> What is Division?
> by Jared Ornstead
I will apologize in advance for the acerbic tone of my
remarks, because I do feel bad. I believe you are making a
logical mistake, which happens, and the correct course of
action would be to try to engage you in a reasoned,
reasonable discussion.
Nonetheless, the entire tone of this essay is presumptuous,
pompous, and filled with a smug certainty that you know far
better than anyone else, including such poor souls as myself
whose hard-earned education actually prevents them from
understanding such leaps of brilliance as your own. Valuing
honesty as I do, I will return your implied insults with
explicit ones.
> What is it? People who perform the function all the time never
> pause to think about it. Is it a memorized times table applied in
> reverse? No, any thinking person must surely agree that is a shortcut;
> the memorization of the results of previously done work.
It's presented with a bit of a sweeping generalization, but
I agree with the argument.
> What, then, is it? Let us imagine for ourselves a small child,
> introduced by his teacher for the first time to the concept of
> dividing. Imagine further that this teacher does not approach via the
> usual shortcut of first teaching multiplication tables and applying
> them in reverse, but instead approaches the concept directly, via long
> division.
I'm going to have to disagree with the assessment of this
method has being a "shortcut." It's actually the most
logical and reasonable way to present division. When it
comes down to it, the simplest way to define division is as
the operation that undoes multiplication. That's it. If
two times three is six, then six divided by three should
give you two, by design and by desire.
This will probably become relevant later. Just a guess.
> Our imaginary teacher begins her lesson by drawing on the board
> something along the lines of:
> ___
> 2)4
>
> thus mystifying the entire class.
Assuming the class was full of the congenital morons that
you apparently assume your readers to be. Had they half the
brains of syphilitic orangutans, they might not be taken by
surprised when they've never before seen something a teacher
is covering for the first time.
> She explains that this is a division problem, and to find the
> answer they have merely to subtract two from four, do it again until
> the number is gone, then make a note of how many times they did it,
> placing that total at the top of the problem.
Rather a painful way of calculating a division, that. But
then, the way it's normally taught wouldn't allow you to
make your point, would it?
> Our child, well versed in subtraction and addition, approaches the
> problem with gusto, subtracts two from four, scribbles a little hash
> mark in the side margin to make note of having done it, then discovers
> there is something left of the four and eagerly vanquishes it with
> another subtraction of two, accompanied by another studious hash mark.
Insipid heroic melodrama, meet scholastic tedium.
Honestly, I'd be more receptive to your argument if you
presented it clearly and straightforwardly, without the
laborious, condescending narrative. I'd like to think I can
grasp your point without it being acted out by Billy The
Bunny-Wunny. So you'll pardon me if I start summarizing
paragraphs.
[introduction of division with remainder]
> It is later that we learn methods perhaps more precise, but let us
> stay with our child for now.
I see. So we're going to ignore the "later," "more precise"
definitions and methods of division in your argument?
One can hardly wait.
> Now let us replace our kind and loving teacher with one of a
> particularly cruel bent, a substitute perchance, while our teacher has
> the sniffles. Let us say this misanthrope desires to bruise the
> childrens' tender thinking and puts the following problem on the board:
>
> ___
> 0)8
>
> Our child, not knowing anything about the reputed impossibility of
> the task, begins eagerly making little hash marks as zero is subtracted
> from eight again and again without ever making the eight any smaller.
> Soon or late, the patience of childhood ends, the bell chimes for
> recess, and the assignment is abandoned.
At least we do know that you were being sarcastic with your
"genius" remarks.
> Now if we, as adults, were to wander in and take note of this
> child's scratch paper, we could amuse ourselves by entertaining the
> notion that theoretically those hash marks would never end; that zero
> could continue to be subtracted from eight forever, spawning hash marks
> off into eternity.
You could, perhaps. I find that I tend not to laugh when
children are taught to be so deficient in mathematical
knowledge and common sense.
> There is even a symbol for that. It is an eight lying on its side.
> But as I lack the higher ascii characters in this program, I beg the
> readers' forbearance and willingness to allow the substitution of two
> asterisks, like so: **
> This represents infinity.
As does, oddly enough, the word "infinity." Is there any
particular reason you desperately need a symbolic shorthand?
> It is here that we as adults would chuckle to ourselves and let the
> matter drop. But if our perky youngster was to return from play at
> exactly that instant to see us and our sideways eight symbol, he would
> solemnly inquire if infinity were all the times that zero could be
> taken out of eight. We would smile in our superior way and reply that,
> yes, infinity was all the times that zero could be subtracted from
> eight.
We would, mind you, be lying.
> With the confidence of youth, he would then scribble:
>
> __**_r8
> 0)8
Allow me, for a second, to engage in a pondering: a
pondering that I shall present to you in a narrative style
(a "story", if you will), which, perhaps, you will find a
little easier to process.
After yon prodigy has so assuredly marked his answer upon
the much-abused piece of paper, it might be that another
student should walk up to the desk before which sits the
young scholastic adventurer, and she (for we should not
doubt the great and oft-greater wisdom of the fair sex, even
in youth) in passing casts a discerning eye upon the
scribbled computations. Not privy to the earlier rigourous
calculations performed by her counterpart, nor to the
imparted wisdom of we, the learned adults, she would
contemplate the result, applying all she knew of the laws
and regulations that form those near-mystic arts that we
choose to label "subtraction" and "division."
At that point, it might very well be that she would choose
to utter, "You can't be done, because zero is less than
eight, so eight can't be the remainder, because you can
still subtract zero some more. Nyah nyah, you're a stupid
poopyhead."
Our intrepid child of learning would have no choice, at that
point, but to admit that she was indeed correct, and fall
down and tremble before her awesome intellect, because, dear
author, we know how you feel about counter-arguments, and so
your young avatar would be most unlikely to provide one.
> I submit this to be our answer. Infinity is, indeed, all the times
> that zero may be taken out of eight. By our own rules that leaves us
> with this answer: Infinity, remainder eight. But, of course, if you
> were to leave it there mathematicians would quibble. They would bring
> forth equations and substitute theories in a frenzy as their world has
> just been challenged.
Well, if by "our own rules," you mean, "Mr. Jared Ornstead's
own rules", then I'm forced to agree with you. Otherwise, I
beg to differ. But, out of fairness, I'll allow you to
finish your argument first.
> But before we give them leave to do that, let me point out that
> there is an error in our mathematical notation. Five divided by two is
> not two with one remainder, we simplify to our error. The true answer
> is two sets of two with one remainder. That is five. Their dispersal or
> consumption is a separate operation. Our system of bandying numbers
> about retains inaccuracy, sometimes to our cost.
"Words, words, words." _Hamlet_, Act II, Scene ii.
I have really no idea what your point is, here. I am left
to conclude that you are using obscurity and inanity to
build up to a conclusion.
> Division is a process wherein we divine, "How many times does this
> go into that?" The answer is a tally of groups. Had I thirteen apples,
> and four kids to be fed while they went and romped through fields, I
> could reserve one for myself, parcel out three to each tyke and there
> would still BE thirteen apples, merely subdivided into groups. If we
> each ate one that is subtraction. If Tommy lost one and Mark stole one
> of Lisa's and little Amy plucked five more from a tree; these are all
> subtraction and addition at the heart of it. Anything else is
> unnecessary complication. Further, our groups are now unequal and so
> the brief answer of a number of groups is completely inadequate to the
> task of tracking apples.
My mistake. You were using obscurity and inanity to build
up to obscurity and inanity. Your point, if I dredge
correctly out of the verbal muck you've dropped it in, is
that division, fundamentally, arises from the real-life need
to be able to partition objects into equally-sized groups,
and that everything else is "unnecessary complication."
Very well. Allow me, if you would be so kind, to offer a
rebuttal. Respecting the utter disregard you've already
expressed for the "equations and substitute theories" of
professional mathematicians, I will try to keep my arguments
on the playing field you have staked out.
Firstly. As my little homily hopefully illustrated, it is
obvious even to a grade-schooler than you can't have a
remainder larger your divisor. If you do, then that means
you can't be done subtracting yet; unless, of course, you've
decided to declare a special case.
Secondly. Suppose I asked you to subtract 1x10^(-100,000)
from eight. The answer is, of course, not eight, but it is
so close to eight that there really is no point in making
any distinction. That would probably be true for the first
thousand or so times you did that subtraction, as well, and
if you were the child in your example, you might very well
conclude that the answer would never be different from
eight. The truth, however, is that eventually the answer
wouldn't be all that close to eight at all; it just takes
doing a hundred thousand subtractions to actually see that.
So who are we to say that, in fact, if you actually sat
around and somehow managed to subtract zero from eight an
infinite number of times that you wouldn't be left with
seven? After all, it's not like anyone can actually even do
a significant fraction of the necessary subtractions.
Thirdly. As I mentioned above, the ideal of division is
that it should undo multiplication. So, if eight divided by
zero is infinity with eight left over, then it must be true
that zero times infinity plus eight will be eight; which, in
turn, means that zero times infinity will be equal to zero.
(I realize that deducing that requires algebra a bit beyond
the grade-school level, and I hope I haven't lost you.)
That makes sense, of course, since anything times zero is
zero. But on the other hand, we also know (since we know
about infinity) that anything times infinity should be
infinity, which implies that zero times infinity should be
infinity, which means that zero times infinity plus eight is
infinity, so where does that leave us?
Fourthly. Your argument only works for the integers. As
you point out, we later learn how to deal with division in
the real numbers, where remainder disappears. How does your
claim generalize, if at all?
Lastly, and most importantly. You've defined division as
"How many times does this go into that?" If I was sitting
around with eight apples, trying to figure out how to parcel
them out fairly, and you came up to me and start babbling
about infinite numbers of sets of size zero, the most I
would acknowledge you would be in trying to get away,
because you know what? Infinity doesn't exist. It is an
"unnecessary complication," to throw your words back in your
face like the sharp objects I wish they were.
Zero itself is, at best, only poorly a "real-world" concept;
you can't really have zero apples, let alone talk about
dividing eight apples into sets of no apples. Infinity, on
the other hand, purely does not exist. Nothing in this
universe is infinite, and nothing can be. Having clearly
(at least to my understanding of the gibberish that was the
previous paragraph) limited your discussion to the real,
tangible world, pulling in a purely imaginary construct to
provide the answer is nothing more than a deceptive way of
saying, "the answer is not defined" -- which happens to be
the answer a mathematician would give.
And that, really, is the crux of it. When it comes to the
real world, which is where you base your arguments, the
whole question of dividing by zero is an irrelevancy, and it
doesn't *matter* what the answer is. The answer only
becomes relevant when you start dealing with mathematics
itself, those theoretical constructs you regard with a sneer
and a dismissive wave. Given that, why should we prefer
your answer over the answer that comes from the theory, and
is therefore internally consistent with it?
> Now lest anyone condemn me, I still regard division as an essential
> tool of math. But, like percentage, it is a tool, a shortcut, merely
> one way of looking at things; a form of notation and nothing more.
> It is an exceedingly useful form of notation. But really, four
> divided into twelve yields three groups of four, not three. Eight
> divided by zero yields infinite groups of zero, and eight. Both
> simplify down to their original forms. Three groups of four is twelve,
> and infinite groups of zero can be added to themselves while still
> yielding zero, leaving eight.
"... full of sound and fury, signifying nothing." _Macbeth_, Act 5, Scene v.
Honestly, in my most generous interpretation, you are
constructing a false dichotomy in order to support your
argument. You're saying that "we say division is this, but
it's really that." However, "this" is "that," at least for
the set of numbers you're using, with the only difference
that "this" is still defined when "that" becomes
meaningless. The net result is that it sort of sounds like
you're making a deep point, when in fact you're repeating
yourself.
In my least generous interpretation, the stinking imbecility
that permeates your words is enough to make blood vomit from
my eye sockets.
More seriously, what is at the crux of your flawed reasoning
is a very common mistake, made by people from first graders
to university students in mathematics: you think you know
what infinity is. Infinity is not the biggest possible
number plus one. Infinity is not a number, period. If your
thinking, at any point, treats infinity as a number, it is
almost certainly flawed thinking. It helps, I find, to
think of infinity as a *process* rather than a quantity.
That is, eight times infinity is not adding eight together
infinity times, but sitting down to add eight to itself and
never, ever stop. And with that thinking, the flaws in your
reasoning become clearer. Being able to say that eight
divided by zero is infinity with any remainder, you had to
stop at some point, and say, "this is infinity." By
stopping, by definition, you can't be at zero.