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-< Mathematics at DEC >-
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Note 1281.0 The Gallows 1 reply
DEC25::ROBERTS "Reason, Purpose, Self-esteem" 98 lines 8-AUG-1990 13:34
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[With thanks to the person who sent a copy of the article to me via
interplant mail. I've misplaced your name, but the appreciation is
still felt. /Dwayne]
The following is the example from _One, Two, Three -- Infinity_:
There was a young and adventurous man who found among his
great-grandfather's papers a piece of parchment that revealed the
location of a hidden treasure. The instructions read:
"Sail to --------- North latitude and ----------- West longitude� where
thou wilt find a deserted island. There lieth a large meadow, not pent,
on the north shore of the island where standeth a lonely oak and a
lonely pine.� There thou wilt see also an old gallows on which we once
were wont to hang traitors. Start thou from the gallows and walk to the
oak counting thy steps. At the oak thou must turn *right* by a right
angle and take the same number of steps. Put here a spike in the
ground. Now must thou return to the gallows and walk to the pine
counting thy steps. At the pine thou must turn *left* by a right angle
and see that thou takest the same number of steps, and put another
spike into the ground. Dig halfway between the spikes; the treasure is
there."
The instructions were quite clear and explicit, so our young man
chartered a ship and sailed to the South Seas. He found the island, the
field, the oak and the pine, but to his great sorrow the gallows was
gone. Too long a time had passed since the document had been written;
rain and sun and wind had disintegrated the wood and returned it to the
soil, leaving no trace even of the place where it once had stood.
Our adventurous young man fell into despair, then in an angry frenzy
began to dig at random all over the field. But all his efforts were in
vain; the island was too big! So he sailed back with empty hands. And
the treasure is probably still there.
A sad story, but what is sadder still is the fact that the fellow might
have had the treasure, if only he had known a bit about mathematics,
and specifically the use of imaginary numbers. Let us see if we can
find the treasure for him, even though it is too late to do him any
good.
�The actual figures of longitude and latitude were given in the
document but are omitted in this text, in order not to give away the
secret.
�The names of the trees are also changed for the same reason as above.
Obviously there would be other varieties of trees on a tropical island.
[For those who wish to solve the problem themselves, stop here. /DSR]
Consider the island as a plane of complex numbers; draw one axis (the
real one) through the base of the two trees, and another axis (the
imaginary one) at right angles to the first, through a point half way
between the trees (Figure 11 [Omitted due to my poor drafting skills.
/DSR]). Taking one half of the distance between the trees as our unit
of length, we can say that the oak is located at the point +1 on the
real axis, and the pine at the point -1. We do not know where the
gallows was so let us denote its hypothetical location by the Greek
letter (capital gamma), which even looks like a gallows. [Since my
keyboard hasn't a gamma key, I'll use the letter T, instead. /DSR]
Since the gallows was not necessarily on one of the two axes T must be
considered as a complex number: T=a+bi, in which the meaning of a and b
is explained by Figure 11.
Now let us do some simple calculations remembering the rules of
imaginary multiplication as stated above. [Omitted. /DSR] If the
gallows is at T and the oak at -1, their separation in distance and
direction may be denoted by (-1)-T=-(1+T). Similarly the separation of
the gallows and the pine is 1-T. To turn these two distances by right
angles clockwise (to the right) and counterclockwise (to the left) we
must, according to the above rules multiply them by -i and by i, thus
finding the location at which we must place our two spikes as follows:
first spike: (-i)[-(1+T)]+1 = i(T+1)+1
second spike: (+i)[1-T)-1 = i(1-T)-1
Since the treasure is halfway between the spikes, we must now find one
half the sum of the two above complex numbers. We get:
�[i(T+1)+1+i(1-T)-1] = �[+iT+i+1+i=iT-1] = �(+2i) = +i.
We now see that the unknown position of the gallows denoted by T fell
out of our calculations somewhere along the way, and that, regardless
of where the gallows stood, the treasure must be located at the point
+i.
And so, if our adventurous young man could have done this simple bit of
mathematics, he would not have needed to dig up the entire island, but
would have looked for the treasure at the point indicated by the cross
in Figure 11, and there would have found the treasure.
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| <<< 2B::NOTES1:[NOTES$LIBRARY]MATH.NOTE;7 >>>
-< Mathematics at DEC >-
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Note 1281.1 The Gallows 1 of 1
CSSE::NEILSEN "I used to be PULSAR::WALLY" 36 lines 9-AUG-1990 16:31
-< how the story really ended >-
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The crew had watched this bozo the whole time. When he gave the order to
sail for home they spread-eagled him on the combing and persuaded him to
give up the instructions. Then the whole crew puzzled over it for a while.
The captain, who had sailed with Charles Darwin, refused to believe that a
gallows could disappear without a trace. So he rowed ashore to find the most
gallows-looking spot and start measuring from there.
The mate (who had had to leave Cambridge in a hurry) jumped up and
explained the whole complex number stuff in .0.
"Complex numbers my *ss!" roared the carpenter, who had once been an engineer
and was sensitive on the subject. "You're saying we don't need to know
where the gallows was, right? So let's just build a gallows and start from
there." He grabbed his tools and rowed for shore.
"First one to the treasure gets it." shouted the bosun, and piled several
of his friends into another boat. He had figured out that it would work
just as well if one crew member stood in a random spot to impersonate a gallows.
He had also been reading about parallel processing in _Byte_, and had figured
out that two crew members could walk to the trees in parallel.
The cabin boy jumped into the bosun's boat but said nothing. He had been
reading George Polya, and figured that if the gallows could be anywhere, he
would pick the most convenient spot: half-way between the two trees.
He started at the pine, walking towards the oak and counting his steps. When
he got to the oak he turned around and walked half the number of steps towards
the pine. Then he turned left and counted out the second half. There he dug.
The bosun, the captain and the carpenter showed up, and the cabin boy wisely
decided to share the treasure.
When they sailed back to the ship the mate sneered: "But where was your proof?"
The cabin boy explained that if it made a difference where the gallows was,
then they might as well sail off, so it was reasonable to assume the contrary
and dig once.
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| well, complex numbers are used heavily in Engineering, in doing the
calculations, it is sometime mathematically simpler to 'transform' your
math into the complex domain, and after you solve the problem, you
switch back the solution to the 'real' (time domain).
example, in a course on intro to Engineering Electromagnetics, we
almost always worked with Maxwell equations written in complex form.
also things like permitivity and permeability (material properties for
magnetic and electric conduction) is "expressed" as a complex quantity,
with real and imaginary parts.
iam not sure what you mean by real world, but i just think of working
in complex domain as just a mathematical tool, most physical responses can
be expressed as sinosoidel functions and compositions of them,
we use then can write down these responses or waves in general in complex
exponential form , using Euler relation (cos w + i sin w = e^iw) , then
things like differentiation becomes multiplications in complex form
i.e. d/dt(e^It) = I*e^T , etc..
it is like mathematical transforms in a way, you can solve a DE equation
by applying a transform the whole equation (Laplace or whatever), this
removes the d/dt operators leaving a simpler algebraic equation that you
solve in terms in the transform domain (S), then you inverse transform
this solution to get the real (time domain solution).
basic thing, i think of complex number just as a math tool, real life
is not complex offcourse :)
on an EM course i took, one student thought that EM waves were complex
because we wrote Maxwell equations in complex form :)
\bye
\nasser
ps. if i wrote something wrong, please note it is been more than 2
weeks since i took the course, so this is my excuse.
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| .-1
a periodic signals (such as AC current) can always be represents as
complex number.
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+ Z
| /| anti clock wise
| / | rotation of Z
|/d |
----------+---+-----
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the speed the tip of the complex vector rotate rotate around the
complex plane in circle represent the frequency (radian per second, W)
of the signal, d is phase shift, so this signal Z can be represented
as Z= (cos(d)+I sin(d)) + ( cos(Wt)+I sin(Wt) ) = e^Id + e^Iwt = e^I(Wt+d)
or if no initial phase shift, signal is just e^Iwt , either way it
is a complex number that can be represented as real part + I imaginary.
>Is this real life ?
electricity is sure is needed in real life :)
\nasser
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