Re: Math & physics.


"Androcles" <Headmaster@xxxxxxxxxxxxxxxx> wrote in message news:Z4k5m.52363$K41.5382@xxxxxxxxxxxxxxxx

You say *there is nothing in a real-valued physical external world
that is physically complex-valued.*
I'd say this: mathematics is a language, and like prose and poetry and
music, it is art. Useful art to the engineer, but art nevertheless.
When Newton wrote his laws he used Latin, the English translation
of which is:
LAW I.
Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by
forces impressed thereon.

LAW II.
The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in
which that force is impressed.

LAW III.
To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal,
and directed to contrary parts.

The mathematical shorthand for laws 1 & 2 (actually Galileo's laws) is:
F = dp/dt.
The mathematical shorthand for law 3 is:
Mv = mV.

We can only use the shorthand if we understand the meaning
of the symbols; F is force, m is mass, v is velocity, p is mass
multiplied by velocity, dp/dt is "alteration of motion". M is a
big mass, v is a small velocity. Beyond that, v is a vector
and is what a "right line" is all about, p we call momentum
but Newton called "motion".


Thanks for development. I think I already understand Newt mechanics well.


Now you know that current leads voltage in a capacitor and
lags in an inductor. You also know that a magnetic field exists
around a current, and an electric field exists across the plates
of a capacitor, I don't need to tell you that. If you disagree
then there is no point in continuing.


This is why I snipped in my previous message. You said there's not point in continuing if we can't agree about what a field "is." I
agree.


But now I want to introduce an idea to you that will seem
strange, but think about it. Minus (-) can be thought off as a
binary operator as in 3-2 = 1, but it is really a unary operator,
as in 3 + (-2) = 1. Mathematicians love shorthand!
Algebraically, a-b=c is identical to a+b = c when b is negative.
A computer will take b, negate it and add a, storing the result
in a location called c.
On a number line, minus is the reverse DIRECTION to plus.

-2........-1.........0.........1.........2........3
<-------- ++++>

Adding 3 to -2 is VECTOR addition. Go back 2, then go
forward 3, arrive at 1.

In a real-valued physical external world I give you three apples,
take two back and you have one apple. 3 forward, 2 back.


I teach essentially exactly this same thing in my elementary mathematics classes. It's amazing how many people coming out of high
school don't understand "subtracting" is simply "adding a negative number." The number line approach makes it clear, to me anyway,
and I hope to my students.


Now... the "number" that you call i = sqrt(-1) isn't a number
at all.

It is most decidedly is a number in mathematics, a complex number, a particular member of the Complex Number Set. I'm sure you know
this. Real numbers are a subset.


It's an operator, just as *, /, +,- are.

No it's not. These operations are performed on complex numbers, so complex numbers are just that. Numbers, not operators.


It represents a vector
that is orthogonal to the number line in the complex plane.
There is '+' going to the right, '-' going to the left, 'i' going
up and, (lo and behold) '-i' going the opposite way to 'i', down.
Multiplying a number by 'i' is a rotation of 90 degrees, doing
that twice is a rotation of 180 degrees; which is the same as a
multiplying by -. It makes sense because it is CONSISTENT.
Multiplying i by - gives -i, a rotation of 270 degrees. i^2
is a rotation by 180, so 'i' is sqrt('-'). I omitted the '1' deliberately.


Yes, complex numbers can provide some amazing mathematical short cuts. I do not at all say we should not use complex numbers and
their manipulations in our modeling. But I very much object to representing a physical state (pos, vel, acc, etc.) as a complex
valued variable.



So where is this all going?

The physical world includes phenomena such as magnetic
and electric fields, and they are orthogonal to each other.
They are VECTORS.
IF
( we use shorthand mathematics to describe magnetic and electric fields)
THEN
there is SOMETHING in a real-valued physical external world
that is physically complex-valued.

Sorry, I can't accept this apparent contradiction in your words. How can something in a real-valued physical external world (your
words) be physically complex-valued? By that I take it you mean the imaginary part is not zero for this "something." So, once again
I ask, where exactly is the imaginary part of a complex-valued position in a real-valued physical external word, like in QM's
Heisenberg picture?



We just need to understand the symbols. Unfortunately the
terms "real" and "imaginary" are confusing, magnetic fields
are as real as electric fields, changing one will cause the other
or motors and generators would not work.
The math is there to describe it, to provide the shorthand
so that we can manipulate phase angles and know what
component to use in a particular application. If you are
designing a microwave oven or a TV, what is the frequency,
what is the power factor? Is it different from radar? Is it safe
or will it start a fire? And the most important question for any
businessman investing capital in it: How much does it COST?



Boy, let me tell you, how much it costs is vastly important. That's why I am now a *retired* engineer. The gov't came in and
cancelled the radar program I was on, all within one month, due to too high a cost for the system.



As I've already said, it is up to engineers to cherry-pick the
bits of theories that make sense and leave out those that do not.
To do that we need to cut through the hyperbole that comes
with schoolmarms trying to describe their own misunderstanding
of the real world. Those that can, do. Those that can't, teach.


I take that as a gigantic insult, me and all teachers everywhere give you the bird.

Steve Bell


.

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