wrote:
why would "equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting..." need to involve "i"?


Well, one is subtracting from a given area, and the numbers subtracted
are therefore 'negative'. As the numbers are _areas_, then we have a
negative number that often needs to have its square root taken.
Mathematically, this does not work. as -1 x -1 = 1. So. " i " is
introduced to make the calculations work.

This was a lecture mind... so here is the issue: feedstock is
expensive, and there are thousands of sizes of boxes. These were
corrugate boxes, with varying dimensions. This plant did not 'stock',
but made to order, to size. The knives were capable of cutting in
either horizontally or vertically and as finenly placed as necessary,
as well as partial cuts. Set-up and roll changing were the costliest
operations (as both involved machine shut-down), waste being the third
largest cost. Several sizes of boxes could be run at once. He
calculated how to run the orders to avoid waste and slivering, and to
minimize the number of different widths of feedstock required. He chose
to use the 'imaginary number i' in his process. It seemed to work as
their scrap-pile was rather tiny to his great pride and joy.

Can you give any more details? Still can't quite see where i or j is required.

--
Eiron

There's something scary about stupidity made coherent - Tom Stoppard.

On Fri, 10 Mar 2006 16:58:53 -0000, "Glenn Booth"
wrote:

"Don Pearce" wrote in message
...
On Fri, 10 Mar 2006 16:37:41 -0000, "Glenn Booth"
wrote:

Hi,

"Don Pearce" wrote in message
.. .

Addition an subtraction are for A level. Multiplication and division
are now optional post grad modules.

With a calculator, naturally.


No other way is currently known - post doc research is underway.

Meanwhile, the term "mental arithmetic", having fallen into disuse,
has been stolen by the medical industry and refers to a head count
in a psychiatric hospital.

What would today's 18 year olds make of a slide rule?

My Faber Castell Novo-Duplex would probably make a nice table
ornament, but my Otis King would be a decent match for those
extensible batons the piggies now use! :-)
--

Stewart Pinkerton | Music is Art - Audio is Engineering

On 10 Mar 2006 07:54:43 -0800, " wrote:

http://mathforum.org/library/drmath/view/58251.html

Wrong side of the pond thing, you Brits? Or is it that you need to be
ahead of us Colonials... and go all the way to j?

Nope, it depends on whether you're a mathematician or an EE.
--

Stewart Pinkerton | Music is Art - Audio is Engineering

OK... how would you come up with the square root of -4? Practical
application, you are starting with so many square feet of feedstock,
you are making 22 boxes each requiring two 4 square foot faces, two
feet on a side and other sides may vary within certain parameters, and
12 boxes each requiring two 1 square foot faces. But the dimensions of
the first box must be calculated to have the correct volume as a
function of dimensions and not preclude the similar values for the
second box. So, you are SUBTRACTING dimensions as square roots of total
areas required for square cuts. As sq.rt. -4 does not calculate, but
sq.rt. 4 x i does... that is how it comes in. Keep in mind that one
*could* reverse the signs in one's head the reality is that all the
areas calculated are *real*, but as there are many sign-changes in the
calculation apart from negative number roots, the chance of error
increases greatly. The elegant part of all this is that the " i " drops
out at the end of the calculations, but it allows the rule of 8 (8
basic axioms of 'real' numbers) to apply during. As others have
suggested, we have computers do this these days. The need for practical
math has been relegated mostly to calculating tips in a restaurant. And
few do even this, it seems.

The history of Negative Numbers, remember?

Peter Wieck
Wyncote, PA

What would today's 18 year olds make of a slide rule?

They are clueless. I have a 14" K&E, double-sided. Hot-Sh*t in its
day... Once upon a time, I could even use it. For calculus even... The
kids (well over 18) only know what it is from me. Their kids? Not at
all.

Peter Wieck
Wyncote, PA

wrote:

OK... how would you come up with the square root of -4? Practical
application, you are starting with so many square feet of feedstock,
you are making 22 boxes each requiring two 4 square foot faces, two
feet on a side and other sides may vary within certain parameters, and
12 boxes each requiring two 1 square foot faces. But the dimensions of
the first box must be calculated to have the correct volume as a
function of dimensions and not preclude the similar values for the
second box. So, you are SUBTRACTING dimensions as square roots of total
areas required for square cuts. As sq.rt. -4 does not calculate, but
sq.rt. 4 x i does... that is how it comes in. Keep in mind that one
*could* reverse the signs in one's head the reality is that all the
areas calculated are *real*, but as there are many sign-changes in the
calculation apart from negative number roots, the chance of error
increases greatly. The elegant part of all this is that the " i " drops
out at the end of the calculations, but it allows the rule of 8 (8
basic axioms of 'real' numbers) to apply during.

Sorry, still don't get it. Perhaps a simple example might help.

--
Eiron

There's something scary about stupidity made coherent - Tom Stoppard.

wrote:

OK... negative numbers by definition do not have square roots. But
negative numbers exist in real life. And in real life, there is a need
for their square root. So, " i " was introduced to fill that need.

Subtracting areas from areas (area X - Area Y) show an area as a
negative number. How big is that area Y if it must be square? Hence the
need for the square root of a negative number.

Simple enough?

Of course not. That isn't an example, and I know all about complex numbers.
I just don't see how they help in calculating your cardboard boxes,
which is why I asked for a simple example. You do know what an example is?
It often has numbers in it.

--
Eiron

There's something scary about stupidity made coherent - Tom Stoppard.

"Hey, what's that.... crap snipped"

Do you ever stop pretending?

Peter Wieck
Wyncote, PA

The real number -1 = -1+0i = (1,180°) has angle 180 degrees (mod 360
degrees) and length 1. The purely imaginary number [0,1] = 0+i1 =
(1,90°) has angle 90 degrees and length 1. Multiplying this point or
number by itself, that is, squaring it, gives the point with length 1
×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1.
We call i, the principal square root of -1.

A second square root of -1 is obtained as follows. The imaginary number
(0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1.
Multiplying this point or number by itself, that is squaring it, gives
the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° =
180° (mod 360°). So this product equals -1+0i = -1 as well.

This provides two square roots of -1 as both (1,+90°)2 = (1,+180°) =
-1 and (1,-90°)2 = (1,-180°) = -1.

I cannot reproduce a diagram. But what you get is four points on a
graph. Set your knives to those points, and you can cut a straight cut
on moving stock. Add other calculations, and you can graph other cuts
to minimize waste. Remember, this was pre-desk-top-computer... well
over 30 years ago and I was NOT the one making the calculations. The
above are unashamedly cribbed from a website that also cribbed from
another website... but also is dedicated to topology. Items like the
Klein Bottle and the Mobius strip can be described mathematically. As I
remember, both also use ' i ' as there are 'imaginary' conditions to be
described as points in space or points on a plain.

Peter Wieck
Wyncote, PA

wrote:

Definition of i snipped

I cannot reproduce a diagram. But what you get is four points on a
graph. Set your knives to those points, and you can cut a straight cut
on moving stock. Add other calculations, and you can graph other cuts
to minimize waste. Remember, this was pre-desk-top-computer... well
over 30 years ago and I was NOT the one making the calculations. The
above are unashamedly cribbed from a website that also cribbed from
another website... but also is dedicated to topology. Items like the
Klein Bottle and the Mobius strip can be described mathematically. As I
remember, both also use ' i ' as there are 'imaginary' conditions to be
described as points in space or points on a plain.

As I said, I know all about complex numbers but don't see how they help
in calculating your cardboard boxes; and neither, apparently, do you.

You seem to be suggesting above that a position in 2-dimensional space
can be described by x,y coordinates but we all knew that anyway.

Where's the example showing that use of a rectangle of negative area and
imaginary sides helps in your calculations? It sounds fascinating and
many of us are waiting with bated breath.

--
Eiron

There's something scary about stupidity made coherent - Tom Stoppard.