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[Phys-L] Re: Division via subtractions



John, I have to break in here after a week or two of this bantering exchange
and voice a strong opposition to your statements. I think what you are
saying is, at some level, true. However, it's not actually addressing the
issue. I'm just a backwoods high school teacher, but, being a Math major
before seeing the 'light', pun intended, and becoming a Physics major, a
grasp of number theory is one of those 'things' I have a grasp of. I think.

Division IS shorthand subtraction. Period. That's its definition. Since
subtraction is simply addition using negative numbers, we DO say that
division is, therefore, shorthand addition (of negative numbers, of course).
By the same token, multiplication IS shorthand addition (of positive
numbers, of course). Period. 4x3 is 12 BECAUSE you add 4 three times to get
12. 12/3=4 BECAUSE we can subtract (or add -3, if you'd prefer) 3 from
twelve four time before running out of numbers. (Remainders and such just
add digits past the decimal, so don't start a writing campaign to the
producer...)

These definitions do not, as you are fond of stating, include 'loops' and
'iterations', 'implicit functions' and 'analog electronics'. Those of us 'in
the trenches', as well as our students, aren't really interested in how a
computer or a TI-92 will perform iterations to do a division. That's a story
for another day and has a totally different and appreciative meaning. That
may very well be YOUR job. It's not mine. My job is to instill an underlying
comprehension of all the concepts we encounter in physics class. And, yes,
this has come up before. This year, in fact.

My assertion is that there is one and ONLY one true mathematical function.
That is addition. All others can be reduced to some sort of addition
problem. Period. Even Derivatives and Integrals...

I think this exchange is being attacked from two very different directions.
This is not a bad thing in itself. Some of us are dealing with numbers that
pop up in class and trying to solve problems and trying to show the kids
where all this stuff fits in. Some others are trying to show how, at a much
higher or deeper level, these operations can be performed or, indeed, how
they MUST be performed by computers. However, I see no reason and no
argument to make me avert my eyes from my understanding of the definitions
of multiplication and division.

So, yes, we CAN say that division and multiplication are related by a simple
addition operation. Even though you typed "not" five times...


Daryl L. Taylor, Fizzix Guy
Greenwich HS, CT
PAEMST '96
International Internet Educator of the Year '03
NASA SEU Educator Ambassador
www.DarylScience.com

This email prepared and transmitted using 100% recycled electrons!



-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu]On
Behalf Of John Denker
Sent: Tuesday, February 08, 2005 7:15 PM
To: PHYS-L@LISTS.NAU.EDU
Subject: Re: Division via subtractions


Ludwik Kowalski wrote:

Division of positive numbers by positive numbers can be done
by consecutive subtractions.

Yeah ... but why bring it up? The original claim was
that
division is iterated subtraction
in the same way that
multiplication is iterated addition
... and this claim is false. It is entirely incompatible
with any real understanding of what is going on.

If you are going to claim
multiplication is iterated addition
division is iterated subtraction
I am going to insist that
division is iterated addition
so at this level of understanding, division is not the
inverse of multiplication -- it is word-for-word the same
as multiplication.

Please, folks:
The inverse of an iteration is not
the iteration of the inverse.

Maybe you wish it were so, but wishing will not make it so.

If you really want to know what's going on, the key idea
in the iterative subtraction algorithm is not the subtraction
(since that can perfectly well be replaced by an addition).
Rather, the key idea is replacing a loop where the number
of operations is determined in advance with a loop where
the number of operations is counted on the fly. That is:
Multiplication: add B to itself A times.
Division: add B to itself an indefinite number of times,
and see how many times it takes before the sum reaches C.

To understand this in greater generality, view it as a feedback
loop. The same key idea is commonly used in analog electronics.
A good example is using the exponential I(V) curve of a diode
to create a circuit to compute logarithms, by placing the
diode in the feedback side of a closed loop.

To say the same thing in mathematical terms, we write
multiplication as an explicit function
for all A and B, C = A * B
and then write division as an _implicit function_
for all C and B, find A such that C = A * B.

I cannot emphasize too strongly that changing addition to
subtraction is not *not* _not_ NOT !NOT! what is required
to change multiplication to division. Iterated addition
gives you A*B. Iterated subtraction gives you -A*B.
-- Neither of those is division.
-- Either of those *placed inside a feedback path*
can give you division.

... too tedious for humans, unless numbers are small.

This is another reason for asking, why bring it up?
Conceptually and pedagogically it is worse than nothing,
and as a practical algorithm it is laughable, so what's
the point?

For numbers of size N, iterated subtraction is _exponentially_
more laborious than long division. That's about as bad as
anything can get.

People make acerbic jokes about the bubble-sort algorithm,
because its workload goes like N^2 while competing
algorithms are only N*log(N) ... which is a very
significant difference in many practical situations.

Imagine how rude the jokes become when the subject is
algorithms that go like 10^N while competing algorithms
are linear in N.

===========================

Pedagogically speaking: I don't expect you to teach nine-
year-olds about feedback loops or implicit functions.

But for crying out loud, don't teach them that you can
invert an iterated operation by iterating the inverse
operation. It's just wrong. It's totally wrong. It's
worse than nothing. Lots worse.