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Miscellany/!4dMandia/docums/Teknic
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
| Tape/disk: | Home » Archimedes archive » Acorn Computing » 1993 09 Mega Disk.adf » 93_09 |
| Filename: | Miscellany/!4dMandia/docums/Teknic |
| Read OK: | ✔ |
| File size: | 33FA bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
Technical Stuff .....
I HATE proof reading, so if the following looses sense at any time
I must apologise. I wrote this some time ago and have not checked
it since then...
This explanation is not really technical, but does require a
reasonable understanding of *Basic* maths. Nothing else is
assumed, which means what follows is suitable to anyone who is a
beginner in fractals. So if you're just curious or are
interested in how such fractals work ... read on!
1.What is a fractal?
Well put simply it is just a mathematical equation into which
you feed a couple (or more) values, and feed the result back into
the equation. You run this process a number of times, forever, or
until a limit is reached. Depending on the final, or each result
the equation gives, the output picture (or anything else for that
matter) is obtained. For the Julia Set fractal, this process is
repeated for every pixel on the screen. Another feature of
fractals is that if you zoom into a fractal image, it never looses
it's complexity.
In life there are an amazing number of examples of fractal
geometry. The best one is a coastline: Look at a map of England
(or Norway ) and examine the coastline - all crinkly and bumpy.
Now get a map which covers a smaller area, for example of
Humberside, and examine the coastline ... it's still crinkly and
uneven. You can carry on zooming in, even to the side of a rock
pool and it will still be bumpy. This is the fractal geometry of
nature. (Of course you can not zoom in and look further than
atomic level due to the size and energy of photons!)
Other places include plants, for instance a fern: on each branch
is a smaller branch which is a copy of the parent, and on each
branch on each branch there is a copy of the parent ... this does
not go on forever, but quite some way.
My carpet (it's one of those with curly patterns) even looks
like it was designed using a fractal (julia set) generator. Life
is even based on a fractal type mechanism. In nature fractals can
be found every where.
Have you looked at a butterfly, all similar for each species,
but slightly different. To me they look like beautiful examples
of how beautiful fractal type pictures look.
'Before' the universe scientists believe that there was nothing,
but chaos..................
You might say '... but what real use do fractals have?'. The
correct answer is: Many, such as image enhancing, image
compressing, simulations of life (eg plants), encoding, and many
more.
With a scanned sheet of text, the quality is not always perfect
and for correct OCR (Optical Character Recognition) the quality
needs to be improved. This can be done by using a fractal
algorithm to tidy up the image.
A large picture can take anything up to, and more than, one mega
byte (1048576 bytes) and a floppy can be filled up Very quickly.
Also if you try to zoom in resolution is lost. But if the screen
is turned into a fractal algorithm, only a few bytes will be
taken up to store it, and if you zoom in (up to a point where it
usually becomes inaccurate) resolution will not be lost.
Biologists now believe that DNA in out bodies is not just a blue
print for ourselves, but a fractal algorithm which is run to
grow us. Some fractals can very well simulate plant growth and
come up with a very natural looking specimen.
(I presume you know what a pixel is - for those who do not know
it is just a dot on the screen. This whole display is made up of
many pixels, usually around 160000.)
2.Your first fractal.
Bifurcate is the original chaos formula and is made up of a very
simple formal which was first used to describe animal
populations:
New popul = Growth Rate * Old popul * ( 1 - Old popul )
where popul(ation) is a value between 0 and 1. With growth rates
less than 200%. Over 200% and the equation splits (bifurcates)
into 2, then 4, then becomes chaotic. This can also be used as a
pseudo-random number generator, with the best random output growth
rate of 4 (400%). Some simple 8-bit computers (eg. Spectrum etc.)
actually use this method for generating random numbers.
The formula is plotted with time across the bottom axis (X) and
population on the side axis (Y).
This is a very simple fractal, but some can be very complex. We
shall not go into these as this is only an introduction into
basic fractals.
3.Complex numbers.
These are not as they sound, but the idea is strange. Complex
numbers were invented to solve a problem: You can not work out
the square root of a negative number. As the square root is the
number multiplied by itself to get the first number, and any
number (even negative) multiplied by itself always gives a
positive result, then the square root of -1 is impossible
(imaginary). To over come this difficulty it was decided that
the imaginary number (represented by the letter 'i') squared was
equal to negative 1. I.E. i*i=-1 and i^2=-1. That is the only difficult
bit!
Here are some examples of what complex numbers look like when
written down: 4+7i (where 4 is real part and 7i the imaginary),
99-42i, 0+2i, 0+0i, 7-0i, a+bi (where a and b are both real
integers).
The other thing to know is that every complex number is made up
of two other numbers, one the real, one the imaginary. These two
names have no real meaning, it is just to distinguish between
them - they could be called any thing e.g. Starsky and Hutch.
Also a complex number is usually represented by the letter 'Z'
although it could be any letter. To keep it simple I usually take
Z to be made out of the two numbers 'a' and 'bi' where the 'i'
represents an imaginary number.
Adding and multiplying two complex numbers is easy:
To add, you just add (or multiply) the parts separately, this is
shown here:
3+4i + 6+2i = 9+6i
9-6i + 2+2i = 11-4i
-2-3i + 4-i = 2-4i
Multiplying is only slightly more difficult, you must remember
that i^2=-1. Here is a worked example:
(3-2i)*(7+4i) = 21+12i-14i-8i^2
= 21-2i-8i^2 and as i^2=-1 then (3-2i)*(7+4i)=21-2i-(8*-1)= 29-2i
If you want to square a complex number, then you just multiply it
by itself. This is shown here:
(3+4i)^2 = 9+12i+12i+16i^2 = 9+24i+16i^2 and as i^2=-1 then
(3+4i)^2 = 9+24i+(16*-1) = -7+24i
A general formula would then be:
(a+bi)^2 = a^2+abi+abi+(b^2*i^2) as i^2=-1 then =a^2+2abi-b^2
group the real and imaginary terms to form the general terms:
a(n+1)=a(n)^2-b(n)^2
b(n+1)=2*a(n)*b(n)i
One more thing to do with complex numbers and fractals is, size.
When we talk about size, we mean, on the two planes real and
imaginary, the distance between the position and 0,0. You could
look as the real and imaginary parts as sides of a right-angle
triangle where the distance is the hypotenuse. To find the
hypotenuse you can use Pythagoras' theorem (a^2=b^2+c^2). So to
find the size of a complex number, you square both the real and
imaginary parts, add them together, and then square root this.
A straight forward result in complex-number theory iterations
guarantees that the iterations will drive Z to infinity, if and
only at some stage Z reaches a size of 2 or greater. Very many
points will reach 2 after only a few iterations, the ones that do
not belong to the Mandlebrot Set.
Do not worry if you did not follow all that, it will become
clear to you in time. Even if you could not possibly understand
it all you only need to know of the general term for squaring a
complex number (and the bit about size) for most fractals. Some
fractals such as the cubic Julia Set use cubing complex numbers,
but we will not go into those in great detail. Above only covers
SOME of the complex-number theory, and if you want to further
your knowledge then read some old A-level or degree mathematics
books on the subject.
4.The Julia Set
This is the fractal that I choose because I have seen very few
COLOUR julia sets, I have only seen ones which use inverse
iteration methods (I will explain later) which only give
monochrome images. The other reason is that this one shows how to
plot a basic fractal using an individual pixel iteration method,
and the Julia Set is an easy one to understand.
Basically the Julia Set uses a Z^2+C formula which is iterated
until a limit is reached, this is done for each pixel. There are
an infinite number of different sets, each one is defined by the
complex number 'C'. Both it's real and imaginary parts define the
shape of the Julia Set and are usually (roughly) between -2 and
2. There are two types of Julia Set - Wholly Connected and Wholly
Dis-connected. Wholly Connected is usually with small parts to
the complex-number 'C'.
To plot a particular Julia Set you have the screen representing
the real(X) and imaginary(Y) planes from -2 to 2. Then set a
complex-number variable 'Z' to the co-ordinates on the screen.
Set a variable count to zero. Set the complex-number variable 'C'
to the value you want for the particular set. Now carry out the
following loop:
.loop Z=Z^2
Z=Z+C
count=count+1
IF count>256 OR SIZE(Z)>2 GOTO end
GOTO loop
Assign the value of count to the colour of the pixel. If the
size of Z has not risen equal to, or above 2, then the pixel is
black, and belongs to the actual Julia Set. Do this for every
pixel and the image will build up.
An example program written in ARM BBC BASIC V is shown here:
(example 2)
1 PROCinit
10 realMIN=-2
20 imagMIN=-2
30 realMAX=2
40 imagMAX=2
50 realC=-.75
60 imagC=0
70 xySIZE=256
80 realINC=(realMAX-realMIN)/xySIZE
90 imagINC=(imagMAX-imagMIN)/xySIZE
100 imagZ=imagMIN
110 Y=0
120 REPEAT
130 realZ=realMIN
140 X=0
150 REPEAT
160 realZZ=realZ
170 imagZZ=imagZ
180 count=0
190 REPEAT
200 s=realZZ^2-imagZZ^2
210 imagZZ=2*realZZ*imagZZ
220 realZZ=s+realC
230 imagZZ=imagZZ+imagC
240 count+=1
250 UNTIL (realZZ^2+imagZZ^2)>=2 OR count>=256
260 PROCplot(X,Y,count)
270 X+=1
280 realZ+=realINC
290 UNTIL X>=xySIZE
300 Y+=1
310 imagZ+=imagINC
320 UNTIL Y>=xySIZE
330 END
340 :
350 DEFPROCinit
360 MODE13
370 link=14:pc=15
380 DIM MC &100
390 FOR pass%=0 TO 2 STEP 2
400 P%=MC
410 [OPT pass%
420 .vdu EQUD 148:EQUD -1
430 .plot LDR r3,vdu
440 ADD r3,r3,r1,LSL #8
450 ADD r3,r3,r1,LSL #6
460 STRB r2,[r3,r0]
470 MOV pc,link
480 ]
490 NEXT
500 SYS "OS_ReadVduVariables",vdu,vdu
510 ENDPROC
520 :
530 DEFPROCplot(X,Y,count)
540 A%=X:B%=Y
550 C%=count MOD 256
560 CALLplot
570 ENDPROC
You must realise that understanding how to program fractals can
be slightly more difficult than understanding them, so if you do
not feel that you are up to doing some more on programs to do
with fractals
The method for plotting can be what ever you like, here I choose
to directly write to screen memory as you do not need to bother
dealing with there not being the same number of pixels as plot
numbers. (ie a plot command can take roughly 0-1024 as a co-
ordinate, but the screen in mode 13 has only roughly 256 pixels
across and up).
In 256 colour screen modes such as MODE 13 it is easy to draw by
directly writing to the screen as each pixel takes one byte. You
just select the colour by writing a number between 0 and 256. All
very simple compared with other number of colours modes.
Normally it is a bad idea to write directly to hardware without
going through the operating system, and on an Archimedes the
screen memory does not stay in the same place as on a BBC, but
Acorn have provided a nice little operating system call to find
out exactly where to write to.
In fact the above listing (example2) is a long version just to
make how it works more clear. Don't bother trying to understand
the machine code plotting routine if you have not come across
this sort of thing before as it will just make it more difficult
for you to understand. Here are a few example BASIC one liner's,
the first one is the same as the above Julia Set plotter, but
changed and compacted onto one line:
One line Julia Set plotter:
For Archi, but will have to type in using BASIC Editor.
10MO.13:C1=-.75:C2=0:S=245:rn=-2:in=-2:rx=2:ix=2:rin=(rx-rn)/S:
iin=(ix-in)/S:iZ=in:Y=0:REP.:rZ=rn:X=0:REP.:A=rZ:B=iZ:c=0:REP.:
s=A^2-B^2+C1:B=2*A*B+C2:A=s:c+=1:UN.(A^2+B^2)>=2ORc>=256:GCOL0,c
MOD 64 TINT c DIV4:POINT4*X,4*Y:X+=1:rZ+=rin:UN.X>=S:Y+=1:
iZ+=iin:UN.Y>=S
Less easily changed Archi version, and using only 64 colours.
10MODE13:C1=-.75:C2=0:S=245:inc=4/S:Y=0:iZ=-2:REP.:rZ=-2:X=0:
REP.:A=rZ:B=iZ:c=0:REP.:s=A^2-B^2+C1:B=2*A*B:A=s:c+=1:UNTIL
(A^2+B^2)>=2ORc>=256:GCOL0,c MOD 64:POINT4*X,4*Y:X+=1:rZ+=inc:
UNTILX>=S:Y+=1:iZ+=inc:UNTILY>=S
In both of the above, change S to the size of image required.
That concludes what I am going to say for now, but if anyone
wants me to continue, send me �5 with S.A.E., or send me a
letter, and I will put it in the public domain for you.
Bye .... for now ........
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T| 000000b0 68 69 73 20 65 78 70 6c 61 6e 61 74 69 6f 6e 20 |his explanation | 000000c0 69 73 20 6e 6f 74 20 72 65 61 6c 6c 79 20 74 65 |is not really te| 000000d0 63 68 6e 69 63 61 6c 2c 20 62 75 74 20 64 6f 65 |chnical, but doe| 000000e0 73 20 72 65 71 75 69 72 65 20 61 20 0a 72 65 61 |s require a .rea| 000000f0 73 6f 6e 61 62 6c 65 20 75 6e 64 65 72 73 74 61 |sonable understa| 00000100 6e 64 69 6e 67 20 6f 66 20 2a 42 61 73 69 63 2a |nding of *Basic*| 00000110 20 6d 61 74 68 73 2e 20 4e 6f 74 68 69 6e 67 20 | maths. Nothing | 00000120 65 6c 73 65 20 69 73 0a 61 73 73 75 6d 65 64 2c |else is.assumed,| 00000130 20 77 68 69 63 68 20 6d 65 61 6e 73 20 77 68 61 | which means wha| 00000140 74 20 66 6f 6c 6c 6f 77 73 20 69 73 20 73 75 69 |t follows is sui| 00000150 74 61 62 6c 65 20 74 6f 20 61 6e 79 6f 6e 65 20 |table to anyone | 00000160 77 68 6f 20 69 73 20 61 0a 62 65 67 69 6e 6e 65 |who is a.beginne| 00000170 72 20 69 6e 20 66 72 61 63 74 61 6c 73 2e 20 53 |r in fractals. S| 00000180 6f 20 69 66 20 79 6f 75 27 72 65 20 6a 75 73 74 |o if you're just| 00000190 20 63 75 72 69 6f 75 73 20 6f 72 20 61 72 65 0a | curious or are.| 000001a0 69 6e 74 65 72 65 73 74 65 64 20 69 6e 20 68 6f |interested in ho| 000001b0 77 20 73 75 63 68 20 66 72 61 63 74 61 6c 73 20 |w such fractals | 000001c0 77 6f 72 6b 20 2e 2e 2e 20 72 65 61 64 20 6f 6e |work ... read on| 000001d0 21 0a 0a 20 31 2e 57 68 61 74 20 69 73 20 61 20 |!.. 1.What is a | 000001e0 66 72 61 63 74 61 6c 3f 0a 20 57 65 6c 6c 20 70 |fractal?. Well p| 000001f0 75 74 20 73 69 6d 70 6c 79 20 69 74 20 69 73 20 |ut simply it is | 00000200 6a 75 73 74 20 61 20 6d 61 74 68 65 6d 61 74 69 |just a mathemati| 00000210 63 61 6c 20 65 71 75 61 74 69 6f 6e 20 69 6e 74 |cal equation int| 00000220 6f 20 77 68 69 63 68 20 0a 79 6f 75 20 66 65 65 |o which .you fee| 00000230 64 20 61 20 63 6f 75 70 6c 65 20 28 6f 72 20 6d |d a couple (or m| 00000240 6f 72 65 29 20 76 61 6c 75 65 73 2c 20 61 6e 64 |ore) values, and| 00000250 20 66 65 65 64 20 74 68 65 20 72 65 73 75 6c 74 | feed the result| 00000260 20 62 61 63 6b 20 69 6e 74 6f 20 0a 74 68 65 20 | back into .the | 00000270 65 71 75 61 74 69 6f 6e 2e 20 59 6f 75 20 72 75 |equation. You ru| 00000280 6e 20 74 68 69 73 20 70 72 6f 63 65 73 73 20 61 |n this process a| 00000290 20 6e 75 6d 62 65 72 20 6f 66 20 74 69 6d 65 73 | number of times| 000002a0 2c 20 66 6f 72 65 76 65 72 2c 20 6f 72 20 0a 75 |, forever, or .u| 000002b0 6e 74 69 6c 20 61 20 6c 69 6d 69 74 20 69 73 20 |ntil a limit is | 000002c0 72 65 61 63 68 65 64 2e 20 44 65 70 65 6e 64 69 |reached. Dependi| 000002d0 6e 67 20 6f 6e 20 74 68 65 20 66 69 6e 61 6c 2c |ng on the final,| 000002e0 20 6f 72 20 65 61 63 68 20 72 65 73 75 6c 74 20 | or each result | 000002f0 0a 74 68 65 20 65 71 75 61 74 69 6f 6e 20 67 69 |.the equation gi| 00000300 76 65 73 2c 20 74 68 65 20 6f 75 74 70 75 74 20 |ves, the output | 00000310 70 69 63 74 75 72 65 20 28 6f 72 20 61 6e 79 74 |picture (or anyt| 00000320 68 69 6e 67 20 65 6c 73 65 20 66 6f 72 20 74 68 |hing else for th| 00000330 61 74 20 0a 6d 61 74 74 65 72 29 20 69 73 20 6f |at .matter) is o| 00000340 62 74 61 69 6e 65 64 2e 20 46 6f 72 20 74 68 65 |btained. For the| 00000350 20 4a 75 6c 69 61 20 53 65 74 20 66 72 61 63 74 | Julia Set fract| 00000360 61 6c 2c 20 74 68 69 73 20 70 72 6f 63 65 73 73 |al, this process| 00000370 20 69 73 20 0a 72 65 70 65 61 74 65 64 20 66 6f | is .repeated fo| 00000380 72 20 65 76 65 72 79 20 70 69 78 65 6c 20 6f 6e |r every pixel on| 00000390 20 74 68 65 20 73 63 72 65 65 6e 2e 20 41 6e 6f | the screen. Ano| 000003a0 74 68 65 72 20 66 65 61 74 75 72 65 20 6f 66 0a |ther feature of.| 000003b0 66 72 61 63 74 61 6c 73 20 69 73 20 74 68 61 74 |fractals is that| 000003c0 20 69 66 20 79 6f 75 20 7a 6f 6f 6d 20 69 6e 74 | if you zoom int| 000003d0 6f 20 61 20 66 72 61 63 74 61 6c 20 69 6d 61 67 |o a fractal imag| 000003e0 65 2c 20 69 74 20 6e 65 76 65 72 20 6c 6f 6f 73 |e, it never loos| 000003f0 65 73 0a 69 74 27 73 20 20 63 6f 6d 70 6c 65 78 |es.it's complex| 00000400 69 74 79 2e 0a 0a 20 49 6e 20 6c 69 66 65 20 74 |ity... In life t| 00000410 68 65 72 65 20 61 72 65 20 61 6e 20 61 6d 61 7a |here are an amaz| 00000420 69 6e 67 20 6e 75 6d 62 65 72 20 6f 66 20 65 78 |ing number of ex| 00000430 61 6d 70 6c 65 73 20 6f 66 20 66 72 61 63 74 61 |amples of fracta| 00000440 6c 20 0a 67 65 6f 6d 65 74 72 79 2e 20 54 68 65 |l .geometry. The| 00000450 20 62 65 73 74 20 6f 6e 65 20 69 73 20 61 20 63 | best one is a c| 00000460 6f 61 73 74 6c 69 6e 65 3a 20 4c 6f 6f 6b 20 61 |oastline: Look a| 00000470 74 20 61 20 6d 61 70 20 6f 66 20 45 6e 67 6c 61 |t a map of Engla| 00000480 6e 64 20 0a 28 6f 72 20 4e 6f 72 77 61 79 20 29 |nd .(or Norway )| 00000490 20 61 6e 64 20 65 78 61 6d 69 6e 65 20 74 68 65 | and examine the| 000004a0 20 63 6f 61 73 74 6c 69 6e 65 20 2d 20 61 6c 6c | coastline - all| 000004b0 20 20 63 72 69 6e 6b 6c 79 20 61 6e 64 20 62 75 | crinkly and bu| 000004c0 6d 70 79 2e 0a 4e 6f 77 20 67 65 74 20 61 20 6d |mpy..Now get a m| 000004d0 61 70 20 77 68 69 63 68 20 63 6f 76 65 72 73 20 |ap which covers | 000004e0 61 20 73 6d 61 6c 6c 65 72 20 61 72 65 61 2c 20 |a smaller area, | 000004f0 66 6f 72 20 20 65 78 61 6d 70 6c 65 20 6f 66 0a |for example of.| 00000500 48 75 6d 62 65 72 73 69 64 65 2c 20 61 6e 64 20 |Humberside, and | 00000510 65 78 61 6d 69 6e 65 20 74 68 65 20 63 6f 61 73 |examine the coas| 00000520 74 6c 69 6e 65 20 2e 2e 2e 20 69 74 27 73 20 73 |tline ... it's s| 00000530 74 69 6c 6c 20 20 63 72 69 6e 6b 6c 79 20 61 6e |till crinkly an| 00000540 64 0a 75 6e 65 76 65 6e 2e 20 59 6f 75 20 63 61 |d.uneven. You ca| 00000550 6e 20 63 61 72 72 79 20 6f 6e 20 7a 6f 6f 6d 69 |n carry on zoomi| 00000560 6e 67 20 69 6e 2c 20 65 76 65 6e 20 74 6f 20 74 |ng in, even to t| 00000570 68 65 20 73 69 64 65 20 20 6f 66 20 61 20 72 6f |he side of a ro| 00000580 63 6b 0a 70 6f 6f 6c 20 61 6e 64 20 69 74 20 77 |ck.pool and it w| 00000590 69 6c 6c 20 73 74 69 6c 6c 20 62 65 20 62 75 6d |ill still be bum| 000005a0 70 79 2e 20 54 68 69 73 20 69 73 20 74 68 65 20 |py. This is the | 000005b0 66 72 61 63 74 61 6c 20 67 65 6f 6d 65 74 72 79 |fractal geometry| 000005c0 20 6f 66 0a 6e 61 74 75 72 65 2e 20 28 4f 66 20 | of.nature. (Of | 000005d0 63 6f 75 72 73 65 20 79 6f 75 20 63 61 6e 20 6e |course you can n| 000005e0 6f 74 20 7a 6f 6f 6d 20 69 6e 20 61 6e 64 20 6c |ot zoom in and l| 000005f0 6f 6f 6b 20 66 75 72 74 68 65 72 20 74 68 61 6e |ook further than| 00000600 0a 61 74 6f 6d 69 63 20 6c 65 76 65 6c 20 64 75 |.atomic level du| 00000610 65 20 74 6f 20 74 68 65 20 73 69 7a 65 20 61 6e |e to the size an| 00000620 64 20 65 6e 65 72 67 79 20 6f 66 20 70 68 6f 74 |d energy of phot| 00000630 6f 6e 73 21 29 0a 20 4f 74 68 65 72 20 70 6c 61 |ons!). Other pla| 00000640 63 65 73 20 69 6e 63 6c 75 64 65 20 70 6c 61 6e |ces include plan| 00000650 74 73 2c 20 66 6f 72 20 69 6e 73 74 61 6e 63 65 |ts, for instance| 00000660 20 61 20 66 65 72 6e 3a 20 6f 6e 20 65 61 63 68 | a fern: on each| 00000670 20 62 72 61 6e 63 68 20 0a 69 73 20 61 20 73 6d | branch .is a sm| 00000680 61 6c 6c 65 72 20 62 72 61 6e 63 68 20 77 68 69 |aller branch whi| 00000690 63 68 20 69 73 20 61 20 63 6f 70 79 20 6f 66 20 |ch is a copy of | 000006a0 74 68 65 20 70 61 72 65 6e 74 2c 20 61 6e 64 20 |the parent, and | 000006b0 6f 6e 20 65 61 63 68 20 0a 62 72 61 6e 63 68 20 |on each .branch | 000006c0 6f 6e 20 65 61 63 68 20 62 72 61 6e 63 68 20 74 |on each branch t| 000006d0 68 65 72 65 20 69 73 20 61 20 63 6f 70 79 20 6f |here is a copy o| 000006e0 66 20 74 68 65 20 70 61 72 65 6e 74 20 2e 2e 2e |f the parent ...| 000006f0 20 74 68 69 73 20 64 6f 65 73 20 0a 6e 6f 74 20 | this does .not | 00000700 67 6f 20 6f 6e 20 66 6f 72 65 76 65 72 2c 20 62 |go on forever, b| 00000710 75 74 20 71 75 69 74 65 20 73 6f 6d 65 20 77 61 |ut quite some wa| 00000720 79 2e 0a 20 4d 79 20 63 61 72 70 65 74 20 28 69 |y.. My carpet (i| 00000730 74 27 73 20 6f 6e 65 20 6f 66 20 74 68 6f 73 65 |t's one of those| 00000740 20 77 69 74 68 20 63 75 72 6c 79 20 70 61 74 74 | with curly patt| 00000750 65 72 6e 73 29 20 65 76 65 6e 20 6c 6f 6f 6b 73 |erns) even looks| 00000760 20 0a 6c 69 6b 65 20 69 74 20 77 61 73 20 64 65 | .like it was de| 00000770 73 69 67 6e 65 64 20 75 73 69 6e 67 20 61 20 66 |signed using a f| 00000780 72 61 63 74 61 6c 20 28 6a 75 6c 69 61 20 73 65 |ractal (julia se| 00000790 74 29 20 67 65 6e 65 72 61 74 6f 72 2e 20 4c 69 |t) generator. Li| 000007a0 66 65 20 0a 69 73 20 65 76 65 6e 20 62 61 73 65 |fe .is even base| 000007b0 64 20 6f 6e 20 61 20 66 72 61 63 74 61 6c 20 74 |d on a fractal t| 000007c0 79 70 65 20 6d 65 63 68 61 6e 69 73 6d 2e 20 49 |ype mechanism. I| 000007d0 6e 20 6e 61 74 75 72 65 20 66 72 61 63 74 61 6c |n nature fractal| 000007e0 73 20 63 61 6e 20 0a 62 65 20 66 6f 75 6e 64 20 |s can .be found | 000007f0 65 76 65 72 79 20 77 68 65 72 65 2e 0a 20 48 61 |every where.. Ha| 00000800 76 65 20 79 6f 75 20 6c 6f 6f 6b 65 64 20 61 74 |ve you looked at| 00000810 20 61 20 62 75 74 74 65 72 66 6c 79 2c 20 61 6c | a butterfly, al| 00000820 6c 20 73 69 6d 69 6c 61 72 20 66 6f 72 20 65 61 |l similar for ea| 00000830 63 68 20 73 70 65 63 69 65 73 2c 20 0a 62 75 74 |ch species, .but| 00000840 20 73 6c 69 67 68 74 6c 79 20 64 69 66 66 65 72 | slightly differ| 00000850 65 6e 74 2e 20 54 6f 20 6d 65 20 74 68 65 79 20 |ent. To me they | 00000860 6c 6f 6f 6b 20 6c 69 6b 65 20 62 65 61 75 74 69 |look like beauti| 00000870 66 75 6c 20 65 78 61 6d 70 6c 65 73 20 0a 6f 66 |ful examples .of| 00000880 20 68 6f 77 20 62 65 61 75 74 69 66 75 6c 20 66 | how beautiful f| 00000890 72 61 63 74 61 6c 20 74 79 70 65 20 70 69 63 74 |ractal type pict| 000008a0 75 72 65 73 20 6c 6f 6f 6b 2e 0a 20 27 42 65 66 |ures look.. 'Bef| 000008b0 6f 72 65 27 20 74 68 65 20 75 6e 69 76 65 72 73 |ore' the univers| 000008c0 65 20 73 63 69 65 6e 74 69 73 74 73 20 62 65 6c |e scientists bel| 000008d0 69 65 76 65 20 74 68 61 74 20 74 68 65 72 65 20 |ieve that there | 000008e0 77 61 73 20 6e 6f 74 68 69 6e 67 2c 20 0a 62 75 |was nothing, .bu| 000008f0 74 20 63 68 61 6f 73 2e 2e 2e 2e 2e 2e 2e 2e 2e |t chaos.........| 00000900 2e 2e 2e 2e 2e 2e 2e 2e 2e 0a 0a 20 59 6f 75 20 |........... You | 00000910 6d 69 67 68 74 20 73 61 79 20 27 2e 2e 2e 20 62 |might say '... b| 00000920 75 74 20 77 68 61 74 20 72 65 61 6c 20 75 73 65 |ut what real use| 00000930 20 64 6f 20 66 72 61 63 74 61 6c 73 20 68 61 76 | do fractals hav| 00000940 65 3f 27 2e 20 54 68 65 20 0a 63 6f 72 72 65 63 |e?'. The .correc| 00000950 74 20 61 6e 73 77 65 72 20 69 73 3a 20 4d 61 6e |t answer is: Man| 00000960 79 2c 20 73 75 63 68 20 61 73 20 69 6d 61 67 65 |y, such as image| 00000970 20 65 6e 68 61 6e 63 69 6e 67 2c 20 69 6d 61 67 | enhancing, imag| 00000980 65 20 0a 63 6f 6d 70 72 65 73 73 69 6e 67 2c 20 |e .compressing, | 00000990 73 69 6d 75 6c 61 74 69 6f 6e 73 20 6f 66 20 6c |simulations of l| 000009a0 69 66 65 20 28 65 67 20 70 6c 61 6e 74 73 29 2c |ife (eg plants),| 000009b0 20 65 6e 63 6f 64 69 6e 67 2c 20 61 6e 64 20 6d | encoding, and m| 000009c0 61 6e 79 0a 6d 6f 72 65 2e 0a 20 57 69 74 68 20 |any.more.. With | 000009d0 61 20 73 63 61 6e 6e 65 64 20 73 68 65 65 74 20 |a scanned sheet | 000009e0 6f 66 20 74 65 78 74 2c 20 74 68 65 20 71 75 61 |of text, the qua| 000009f0 6c 69 74 79 20 69 73 20 6e 6f 74 20 61 6c 77 61 |lity is not alwa| 00000a00 79 73 20 70 65 72 66 65 63 74 0a 61 6e 64 20 66 |ys perfect.and f| 00000a10 6f 72 20 63 6f 72 72 65 63 74 20 4f 43 52 20 28 |or correct OCR (| 00000a20 4f 70 74 69 63 61 6c 20 43 68 61 72 61 63 74 65 |Optical Characte| 00000a30 72 20 52 65 63 6f 67 6e 69 74 69 6f 6e 29 20 74 |r Recognition) t| 00000a40 68 65 20 71 75 61 6c 69 74 79 0a 6e 65 65 64 73 |he quality.needs| 00000a50 20 74 6f 20 62 65 20 69 6d 70 72 6f 76 65 64 2e | to be improved.| 00000a60 20 54 68 69 73 20 63 61 6e 20 62 65 20 64 6f 6e | This can be don| 00000a70 65 20 62 79 20 75 73 69 6e 67 20 61 20 66 72 61 |e by using a fra| 00000a80 63 74 61 6c 0a 61 6c 67 6f 72 69 74 68 6d 20 74 |ctal.algorithm t| 00000a90 6f 20 74 69 64 79 20 75 70 20 74 68 65 20 69 6d |o tidy up the im| 00000aa0 61 67 65 2e 0a 20 41 20 6c 61 72 67 65 20 70 69 |age.. A large pi| 00000ab0 63 74 75 72 65 20 63 61 6e 20 74 61 6b 65 20 61 |cture can take a| 00000ac0 6e 79 74 68 69 6e 67 20 75 70 20 74 6f 2c 20 61 |nything up to, a| 00000ad0 6e 64 20 6d 6f 72 65 20 74 68 61 6e 2c 20 6f 6e |nd more than, on| 00000ae0 65 20 6d 65 67 61 0a 62 79 74 65 20 28 31 30 34 |e mega.byte (104| 00000af0 38 35 37 36 20 62 79 74 65 73 29 20 61 6e 64 20 |8576 bytes) and | 00000b00 61 20 66 6c 6f 70 70 79 20 63 61 6e 20 62 65 20 |a floppy can be | 00000b10 66 69 6c 6c 65 64 20 75 70 20 56 65 72 79 20 71 |filled up Very q| 00000b20 75 69 63 6b 6c 79 2e 0a 41 6c 73 6f 20 69 66 20 |uickly..Also if | 00000b30 79 6f 75 20 74 72 79 20 74 6f 20 7a 6f 6f 6d 20 |you try to zoom | 00000b40 69 6e 20 72 65 73 6f 6c 75 74 69 6f 6e 20 69 73 |in resolution is| 00000b50 20 6c 6f 73 74 2e 20 42 75 74 20 69 66 20 74 68 | lost. But if th| 00000b60 65 20 73 63 72 65 65 6e 0a 69 73 20 74 75 72 6e |e screen.is turn| 00000b70 65 64 20 69 6e 74 6f 20 61 20 66 72 61 63 74 61 |ed into a fracta| 00000b80 6c 20 61 6c 67 6f 72 69 74 68 6d 2c 20 6f 6e 6c |l algorithm, onl| 00000b90 79 20 61 20 66 65 77 20 62 79 74 65 73 20 77 69 |y a few bytes wi| 00000ba0 6c 6c 20 62 65 0a 74 61 6b 65 6e 20 75 70 20 74 |ll be.taken up t| 00000bb0 6f 20 73 74 6f 72 65 20 69 74 2c 20 61 6e 64 20 |o store it, and | 00000bc0 69 66 20 79 6f 75 20 7a 6f 6f 6d 20 69 6e 20 28 |if you zoom in (| 00000bd0 75 70 20 74 6f 20 61 20 70 6f 69 6e 74 20 77 68 |up to a point wh| 00000be0 65 72 65 20 69 74 0a 75 73 75 61 6c 6c 79 20 62 |ere it.usually b| 00000bf0 65 63 6f 6d 65 73 20 69 6e 61 63 63 75 72 61 74 |ecomes inaccurat| 00000c00 65 29 20 72 65 73 6f 6c 75 74 69 6f 6e 20 77 69 |e) resolution wi| 00000c10 6c 6c 20 6e 6f 74 20 62 65 20 6c 6f 73 74 2e 20 |ll not be lost. | 00000c20 0a 20 42 69 6f 6c 6f 67 69 73 74 73 20 6e 6f 77 |. Biologists now| 00000c30 20 62 65 6c 69 65 76 65 20 74 68 61 74 20 44 4e | believe that DN| 00000c40 41 20 69 6e 20 6f 75 74 20 62 6f 64 69 65 73 20 |A in out bodies | 00000c50 69 73 20 6e 6f 74 20 6a 75 73 74 20 61 20 62 6c |is not just a bl| 00000c60 75 65 20 0a 70 72 69 6e 74 20 66 6f 72 20 6f 75 |ue .print for ou| 00000c70 72 73 65 6c 76 65 73 2c 20 62 75 74 20 61 20 66 |rselves, but a f| 00000c80 72 61 63 74 61 6c 20 61 6c 67 6f 72 69 74 68 6d |ractal algorithm| 00000c90 20 77 68 69 63 68 20 69 73 20 72 75 6e 20 74 6f | which is run to| 00000ca0 20 0a 67 72 6f 77 20 75 73 2e 20 53 6f 6d 65 20 | .grow us. Some | 00000cb0 66 72 61 63 74 61 6c 73 20 63 61 6e 20 76 65 72 |fractals can ver| 00000cc0 79 20 77 65 6c 6c 20 73 69 6d 75 6c 61 74 65 20 |y well simulate | 00000cd0 70 6c 61 6e 74 20 67 72 6f 77 74 68 20 61 6e 64 |plant growth and| 00000ce0 20 0a 63 6f 6d 65 20 75 70 20 77 69 74 68 20 61 | .come up with a| 00000cf0 20 76 65 72 79 20 6e 61 74 75 72 61 6c 20 6c 6f | very natural lo| 00000d00 6f 6b 69 6e 67 20 73 70 65 63 69 6d 65 6e 2e 0a |oking specimen..| 00000d10 0a 28 49 20 70 72 65 73 75 6d 65 20 79 6f 75 20 |.(I presume you | 00000d20 6b 6e 6f 77 20 77 68 61 74 20 61 20 70 69 78 65 |know what a pixe| 00000d30 6c 20 69 73 20 2d 20 66 6f 72 20 74 68 6f 73 65 |l is - for those| 00000d40 20 77 68 6f 20 64 6f 20 6e 6f 74 20 6b 6e 6f 77 | who do not know| 00000d50 20 0a 69 74 20 69 73 20 6a 75 73 74 20 61 20 64 | .it is just a d| 00000d60 6f 74 20 6f 6e 20 74 68 65 20 73 63 72 65 65 6e |ot on the screen| 00000d70 2e 20 54 68 69 73 20 77 68 6f 6c 65 20 64 69 73 |. This whole dis| 00000d80 70 6c 61 79 20 69 73 20 6d 61 64 65 20 75 70 20 |play is made up | 00000d90 6f 66 20 0a 6d 61 6e 79 20 70 69 78 65 6c 73 2c |of .many pixels,| 00000da0 20 75 73 75 61 6c 6c 79 20 61 72 6f 75 6e 64 20 | usually around | 00000db0 31 36 30 30 30 30 2e 29 0a 0a 0a 32 2e 59 6f 75 |160000.)...2.You| 00000dc0 72 20 66 69 72 73 74 20 66 72 61 63 74 61 6c 2e |r first fractal.| 00000dd0 0a 0a 20 42 69 66 75 72 63 61 74 65 20 69 73 20 |.. Bifurcate is | 00000de0 74 68 65 20 6f 72 69 67 69 6e 61 6c 20 63 68 61 |the original cha| 00000df0 6f 73 20 66 6f 72 6d 75 6c 61 20 61 6e 64 20 69 |os formula and i| 00000e00 73 20 6d 61 64 65 20 75 70 20 6f 66 20 61 20 76 |s made up of a v| 00000e10 65 72 79 20 0a 73 69 6d 70 6c 65 20 66 6f 72 6d |ery .simple form| 00000e20 61 6c 20 77 68 69 63 68 20 77 61 73 20 66 69 72 |al which was fir| 00000e30 73 74 20 75 73 65 64 20 74 6f 20 64 65 73 63 72 |st used to descr| 00000e40 69 62 65 20 61 6e 69 6d 61 6c 20 0a 70 6f 70 75 |ibe animal .popu| 00000e50 6c 61 74 69 6f 6e 73 3a 0a 0a 20 20 20 20 20 20 |lations:.. | 00000e60 20 20 4e 65 77 20 70 6f 70 75 6c 20 3d 20 47 72 | New popul = Gr| 00000e70 6f 77 74 68 20 52 61 74 65 20 2a 20 4f 6c 64 20 |owth Rate * Old | 00000e80 70 6f 70 75 6c 20 2a 20 28 20 31 20 2d 20 4f 6c |popul * ( 1 - Ol| 00000e90 64 20 70 6f 70 75 6c 20 29 0a 0a 77 68 65 72 65 |d popul )..where| 00000ea0 20 70 6f 70 75 6c 28 61 74 69 6f 6e 29 20 69 73 | popul(ation) is| 00000eb0 20 61 20 76 61 6c 75 65 20 62 65 74 77 65 65 6e | a value between| 00000ec0 20 30 20 61 6e 64 20 31 2e 20 57 69 74 68 20 67 | 0 and 1. With g| 00000ed0 72 6f 77 74 68 20 72 61 74 65 73 20 0a 6c 65 73 |rowth rates .les| 00000ee0 73 20 74 68 61 6e 20 32 30 30 25 2e 20 4f 76 65 |s than 200%. Ove| 00000ef0 72 20 32 30 30 25 20 61 6e 64 20 74 68 65 20 65 |r 200% and the e| 00000f00 71 75 61 74 69 6f 6e 20 73 70 6c 69 74 73 20 28 |quation splits (| 00000f10 62 69 66 75 72 63 61 74 65 73 29 20 0a 69 6e 74 |bifurcates) .int| 00000f20 6f 20 32 2c 20 74 68 65 6e 20 34 2c 20 74 68 65 |o 2, then 4, the| 00000f30 6e 20 62 65 63 6f 6d 65 73 20 63 68 61 6f 74 69 |n becomes chaoti| 00000f40 63 2e 20 54 68 69 73 20 63 61 6e 20 61 6c 73 6f |c. This can also| 00000f50 20 62 65 20 75 73 65 64 20 61 73 20 61 20 0a 70 | be used as a .p| 00000f60 73 65 75 64 6f 2d 72 61 6e 64 6f 6d 20 6e 75 6d |seudo-random num| 00000f70 62 65 72 20 67 65 6e 65 72 61 74 6f 72 2c 20 77 |ber generator, w| 00000f80 69 74 68 20 74 68 65 20 62 65 73 74 20 72 61 6e |ith the best ran| 00000f90 64 6f 6d 20 6f 75 74 70 75 74 20 67 72 6f 77 74 |dom output growt| 00000fa0 68 20 0a 72 61 74 65 20 6f 66 20 34 20 28 34 30 |h .rate of 4 (40| 00000fb0 30 25 29 2e 20 53 6f 6d 65 20 73 69 6d 70 6c 65 |0%). Some simple| 00000fc0 20 38 2d 62 69 74 20 63 6f 6d 70 75 74 65 72 73 | 8-bit computers| 00000fd0 20 28 65 67 2e 20 53 70 65 63 74 72 75 6d 20 65 | (eg. Spectrum e| 00000fe0 74 63 2e 29 20 0a 61 63 74 75 61 6c 6c 79 20 75 |tc.) .actually u| 00000ff0 73 65 20 74 68 69 73 20 6d 65 74 68 6f 64 20 66 |se this method f| 00001000 6f 72 20 67 65 6e 65 72 61 74 69 6e 67 20 72 61 |or generating ra| 00001010 6e 64 6f 6d 20 6e 75 6d 62 65 72 73 2e 0a 20 54 |ndom numbers.. T| 00001020 68 65 20 66 6f 72 6d 75 6c 61 20 69 73 20 70 6c |he formula is pl| 00001030 6f 74 74 65 64 20 77 69 74 68 20 74 69 6d 65 20 |otted with time | 00001040 61 63 72 6f 73 73 20 74 68 65 20 62 6f 74 74 6f |across the botto| 00001050 6d 20 61 78 69 73 20 28 58 29 20 61 6e 64 20 0a |m axis (X) and .| 00001060 70 6f 70 75 6c 61 74 69 6f 6e 20 6f 6e 20 74 68 |population on th| 00001070 65 20 73 69 64 65 20 61 78 69 73 20 28 59 29 2e |e side axis (Y).| 00001080 0a 20 54 68 69 73 20 69 73 20 61 20 76 65 72 79 |. This is a very| 00001090 20 73 69 6d 70 6c 65 20 66 72 61 63 74 61 6c 2c | simple fractal,| 000010a0 20 62 75 74 20 73 6f 6d 65 20 63 61 6e 20 62 65 | but some can be| 000010b0 20 76 65 72 79 20 63 6f 6d 70 6c 65 78 2e 20 57 | very complex. W| 000010c0 65 20 0a 73 68 61 6c 6c 20 6e 6f 74 20 67 6f 20 |e .shall not go | 000010d0 69 6e 74 6f 20 74 68 65 73 65 20 61 73 20 74 68 |into these as th| 000010e0 69 73 20 69 73 20 6f 6e 6c 79 20 61 6e 20 69 6e |is is only an in| 000010f0 74 72 6f 64 75 63 74 69 6f 6e 20 69 6e 74 6f 20 |troduction into | 00001100 0a 62 61 73 69 63 20 66 72 61 63 74 61 6c 73 2e |.basic fractals.| 00001110 0a 0a 0a 33 2e 43 6f 6d 70 6c 65 78 20 6e 75 6d |...3.Complex num| 00001120 62 65 72 73 2e 0a 0a 20 54 68 65 73 65 20 61 72 |bers... These ar| 00001130 65 20 6e 6f 74 20 61 73 20 74 68 65 79 20 73 6f |e not as they so| 00001140 75 6e 64 2c 20 62 75 74 20 74 68 65 20 69 64 65 |und, but the ide| 00001150 61 20 69 73 20 73 74 72 61 6e 67 65 2e 20 43 6f |a is strange. Co| 00001160 6d 70 6c 65 78 20 0a 6e 75 6d 62 65 72 73 20 77 |mplex .numbers w| 00001170 65 72 65 20 69 6e 76 65 6e 74 65 64 20 74 6f 20 |ere invented to | 00001180 73 6f 6c 76 65 20 61 20 70 72 6f 62 6c 65 6d 3a |solve a problem:| 00001190 20 59 6f 75 20 63 61 6e 20 6e 6f 74 20 77 6f 72 | You can not wor| 000011a0 6b 20 6f 75 74 20 0a 74 68 65 20 73 71 75 61 72 |k out .the squar| 000011b0 65 20 72 6f 6f 74 20 6f 66 20 61 20 6e 65 67 61 |e root of a nega| 000011c0 74 69 76 65 20 6e 75 6d 62 65 72 2e 20 41 73 20 |tive number. As | 000011d0 74 68 65 20 73 71 75 61 72 65 20 72 6f 6f 74 20 |the square root | 000011e0 69 73 20 74 68 65 20 0a 6e 75 6d 62 65 72 20 6d |is the .number m| 000011f0 75 6c 74 69 70 6c 69 65 64 20 62 79 20 69 74 73 |ultiplied by its| 00001200 65 6c 66 20 74 6f 20 67 65 74 20 74 68 65 20 66 |elf to get the f| 00001210 69 72 73 74 20 6e 75 6d 62 65 72 2c 20 61 6e 64 |irst number, and| 00001220 20 61 6e 79 0a 6e 75 6d 62 65 72 20 28 65 76 65 | any.number (eve| 00001230 6e 20 6e 65 67 61 74 69 76 65 29 20 6d 75 6c 74 |n negative) mult| 00001240 69 70 6c 69 65 64 20 62 79 20 69 74 73 65 6c 66 |iplied by itself| 00001250 20 61 6c 77 61 79 73 20 67 69 76 65 73 20 61 20 | always gives a | 00001260 0a 70 6f 73 69 74 69 76 65 20 72 65 73 75 6c 74 |.positive result| 00001270 2c 20 74 68 65 6e 20 74 68 65 20 73 71 75 61 72 |, then the squar| 00001280 65 20 72 6f 6f 74 20 6f 66 20 2d 31 20 69 73 20 |e root of -1 is | 00001290 69 6d 70 6f 73 73 69 62 6c 65 20 0a 28 69 6d 61 |impossible .(ima| 000012a0 67 69 6e 61 72 79 29 2e 20 54 6f 20 6f 76 65 72 |ginary). To over| 000012b0 20 63 6f 6d 65 20 74 68 69 73 20 64 69 66 66 69 | come this diffi| 000012c0 63 75 6c 74 79 20 69 74 20 77 61 73 20 64 65 63 |culty it was dec| 000012d0 69 64 65 64 20 74 68 61 74 0a 74 68 65 20 69 6d |ided that.the im| 000012e0 61 67 69 6e 61 72 79 20 6e 75 6d 62 65 72 20 28 |aginary number (| 000012f0 72 65 70 72 65 73 65 6e 74 65 64 20 62 79 20 74 |represented by t| 00001300 68 65 20 6c 65 74 74 65 72 20 27 69 27 29 20 73 |he letter 'i') s| 00001310 71 75 61 72 65 64 20 77 61 73 20 0a 65 71 75 61 |quared was .equa| 00001320 6c 20 74 6f 20 6e 65 67 61 74 69 76 65 20 31 2e |l to negative 1.| 00001330 20 49 2e 45 2e 20 69 2a 69 3d 2d 31 20 61 6e 64 | I.E. i*i=-1 and| 00001340 20 69 5e 32 3d 2d 31 2e 20 54 68 61 74 20 69 73 | i^2=-1. That is| 00001350 20 74 68 65 20 6f 6e 6c 79 20 64 69 66 66 69 63 | the only diffic| 00001360 75 6c 74 20 0a 62 69 74 21 0a 20 48 65 72 65 20 |ult .bit!. Here | 00001370 61 72 65 20 73 6f 6d 65 20 65 78 61 6d 70 6c 65 |are some example| 00001380 73 20 6f 66 20 77 68 61 74 20 63 6f 6d 70 6c 65 |s of what comple| 00001390 78 20 6e 75 6d 62 65 72 73 20 6c 6f 6f 6b 20 6c |x numbers look l| 000013a0 69 6b 65 20 77 68 65 6e 20 0a 77 72 69 74 74 65 |ike when .writte| 000013b0 6e 20 64 6f 77 6e 3a 20 34 2b 37 69 20 28 77 68 |n down: 4+7i (wh| 000013c0 65 72 65 20 34 20 69 73 20 72 65 61 6c 20 70 61 |ere 4 is real pa| 000013d0 72 74 20 61 6e 64 20 37 69 20 74 68 65 20 69 6d |rt and 7i the im| 000013e0 61 67 69 6e 61 72 79 29 2c 20 0a 39 39 2d 34 32 |aginary), .99-42| 000013f0 69 2c 20 30 2b 32 69 2c 20 30 2b 30 69 2c 20 37 |i, 0+2i, 0+0i, 7| 00001400 2d 30 69 2c 20 61 2b 62 69 20 28 77 68 65 72 65 |-0i, a+bi (where| 00001410 20 61 20 61 6e 64 20 62 20 61 72 65 20 62 6f 74 | a and b are bot| 00001420 68 20 72 65 61 6c 20 0a 69 6e 74 65 67 65 72 73 |h real .integers| 00001430 29 2e 0a 20 54 68 65 20 6f 74 68 65 72 20 74 68 |).. The other th| 00001440 69 6e 67 20 74 6f 20 6b 6e 6f 77 20 69 73 20 74 |ing to know is t| 00001450 68 61 74 20 65 76 65 72 79 20 63 6f 6d 70 6c 65 |hat every comple| 00001460 78 20 6e 75 6d 62 65 72 20 69 73 20 6d 61 64 65 |x number is made| 00001470 20 75 70 20 0a 6f 66 20 74 77 6f 20 6f 74 68 65 | up .of two othe| 00001480 72 20 6e 75 6d 62 65 72 73 2c 20 6f 6e 65 20 74 |r numbers, one t| 00001490 68 65 20 72 65 61 6c 2c 20 6f 6e 65 20 74 68 65 |he real, one the| 000014a0 20 69 6d 61 67 69 6e 61 72 79 2e 20 54 68 65 73 | imaginary. Thes| 000014b0 65 20 74 77 6f 20 0a 6e 61 6d 65 73 20 68 61 76 |e two .names hav| 000014c0 65 20 6e 6f 20 72 65 61 6c 20 6d 65 61 6e 69 6e |e no real meanin| 000014d0 67 2c 20 69 74 20 69 73 20 6a 75 73 74 20 74 6f |g, it is just to| 000014e0 20 64 69 73 74 69 6e 67 75 69 73 68 20 62 65 74 | distinguish bet| 000014f0 77 65 65 6e 20 0a 74 68 65 6d 20 2d 20 74 68 65 |ween .them - the| 00001500 79 20 63 6f 75 6c 64 20 62 65 20 63 61 6c 6c 65 |y could be calle| 00001510 64 20 61 6e 79 20 74 68 69 6e 67 20 65 2e 67 2e |d any thing e.g.| 00001520 20 53 74 61 72 73 6b 79 20 61 6e 64 20 48 75 74 | Starsky and Hut| 00001530 63 68 2e 20 0a 41 6c 73 6f 20 61 20 63 6f 6d 70 |ch. .Also a comp| 00001540 6c 65 78 20 6e 75 6d 62 65 72 20 69 73 20 75 73 |lex number is us| 00001550 75 61 6c 6c 79 20 72 65 70 72 65 73 65 6e 74 65 |ually represente| 00001560 64 20 62 79 20 74 68 65 20 6c 65 74 74 65 72 20 |d by the letter | 00001570 27 5a 27 20 0a 61 6c 74 68 6f 75 67 68 20 69 74 |'Z' .although it| 00001580 20 63 6f 75 6c 64 20 62 65 20 61 6e 79 20 6c 65 | could be any le| 00001590 74 74 65 72 2e 20 54 6f 20 6b 65 65 70 20 69 74 |tter. 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