This is great, Steve. Back to Eric's strategy of personal experience and
money exchanges.
We are dividing in what I think is a productive way. One way goes
toward the
mathematical foundations,
the other goes toward everyday transactions.
I still want a picture to help me out the way + cartesian coordinates
APPEARS to help us understand
addition and subtraction involving mixture of positive and negative
numbers.
every puzzled in san diego.
mike
On Mon, Apr 27, 2009 at 7:19 PM, Steve Gabosch <stevegabosch@me.com>
wrote:
Thinking in terms of minus and plus along number axes goes to the very
heart of how to think in terms of how a typical CNC milling machine
moves.
These are large milling machines that, without making this more
complicated, essentially move in three axes, x, y and z. The
programs work
by "zeroing out" each axis at a given point in relationship to the part
being cut, and then commanding the cutter to go to specific points
along the
x, y and z axes to create cutter paths that machine the part. CNC mill
operators (in 1980's-90's generation technology) usually have to
manually
move the machine to set cutters, otherwise have to move the machine
around
for a variety of reasons, including sometimes making rather intricate
tooling moves, and adjust cutter paths during cuts when the part is
coming
out too thick or thin, or wide or narrow.
Getting plus and minus right really, really matters. Go the wrong
way -
get minus and plus mixed up - even just a small fraction of an inch,
and you
can scrap the part, break the cutter, damage the tooling surfaces,
wreck
tools, knock the spindles out of alignment, put the machine down for
hours
or days, and otherwise cause thousands or more dollars in damage - and
otherwise ruin your day.
I did a lot of training of operators on spar mills at Boeing over the
years. I was just an operator myself, but enjoyed doing this, and had
something of a knack for it. One of the reasons Boeing needed some
of this
training was to transition a whole bunch of "conventional" milling
machine
operators to CNC (computer numerical controlled) machines. This was
interesting because the idea of running a machine along numerically
designated axes was often new to them.
So I got to look inside the heads of a lot of people who were grappling
with numbers and axes in a new way, and this minus and plus business of
course came up all the time. I am a little off-topic here in that
it is
rarely, if ever, necessary to **multiply** anything by a negative
number
when running a spar mill. But just **adding** and **subtracting**
negative
numbers - especially in complex successions - and moving the machine
exactly
where you wanted to go could get surprisingly complicated to wrap one's
brain around. Mistakes were common, and very experienced operators
could
make them, too.
In classes for the conventional operators who had never been exposed
to CNC
technology, I used the idea of a trip odometer on a car that only moved
forward and backward 100 feet. I explained that you can zero it out
> anywhere, and move back and forth all you want, and you will always
be a
specific distance from where you started. That is basically how the
relative register on the machine works, and most grasped the idea as
second
nature. There is also an absolute register on the machine, which
has a
permanent zero position for each axis, so it is important to know
which one
is which when you are looking at the screen for axis position.
Where and
how that permanent position gets set is still another question.
The biggest trick in running a spar mill or any such CNC machine is
to be
able to think in terms not of just moving in one, but *three* axes
at once.
In the case of our usual spar mill, we ran with two spindles at
once, each
with their own axis system - which made everything more complicated,
now
having to keep in mind things like whether your spindles are being
programmed in "mirror mode" so the two parts come out as mirror
images of
each other.
The operators have to routinely keep precise track of which
direction is
minus and plus, and precisely which axis or axes they are concerned
with in
a given circumstance. Running a mill is something of an ongoing
multi-dimensional puzzle with lots of little tricks and
misdirections to
fool the operator into thinking one thing when something else is
true, or
looking in one direction and forgetting to look in another. Believe
me,
under pressure, in these conditions of complexity, and where can be so
little or no room for error, keeping minus and plus straight all day
is not
always so easy!
Working with many operators over the years, I could see that some
were very
sharp about how the CNC axis system worked, could read programs
fluently,
write their own, etc. And then there were a few who had trouble even
grasping the basic idea of negative numbers.
There was one fellow, then in his 40's, who was having a hell of a time
with the concept of negative numbers. I spent a few sessions with him
trying to figure out a way to get negative numbers to make sense to
him as a
"theory" or "concept" so he could do some routine things more
confidently,
and not run into trouble, as he not infrequently did - often by
playing it
safe and just not doing some things, which sometimes caused
problems, too.
I tried everything I could, using paper and pencil, moving the machine
around and watching the screen together, using rulers and number
scales and
moving things back and forth over them, and anything else I could
think of.
He would get lost as to "where he was" as the objects or symbols
moved back
and forth along an axis, especially if he started at a negative
numbered
position. Something about the concept just wouldn't stick.
I finally made a breakthrough when I related moving the machine back
and
forth along an axis - to money in his pocket! "Suppose you start
with $100,
and then you spend $30 ... and then I give you $40 ... following me?
(yes)
... and then you spend another $20 ..." Without fail, he always knew
EXACTLY how much money he had! The idea of negative numbers started
to fall
into place after that.
Luckily, I didn't have to try to explain to him why -8 * -8 = 64.
- Steve
On Apr 27, 2009, at 5:34 PM, Wolff-Michael Roth wrote:
Can't you think like this---perhaps it is too much of a physicist's
thinking. We can think of the following general function (operator in
physics) that produces an image y of x operated upon by A.
y = Ax
if x is from the domain of positive integers, then A = -1 would
produce an
image that is opposite to the one when A = +1, the identity operation.
Conceptually you would then not think in terms of a positive times a
negative number, but in terms of a positive number that is projected
opposite of the origin on a number line, and, if the number is
unequal to 1,
like -2, then it is also stretched.
The - would then not be interpreted in the same way as the +
Cheers,
Michael
On 27-Apr-09, at 4:16 PM, Ed Wall wrote:
>>
Mike
It is simply (of course, it isn't simple by the way) because, the
negative integers (and, if you wish, zero) were added to the
natural numbers
in a way that preserves (in a sense) their (the natural numbers) usual
arithmetical regularities. It would be unfortunate if something
that was
true in the natural numbers was no longer true in the integers,
which is a
extension that includes them. Perhaps the easiest way to the
negative x
positive business is as follows (and, of course, this can be made
opaquely
precise - smile):
3 x 1 = 3
2 x 1 = 2
1 x 1 = 1
0 x 1 = 0
so what, given regularity in the naturals + zero) do you think happens
next? This thinking works for, of course, for negative times
negative. The
opaque proof is more or less as follows.
Negative numbers are solutions to natural number equations of the form
(I'm simplifying all this a little)
x + a = 0 ('a' a natural number)
and likewise positive numbers are solutions to natural number
equations
of the form
y = b ('b' a natural number)
Multiplying these two equations in the usual fashion within the
natural
numbers gives
xy + ay = 0
or substituting for y
xy + ab = 0
so, by definition, xy is a negative number.
Notice how all this hinges on the structure of the natural numbers
(which
I've somewhat assumed in all this).
Ed
On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:
Since we have some mathematically literate folks on xmca, could
someone
please post an explanation of why
multiplying a negative number by a positive numbers yields a negative
number? What I would really love is an explanation
that is representable in a manner understandable to old college
professors
and young high school students alike.
mike
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