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Re: [xmca] a minus times a plus

Personally, Jim and Linda, I would find ideas like negative as reflection or negative as contradiction, easy to understand if *you are already familiar* with -x-=+ but I would have thought that the leap from visual or rhetorical relationships to numbers is asking too much of children who have not yet learnt to count backwards from zero.

Why is a number line easier to understand than the symmetries of topological or rhetorical transformations? I don't know, but intuitively I think it is. But different people think differently. Maybe the Linda's rhetorical explanation is easier for a verbal thinker than a spatial thinker? (is this mumbo-jumbo?)


Jim Levin wrote:
Perhaps it would be helpful to think about multiplying by -1 as reflection. So, if you separate out the -1 part of the multiplication, you get:
-a * -b = -1 * a * -1 * b = -1 * -1 * a * b
(an even number of reflections)

-a * b = -1 * a * b
(an odd number of reflections)

a * b = a * b
(no reflections)

Each -1 multiplier "reflects" the result across the 0 point of the number line (imagine putting a mirror at the 0 point - it then looks like the number is on the other side of the zero point). So if you have an odd number of these reflections, you end up on the other side; if you have an even number, you end up on the same side. Perhaps you'll want to draw a number line for your granddaughter, Mike, and find a small mirror?

At 7:49 PM -0700 4/27/09, Mike Cole wrote:
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.

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 +


 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 - 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).


 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
 and young high school students alike.

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Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
From Erythrós Press and Media <http://www.erythrospress.com/>.

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