decimals and fractions
for the purpose of clarity, we write a fractional number in a decimal as
0.a_1 a_2 a_3 a_4 a_5... where a_n is a digit 0 to 9. the representations are infinite, even in cases where numbers end in 0000000..., even though we usually ignore these trailing zeroes when we write out decimal fractions.
for example, we write 31/57 as 0.5438791...
some of these drawbacks are as follows
-some very simple rational numbers such as 1/3 have non-terminating decimal representations and so cannot be expressed exactly by decimals.
-approximating decimals by fractions is not straightforward. eg we approximate 0.143 by 143/1000, and have no obvious way of getting the simpler 1/7
-representations are not unique: 1/2 can be written as 0.500000... or as 0.49999...
if you're not convinced that these two are equal, try subtracting the latter from the former, and stop when you get convinced.
-working out the decimal representations of rational numbers, ie long division, is a pain in the arse.
the egyptians used a counting system in which the natural numbers (1, 2, 3..) and their reciprocals (1, 1/2, 1/3..) were considered as equally essential, while nowadays in mathematics the natural numbers are often seen as being more basic. the egyptians wrote fractional numbers as the sums of reciprocals of natural numbers.
for example 31/57 = 1/2 + 1/23 + 1/2622
the way i worked this out was simply to take the largest reciprocal smaller than 31/57, (= 1/2) subtract that from 31/57 to get 5/114, and repeat the process on that.
this isn't a great system. although we can write any rational number finitely, these are not unique, and it's not easy to find the best one for a given number. the 'best one' here might mean the one with the fewest terms, or the one with the smallest denominators. when we add or multiply this type of number, we do obtain a sum of reciprocals as the answer, (except when we add 1/n to 1/n, but 2/n is in general not hard to simplify) but it is unlikely to be a very simple form.
it is probably redundant to say that the egyptians adopted this way of writing numbers for aesthetic and ideological reasons. most of the concepts we evolve in mathematics are based on an implicit view of the world as it is, however much mathematicians pretend otherwise.
however, i don't think the egyptians were completely off the right track. perhaps if they had refined their system a little, they might have arrived at a method of writing fractional numbers that i believe would have been too useful ever to be properly superseded by decimal fractions. they could have done this by inventing continued fractions.
my filing cabinet
"The final dying sounds of their dress rehearsal left the Laurel Players with nothing to do but stand there, silent and helpless, blinking out over the footlights of an empty auditorium."
(having opened it again to write this, i'm not sure exactly why i was so put off by the first time...)
"the curious incident of the dog in the night-time"
is a quotation from a Sherlock Holmes story, i believe 'The Hound of the Baskervilles'. the latter is mentioned in the former.
'fray' is a comic book written by Joss Whedon, creator, i.e. God, of Buffy the Vampire Slayer and her attendant Buffyverse. Yes unlike many other Buffy comic books 'created' by Joss Whedon this one is actually written by him (although some others are very good, and some very good ones are written by him including some of the 'tales of the slayers' iirc). fray is the story of a vampire slayer discovering her destiny in Whedon's very own dystopic future, complete with flying cars, and is very good indeed.
i recently lent my friend 2 novels, chosen from a pile of my books, one by him and one by me, fairly randomly. when he got to the end of them both though, he observed a connection between the two.
the books were
'myra beckinridge' by gore vidal
'the wasp factory' by ian banks
today i borrowed a book from the library on 'outsider art' (colin rhodes, thames & hudson). a few years ago someone gave me another thames & hudson art book 'photomontage' in response to some collages of mine i had sent her, and this book has since become very important to me. i wondered how my life would have been different if she had bought me the book on outsider art instead.
my new webcam
alice, bob and the coins
one chess board has a single pawn on it in a random position. the other one has an identical coin on each square. the coins show heads and tails at random.
alice looks at the chessboard with the pawn on it, then removes it from the chess board. then she turns over a single coin on the other board.
then alice goes into the soundproof room.
then bob comes in. he looks at the chess board with the coins on it, thinks long and hard, then removes a pawn from his pocket and puts it down on the other board in the same spot that alice removed it from!
how did they do it?
Rick Scannell, 'special advocate' for SIAC, resigns
Mr Scannell said in his resignation letter: "In light of the government's continued maintenance of detention under ACSA powers and the government's apparent determination to extend those powers I consider my continued participation in a system condemned by the House of Lords to be wholly untenable."
They may challenge their detention before the Special Immigration Appeals Commission (SIAC) but neither they nor their lawyers are allowed to see the secret evidence against them. That evidence can be seen only by the special advocate appointed to represent the detainee's interests, who may not discuss it with the detainee or his lawyers.
Ian Macdonald QC, who has also seen secret intelligence reports on the detainees, is one of 15 special advocates appointed by the government to represent the detainees' interests in secret sessions of the Special Immigration Appeals Commission, Siac. He has resigned his post, describing the law which gives the government the power to detain indefinitely without trial as odious. He says he feels he was being used to justify a system in a way he now finds intolerable.
In 1997, I accepted the offer of appointment as a lay member of the new Siac. I thought then, and still think, that the Siac system (under which a special advocate represents the appellant at closed hearings of highly sensitive evidence which cannot safely be disclosed to him), although imperfect, was the least objectionable way of protecting both those appealing against deportation and the sources of information essential to the effective functioning of the security services. I was a member of the Siac panel for its first case, and nothing about that experience changed my mind. But subsequent developments forced me to conclude that I could not conscientiously play any further part, and in January I resigned.
vietnam: the ten thousand day war
this the name of a book i have been reading by a journalist and documentary maker, Michael Maclear, who also made a film of the same name. so far i've read almost half of this book, which deals with the conflict in a lot of detail and background, starting with the beginning of the Vietnamese liberation war against the French after WWII. i'm now up to 1965 - the start of the US land war. as you might expect, this book is to a certain extent centred on the demands and effects of the war on US and South Vietnamese history and society. i chose this book because while i had been reading some history about Southeast Asia in this period, i didn't know that much about the actual history of the conflict.
this is the problem with history - once you have become interested in something, you can't simply read one book on it - there have to be several viewpoints before you can make any sense of or form an opinion on anything. a very interesting book about the wider issues and conflicts in South East Asia during this period is "An Eye for the Dragon" by Dennis Bloodworth. Dennis Bloodworth was the London Observer correspondent to the Far East from 1956. This description of his book was taken from Pointer, the journal of the singaporean armed forces:
In his second book An Eye for the Dragon, Dennis Bloodworth writes on the turbulent events in Indonesia, Cambodia, Vietnam, Philippines and Singapore in the period 1954 to 1970. He tells of the megalomaniac Sukarno of Indonesia, Cambodia's Prince Sihanouk, the love-hate relationship the Prime Minister of Singapore has with his former British masters and the story of how the colonial masters the British, French, Dutch have been involved in this region.
prior to this i read "At War with Asia" by Noam Chomsky, which deals in 3 sections with the US aggression against Vietnam, Laos and Cambodia. as usual in his books, Chomsky focuses on a critique of the actions and propaganda of the US government and military in this sphere.
my initial inspiration to read all these books, and generally become interested in Vietnam, was the brillian spy novel "An Honourable Schoolboy" by John le Carré. this is the second in his "smiley vs. karla" trilogy, begun by the well-known "tinker, tailor, soldier, spy". while the first book in the trilogy, is a more straightforward whodunnit, and the third is mainly concerned with the resolution of the conflict between the British and Russian spymasters, this second book draws a wide journalistic arc outside of le Carré's usual Cold War stories, and through the entire South East Asian region at the time of the American pull out. Jerry Westerby, the book's genial antihero, is sent on a mission which takes him through Hong Kong, communist China, war torn Laos, besieged Saigon, Phnom Penh under the Khmer Rouge and a vast American airbase.
like all of le Carré's books, this features a compelling plot of spies, double agents, treason and secret war, however, in a manner most like his more recent books ("The Night Manager", "The Constant Gardener", "Single vs Single", as well as, less recently "The Little Drummer Girl" and others), he seems to use the novelistic form deliberately as a form of journalism, in order to reveal to us the truth about the horrors and absurditites of this war and other political issues. This means the casual reader will pick up a lot about Vietnam from "The Honourable Schoolboy", and I recommend it, but especially with "An Eye for the Dragon" as a 'factual' companion to anyone interested in this period of history.
"If we can say that the law of gravity was valid before Newton formulated it, then the law itself does not rest in the substance of matter. Instead, it only illustrates the way in which the relations of matter present themselves to a specifically organized mind, and the validity of the laws is independent of the fact that matter exists in reality."
Well put, Georg.
try a buying a cd player from sony. then go right out and buy an album on sony records (epic), the greatest hits of a band signed to sony. the 'cd' has some kind of bullsh*t copy protection, is not therefore a cd under the cd standard that sony invented, and does not feature the 'compact disc' logo that sony invented, wrote on my cd player, and told me to look out for, so i could be sure i was buying an actual, proper cd and not a piece of junk imitation ripoff made by incompetent criminals.
the cd does not play in the f*cking cd player!
forget about putting it in your computer to fileshare, the sony cd does not play in the sony f*cking cd player!
what am i doing wrong, sony?
christopher explains the problem well, and the solution as well as, if not better than, at least one account of it i have seen in a popular maths book.
(like excuses, explanations are never quite so convincing when you need more than one of them.) he exposes (and solves) quite a few other interesting maths problems in the book. his purpose is this chapter is given by the opening paragraph(s):
Mr Jeavons said that I liked maths because it was safe. He said I liked maths because it meant solving problems, and these problems were difficult and interesting, but there was always a straightforward answer at the end. And what he meant was that maths wasn't like life because in life there are no straightforward answers at the end. I know he meant this because this is what he said.
This is because Mr Jeavons doesn't understand numbers.
Here is a famous story called The Monty Hall Problem which I have included in this book because it illustrates what I mean.
you can find this problem very easily ("as Eugene Northrop observes in his Riddles in Mathematics, it 'has been used as an illustrative example in almost every subsequent textbook'", The Magical Maze, Ian Stewart) if you want to see if you can understand it and avoid making the obvious mistake that is what the problem is all about. christopher goes on to quote several letters from people at universities, some with Phds, who wrote to tell a popular maths writer her solution to the problem was wrong (when in fact and of course it was correct).
I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error.
Robert Sachs, PhD., George Mason (?) University
these quotes to me make very little sense. the story is such a classic, a cliche and a cautionary legend, that it would maybe need further justification to recount it here. the frets and foibles of PhDs are perhaps a long way removed from the concern of 15 yr-old autistic children, however good at maths they may be. and do we not learn every time we do maths that the more straightforward our answer (and, if it is maths, straightforward it _must_ be), the more complex its relation to the world of real problems?
since my own opinion leans more and more towards the idea that mathematics is simply the study of certain properties of the mind, the imagination, and parts of our cultural heritage, i find it difficult to speculate on its meaning to others.
all i know is that parts of this narration ring slightly false, that the author here seems to burst through the voice of the narrator like a small child who wishes to tell us something, and that those of us who look for solutions in maths, rarely dare to hope for simplicity.
Do Honeybees Defy Dinosaur Extinction Theories?
i blame zero
"The notion of zero also had its effect on our psychological conceptualizations. Ideas such as the finality of death and the fear of it, the separation of heaven and earth, the whole range of existential philosophies based on the despair and absurdity of a world followed by non-being, all owe much to the notion of zero. We saw ourselves as separate individuals moving in a space which was other than ourselves, encountering in that space other beings separate from and other than ourselves. But these concepts are now also loosing their hold. We know now that we exist in groups, determined by various levels of energetic affinities, repelling, exchanging and absorbing through interconnected, subtle energetic communications. And our being extends outward through various energy fields to connect with larger fields. We have had to learn that there is nowhere that we can dispose of the things we have finished using - that there is no zero drain in our sink; there is no factory pipe or hole in the ground that does not lead somewhere. Everything remains here with us; the cycles of growth, utilization and decay are unbroken. There is no throwaway bottle."
'Sacred Geometry' by Robert Lawlor
1982, Thames & Hudson, London.
This is to let you know that I have yet to be paid for 3 of the shifts I worked for you in the week leading up to Christmas.
That week I worked
Tues 21 6pm-close
Wed 22 12-6pm
Thurs 23 6pm-close
Fri 24 6-close
Last week I was paid by [pub] for 11.5 h. This seems to be payment for Xmas Eve - roughly 8 hours at time and a half.
On Friday payday I came to [pub] and gave [deputy manager] details of the 3 shifts I had not been paid for. He told me I would be paid for them next week.
I also asked [deputy manager] if he could advance me some of this pay in cash, since it was New Year's Eve and the amount unpaid is roughly £80-90. He said this would not be possible.
This week my payslip does not show any hours being paid by [pub].
I would therefore like you to:
- guarantee that I will be paid for the 18h you owe me next Friday.
- if possible advance me most or all of this money this weekend. If your cash control procedures do not allow you to do this I understand. However, in the circumstances I will not be able to work for [pub] again.
You can call me to discuss this on [phone number]. If I have not heard from you by tomorrow I will pop into [pub] to see you some time in the afternoon.
Thank you in advance for your help,