonsense at all other times. There is a whole language created to talk about it using words like convergent and divergent and the Greek letters epsilon and delta.
So we began with a method for dividing one number by another. We then extend that method to polynomials. This led to an infinite series where work had to be done to delineate what could be said. It would appear to be the end of the matter. Yet.
Given the theory developed to talk about convergence, the idea that the series S=1-1+1-1+... might be equal to anything appears to be nonsense, but consider the calculation below. It is like a twisted coda at the end of a horror movie, Jason lives:
What was it that Scotty said, “If one man calls you an ass, pay him no mind. If two men do, buy a saddle.”
One can talk about Cesaro sums. These are like running averages of series. The Cesaro sum of S is 1/2.
Let’s go back to the equation 1/(1-x)=1+x+x2+x3+x4+... . Those who know a little calculus will recognize that taking the derivative of each side will yield 1/(1-x)2=1+2x+3x2+4x3+... .
Putting in x=0.5 makes the left hand side equal to 4 and the right hand side 1+2(0.5)+3(0.5)2+4(0.5)3+... can be made as close to 4 as we desire. This is all good. It becomes more interested when we let x=-1. Then the left hand side becomes 1/4 and the right hand side becomes 1-2+3-4+5-... .
But this can be made meaningful as well. There is a theory of divergent series that has been worked out. It is not my point here to expound on it--though if I learn any more I might--but to bring it up as an example of what mathematics sometimes do.
We take a technique in one area where it is clearly defined and makes sense and then apply it--sometimes playfully--in another area, where sometimes we get a mixture of good results and nonsense. We then perform investigations to separate the good results from the nonsense.
It’s not a bad way of doing business.