Re: [xmca] a minus times a plus

[Mike Cole <lchcmike@gmail.com>]

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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>>>
>>>
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