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Re: [xmca] a minus times a plus
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
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.
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
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 +
On 27-Apr-09, at 4:16 PM, Ed Wall wrote:
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 -
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).
On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:
Since we have some mathematically literate folks on xmca, could
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
and young high school students alike.
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