Collaborator

Journal Entries

Infinity

Posted Nov 20, 2004

Forgot the Anumber

A3304216

-Hussassan

Infinity

Posted Nov 20, 2004

This is a Collaborative entry on infinity, with some inspiration drawn from a number of unedited entries on the subject.

Though infinity (written as On some browsers, this may show up as a little square. If it does, then the proper symbol for infinity is an '8' on its side. is often spoken of as a number, especially, by schoolchildren, it is really more a concept. Basically it is the furthest possible number from zero. However, numbers go on forever. No matter how big a number you have, you can always add one to it to get a new one. This is entirely possible, since numbers are not real objects, they are simply abstract concepts which allow people to easily describe amounts of stuff. It is therefore impossible to have of something, unless you happen to be some sort of advanced god. If you are, then it is probably still a bad idea, as where would you put it all?

The Arithmetic of Infinity

As proof that infinity is not a real (or, indeed, an 'imaginary') number, let's have a look at mathematics involving it. If a is any number, then:

a+=
-a=
a - = -
a*=
a/=0
/a=
a=infinity if a is greater than one or less than minus one
a=0 if a is between minus one and one
a=
i^=+/-1, where i=(-1)If you see two little boxes in this point, the second one is supposed to be a square root sign.
At present, ^i is undefined. Nobody is particularly interested in defining it.

David Hilbert proposed the following paradox to explain just how unlike other numbers infinity is. Suppose some god runs a hotel with rooms, all of which are somehow full. However, this manager will have no trouble if a new guest asks for a room. The manager simply moves the person in room 1 to room 2, the person in room 2 to room 3, etc., so that the person in room n goes to room n+1. Thus room 1 is freed up for the new guest, and the rest can all get new rooms - since there are rooms, everyone will always have a higher-numbered room to go to.

What if ten new guests all want rooms? In that case, the person in room n goes to room n+10. The same logic applies.

So what if guests arrive simultaneously, and they all want rooms? Again, no problem. Everyone in the hotel moves to the room twice their own number, so that the person in room n moves to room 2n. This frees up all the odd-numbered rooms, and there are odd numbers and even numbers less than - hence, all the travellers have rooms.

The concept of infinity is considerably older than the concept of some other numbers, such as zero. Infinity is quite important to the motion paradox of the Greek philosopher Zeno. Zeno one day reasoned as follows:

Suppose a warrior sees an enemy archer about to shoot at him. The warrior turns and runs away at a steady speed in the same direction as the arrow is pointing.
Suppose the archer is 100m away from the warrior, and the arrow flies ten times faster than the runner. By the time the arrow has reached the original position of the warrior, the warrior has moved on another 10m.
When the arrow gets to that position, the warrior has moved on another metre.
When the arrow covers that metre, the warrior has moved on another 10cm.
etc.

This logic can also be used to show that Hercules cannot beat a turtle in a race if the turtle has a head start. Common sense tells us that this is rubbish - of course Hercules will overtake the turtle, quite quickly is we use equal divisions of time. Similarly, the warrior is better off to just duck.

In order for Zeno's paradox to work out, time and space must be split into infinitely shorter divisions. If this is possible, then a stage will eventually be reached when the warrior and the arrow will not move at all in the allotted time division, since time will have been split up into pieces and, as the number of divisions gets larger (and the size gets smaller), the distance moved by both the arrow and the warrior decreases. This indicates than any number divided by infinity gives zero. It also indicates that there is a minimum size for divisions of space and time, otherwise how could the arrow ever catch up to the warrior?

Universes and Elementary Particles

Some say that the Universe is infinite in size, others that it is not. An infinitely big Universe supports the Steady State theory, and a finite Universe holds up well with the Big Bang theory. However, could it possibly be both?

Take a piece of paper and imagine that there are a bunch of 2D people living on it. Roll it into a cylinder, and it becomes 3D. Bend the cylinder into a doughnut shape, and the 2D people living on it will never be able to find the edge - hence, it will seem infinite to them, despite being finite. There is a theory in quantum physics that the Universe really is doughnut shaped.

It was once thought that there might be particles in an atom. However, this is clearly bunk. If that was true, then nothing could possibly exist. If each elementary particle contained even the tiniest bit of mass, then of them would take up an infinite amount of space and contain an infinite amount of mass. According to the theory of relativity, as soon as something contains infinite mass, it ceases to exist.

-Hussassan

The Year 2003

Posted May 22, 2004

This is the ChangeLog for the entry about the Year 2003, being written in the Writing-Workshop-Emulation Style.

MIchael Moorcock - Author (A2629190)

Posted May 13, 2004

Michael Moorcock is a prolific British fantasy and science fiction author, probably best known for his fantasy saga The Tale Of The Eternal Champion. This sequence of loosely interconnected books concerns the "Eternal Champion", who has many identities in different times and dimensions, but always fights to maintain the balance between order and chaos, though he may not know that he is an incarnation of the Eternal Champion or that his destiny is to maintain the Balance.

Moorcock's Life

Moorcock has frequently collaborated with the rock band Hawkwind, and has written the lyrics to three songs about Elric for the band Blue Oyster Cult.

The Tale of The Eternal Champion

Elric of Melnibone

Elric is the hereditary emperor of the Melniboneans, a race of long-lived beings who are decadent and totally amoral, often taking a delight in cruelty - Elric is the only one of them to have any moral sense. Because of his sensibilities and his physical weakness (he is an albino who is dependent on drugs to give him any vitality at all) many believe that he does not deserve to be emperor. In the first book of this long saga, his cousin attempts to assassinate him and take the throne. During this book, he acquires the black sword Stormbringer, which steals the souls of its victims and transfers their energy to its wielder. By the end of the book, he is in a position to marry the woman he loves and reclaim his throne, but chooses instead to travel the world having adventures in an attempt to find out who he truly is.

Elric is sometimes described as an "anti-hero", perhaps because he is far more angst-ridden than your typical fantasy hero, and because Stormbringer several times forces him to kill his friends and lovers. However, he often acts in a noble and heroic manner - he shows mercy to the people who planned to kill him in the first book, and on one occasion almost dies rather than kill an innocent person with Stormbringer to restore his energy. Moorcock says in the introduction to the book Elric of Melnibone "I prefer to think of him simply as a hero" and goes on to describe how alienated heroes can be very useful in pointing out the weaknesses of society

Moorcock's Other Works

Ekaterin.

Initial Posting: Mathematical Glossary

Posted May 3, 2004

First attempt at posting this forgot to alter all the internal links, and so got deleted. Here is the entry really intended

Mathematical Glossary (Work in Progress)

This glossary is indexed and linked, rather like a normal Web page. It comprises a list of letters, each linking to a list of words. Each word in the list links to a definition, and each definition links back to the list of words. The definitions may contain links to other words in this glossary.

A
B
C
D
E
F
G
H
I
J
K
L
M

N
O
P
Q
R
S
T
U
V
W
X
Y
Z

Other Guide entries may link directly to the definition of a word in the glossary. The format of such a link is
&nbsp;&nbsp;&nbsp;&nbsp;

where xxx denotes the word exactly as defined here, and nnn denotes the text for the link. For example, in the sentence
&nbsp;&nbsp;&nbsp;&nbsp;

The zeroes of a quadratic may be determined by the use of a formula.

the word 'zeroes' is not an entry in the glossary, but 'zero' is defined. Thus the GuideML required would be

&nbsp;&nbsp;&nbsp;&nbsp;

producing the sentence

&nbsp;&nbsp;&nbsp;&nbsp;

The zeroes of a quadratic may be determined by the use of a formula.

with a link to the definition of 'zero'.

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A

accumulate

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B

binary

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C

corollary

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D

d-word

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E

e-word

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F

field

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G

g-word

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H

h-word

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I

i-word

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J

j-word

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K

k-word

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L

lemma

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M

m-word

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N

n-word

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O

o-word

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P

p-word

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Q

q-word

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R

remainder

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S

s-word

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T

theorem

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U

u-word

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V

v-word

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W

w-word

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X

x-word

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Y

y-word

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Z

zero

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accumulate
U241325
perform addition. Often associated with multiplication, either in the sense of the repeated additions necessary to form a product, or in the sense of the additions needed to form a sum of products.

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binary
U241325
usually an adjective, when binary means related to the number two. In computer contexts, sometimes used as a noun, to denote a program in a machine-readable, executable form.

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corollary
U152404
A corollary is a theorem which is implied by another.

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d-word
U241325
d-word is ...

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e-word
U241325
e-word is ...

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field
U241325
field

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g-word
U241325
g-word is ...

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h-word
U241325
h-word is ...

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i-word
U241325
i-word is ...

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j-word
U241325
j-word is ...

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k-word
U241325
k-word is ...

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lemma
U152404
A mathematically proven statement, usually presented with a proof, which is used as a component in the proof of a theorem. Although not usually of great importance in itself, a lemma may be an essential part of a theorem. It is often said (not entirely in jest) that the lemmas are the hard parts of a theorem!

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m-word
U241325
m-word is ...

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n-word
U241325
n-word is ...

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o-word
U241325
o-word is ...

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p-word
U241325
p-word is ...

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q-word
U241325
q-word is ...

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U241325

remainder
U241325
remainder

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s-word
U241325
s-word is ...

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theorem
U152404
A mathematically proven statement, usually presented with a proof (or a reference to a proof). Sub-theorems which may appear in a proof are usually referred to as lemmas, and further results which follow directly from a theorem are usually referred to as corollaries. The term theorem is therefore usually applied to important statements only.

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u-word
U241325
u-word is ...

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v-word
U241325
v-word is ...

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w-word
U241325
w-word is ...

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x-word
U241325
x-word is ...

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y-word
U241325
y-word is ...

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zero
U241325
the noun zero denotes either a specific number, or a property of a function. In computer contexts, sometimes used as a verb, meaning to make equal to the number zero.

number zero
the unique value, denoted by 0, which acts as an identity element in an addition operation, that is x+0&nbsp;=&nbsp;0+x&nbsp;=&nbsp;x.

zeroes of a function
the zeroes of a function are those values of the indepent variable(s) which cause the value of that function to become zero. A function may have no zeroes (for example ex is non-zero for any finite value of x, real or complex), one or more zeroes (for example x2&ndash;3x+2 is zero if x=1 or if x=2), or an infinity of zeroes (for example sin&nbsp;x has the value zero when x=n&pi; for any integer n</I&gt.

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posted by Old Hairy