THE global arms trade has grown by more than 20% in volume over the past five years, according to a new report by the Stockholm International Peace and Research Institute, a think-tank. Imports to the Middle East have increased by 38%, with the United Arab Emirates now the world's third-largest recipient of large conventional weapons. Israel and Egypt bought more arms than Iran, which is ranked 27th. China and India remain the world's biggest recipients of weapons, with Russia supplying the vast bulk to both countries. However in 2007 and 2008 South Korea was the largest arms importer, as it stocked up with American weapons in response to its increasingly belligerent neighbour, North Korea.
Every genuinely revolutionary technology implants some kind of "aha" moment in your memory -- the moment where you flip a switch and something magical happens, something that tells you in an instant that the rules have changed forever.
I still have vivid memories of many such moments: clicking on my first Web hyperlink in 1994 and instantly transporting to a page hosted on a server in Australia; using Google Earth to zoom in from space directly to the satellite image of my house; watching my 14-month-old master the page-flipping gesture on the iPhone's touch interface.
The latest such moment came courtesy of the Kindle, Amazon.com Inc.'s e-book reader. A few weeks after I bought the device, I was sitting alone in a restaurant in Austin, Texas, dutifully working my way through an e-book about business and technology, when I was hit with a sudden desire to read a novel. After a few taps on the Kindle, I was browsing the Amazon store, and within a minute or two I'd bought and downloaded Zadie Smith's novel "On Beauty." By the time the check arrived, I'd finished the first chapter.
How does Sony's E-Reader hold up against the Amazon Kindle? Stacey Delo reviews.
Aha.
I knew then that the book's migration to the digital realm would not be a simple matter of trading ink for pixels, but would likely change the way we read, write and sell books in profound ways. It will make it easier for us to buy books, but at the same time make it easier to stop reading them. It will expand the universe of books at our fingertips, and transform the solitary act of reading into something far more social. It will give writers and publishers the chance to sell more obscure books, but it may well end up undermining some of the core attributes that we have associated with book reading for more than 500 years.
There is great promise and opportunity in the digital-books revolution. The question is: Will we recognize the book itself when that revolution has run its course?
The Dark Matter
In our always-connected, everything-linked world, we sometimes forget that books are the dark matter of the information universe. While we now possess terabytes of data at our fingertips, we have nonetheless drifted further and further away from mankind's most valuable archive of knowledge: the tens of millions of books that have been published since Gutenberg's day.
That's because the modern infosphere is both organized and navigated through hyperlinked pages of digital text, with the most-linked pages rising to the top of Google Inc.'s all-powerful search-results page. This has led us toward some traditional forms of information, such as newspapers and magazines, as well as toward new forms, such as blogs and Wikipedia. But because books have largely been excluded from Google's index -- distant planets of unlinked analog text -- that vast trove of knowledge can't compete with its hyperlinked rivals.
John Weber
But there is good reason to believe that this strange imbalance will prove to be a momentary blip, and that the blip's moment may be just about over. Credit goes to two key developments: the breakthrough success of Amazon's Kindle e-book reader, and the maturation of the Google Book Search service, which now offers close to 10 million titles, including many obscure and out-of-print works that Google has scanned. As a result, 2009 may well prove to be the most significant year in the evolution of the book since Gutenberg hammered out his original Bible.
If so, if the future is about to be rewritten, the big question becomes: How?
The World of Ideas
For starters, think about what happened because of the printing press: The ability to duplicate, and make permanent, ideas that were contained in books created a surge in innovation that the world had never seen before. Now, the ability to digitally search millions of books instantly will make finding all that information easier yet again. Expect ideas to proliferate -- and innovation to bloom -- just as it did in the centuries after Gutenberg.
Think about it. Before too long, you'll be able to create a kind of shadow version of your entire library, including every book you've ever read -- as a child, as a teenager, as a college student, as an adult. Every word in that library will be searchable. It is hard to overstate the impact that this kind of shift will have on scholarship. Entirely new forms of discovery will be possible. Imagine a software tool that scans through the bibliographies of the 20 books you've read on a specific topic, and comes up with the most-cited work in those bibliographies that you haven't encountered yet.
The Impulse Buy
The magic of that moment in Austin ("I'm in the mood for a novel -- oh, here's a novel right here in my hands!") also tells me that e-book readers are going to sell a lot of books, precisely because there's an impulse-buy quality to the devices that's quite unlike anything the publishing business has ever experienced before.
Geoffrey A. Fowler
SECOND EDITION Soon you'll be able to create a digital, searchable version of your library. It's hard to overstate the impact that will have.
On another occasion, I managed to buy and download a book on a New York City subway train, during a brief two-stop stretch on an elevated platform.
Amazon's early data suggest that Kindle users buy significantly more books than they did before owning the device, and it's not hard to understand why: The bookstore is now following you around wherever you go. A friend mentions a book in passing, and instead of jotting down a reminder to pick it up next time you're at Barnes & Noble, you take out the Kindle and -- voilà! -- you own it.
My impulsive purchase of "On Beauty" has another element to it, though -- one that may not be as welcomed by authors. Specifically: I was in the middle of the other book, and in a matter of seconds, I left it for one of its competitors. The jump was triggered, in this case, by a sudden urge to read fiction, but it could have been triggered by something in the book I was originally reading: a direct quote or reference to another work, or some more indirect suggestion in the text.
In other words, an infinite bookstore at your fingertips is great news for book sales, and may be great news for the dissemination of knowledge, but not necessarily so great for that most finite of 21st-century resources: attention.
Because they have been largely walled off from the world of hypertext, print books have remained a kind of game preserve for the endangered species of linear, deep-focus reading. Online, you can click happily from blog post to email thread to online New Yorker article -- sampling, commenting and forwarding as you go. But when you sit down with an old-fashioned book in your hand, the medium works naturally against such distractions; it compels you to follow the thread, to stay engaged with a single narrative or argument.
Ahead of Amazon's quarterly results, Citi analyst Mark Mahaney says the company's growth potential is robust and explains why he upgraded his rating on the stock to buy. Stacey Delo reports. (April 20)
The Kindle in its current incarnation maintains some of that emphasis on linear focus; it has no dedicated client for email or texting, and its Web browser is buried in a subfolder for "experimental" projects. But Amazon has already released a version of the Kindle software for reading its e-books on an iPhone, which is much more conducive to all manner of distraction. No doubt future iterations of the Kindle and other e-book readers will make it just as easy to jump online to check your 401(k) performance as it is now to buy a copy of "On Beauty."
As a result, I fear that one of the great joys of book reading -- the total immersion in another world, or in the world of the author's ideas -- will be compromised. We all may read books the way we increasingly read magazines and newspapers: a little bit here, a little bit there.
You're Never Alone
Putting books online will also change how we find books -- and talk about them.
Now that books are finally entering the world of networked, digital text, they will undergo the same transformation that Web pages have experienced over the past 15 years. Blogs, remember, were once called "Web logs," cultivated by early digital pioneers who kept a record of information they found online, quoting and annotating as they browsed.
With books becoming part of this universe, "booklogs" will prosper, with readers taking inspiring or infuriating passages out of books and commenting on them in public. Google will begin indexing and ranking individual pages and paragraphs from books based on the online chatter about them. (As the writer and futurist Kevin Kelly says, "In the new world of books, every bit informs another; every page reads all the other pages.") You'll read a puzzling passage from a novel and then instantly browse through dozens of comments from readers around the world, annotating, explaining or debating the passage's true meaning.
Think of it as a permanent, global book club. As you read, you will know that at any given moment, a conversation is available about the paragraph or even sentence you are reading. Nobody will read alone anymore. Reading books will go from being a fundamentally private activity -- a direct exchange between author and reader -- to a community event, with every isolated paragraph the launching pad for a conversation with strangers around the world.
This great flowering of annotating and indexing will alter the way we discover books, too. Web publishers have long recognized that "front doors" matter much less in the Google age, as visitors come directly to individual articles through search. Increasingly, readers will stumble across books through a particularly well-linked quote on page 157, instead of an interesting cover on display at the bookstore, or a review in the local paper.
Imagine every page of every book individually competing with every page of every other book that has ever been written, each of them commented on and indexed and ranked. The unity of the book will disperse into a multitude of pages and paragraphs vying for Google's attention.
In this world, citation will become as powerful a sales engine as promotion is today. An author will write an arresting description of Thomas Edison's controversial invention of the light bulb, and thanks to hundreds of inbound links from bookloggers quoting the passage, those pages will rise to the top of Google's results for anyone searching "invention of light bulb." Each day, Google will deposit a hundred potential book buyers on that page, eager for information about Edison's breakthrough. Those hundred readers might pale compared with the tens of thousands of prospective buyers an author gets from an NPR appearance, but that Google ranking doesn't fade away overnight. It becomes a kind of permanent annuity for the author.
Writing for Google
A world in which search attracts new book readers also will undoubtedly change the way books are written, just as the serial publishing schedule of Dickens's day led to the obligatory cliffhanger ending at the end of each installment. Writers and publishers will begin to think about how individual pages or chapters might rank in Google's results, crafting sections explicitly in the hopes that they will draw in that steady stream of search visitors.
Individual paragraphs will be accompanied by descriptive tags to orient potential searchers; chapter titles will be tested to determine how well they rank. Just as Web sites try to adjust their content to move as high as possible on the Google search results, so will authors and publishers try to adjust their books to move up the list.
What will this mean for the books themselves? Perhaps nothing more than a few strategically placed words or paragraphs. Perhaps entire books written with search engines in mind. We'll have to see.
(One geeky side note here: Before we can get too far in this new world, we need to have a technological standard for organizing digital books. We have the Web today because back in the early 1990s we agreed on a standard, machine-readable way of describing the location of a page: the URL.
But what's the equivalent for books? For centuries, we've had an explicit system for organizing print books in the form of page numbers and bibliographic info. All of that breaks down in this new digital world. The Kindle doesn't even have page numbers -- it has an entirely new system called "locations" because the pagination changes constantly based on the type size you choose to read. If you want to write a comment about page 32 of "On Beauty," what do you link to? The Kindle location? The Google Book Search page? This sounds like a question only a librarian would get excited about, but the truth is, until we figure out a standardized way to link to individual pages -- so that all the data associated with a specific passage from "On Beauty" point to the same location -- books are going to remain orphans in this new world.)
Paying Per Chapter?
The economics of digital books will likely change the conventions of reading and writing as well. Digital distribution makes it a simple matter to offer prospective buyers a "free sample" to entice them to purchase the whole thing. Many books offered for the Kindle, for instance, allow readers to download the first chapter free of charge. The "free sample" component of a book will become as conventional as jacket-flap copy and blurbs; authors will devise a host of stylistic and commercial techniques in crafting these giveaway sections, just as Dickens mastered the cliffhanger device almost two centuries before.
It's not hard to imagine, for instance, how introductions will be transformed in this new world. Right now, introductions are written with the assumption that people have already bought the book. That won't be the case in the future, when the introduction is given away. It will, no doubt, be written more to entice readers to buy the whole book.
Clearly, we are in store for the return of the cliffhanger.
For nonfiction and short-story collections, a la carte pricing will emerge, as it has in the marketplace for digital music. Readers will have the option to purchase a chapter for 99 cents, the same way they now buy an individual song on iTunes. The marketplace will start to reward modular books that can be intelligibly split into standalone chapters.
This fragmentation sounds unnerving -- yet another blow to the deep-focus linearity of the print-book tradition. Breaking the book into detachable parts may sell more books, but there are certain kinds of experiences and arguments that can only be conveyed by the steady, directed immersion that a 400-page book gives you. A playlist of the best chapters from "Middlemarch," "Gravity's Rainbow" and "Beloved" will never work the way a playlist of songs culled from different albums does today.
Yet that modular pricing system will have one interesting, and laudable, side effect: The online marketplace will have established an easy, one-click mechanism for purchasing small quantities of text.
Tellingly, the Kindle already includes blog and newspaper subscriptions that can be purchased in a matter of seconds.
Skeptics may ask why anyone would pay for something that was elsewhere available at no charge, but that's precisely what they said when Steve Jobs launched the iTunes Music Store, competing with the free offerings on Napster. We've seen how that turned out. If the Kindle payment architecture takes off, it may ultimately lead the way toward the standardized micropayment system whose nonexistence has caused so much turmoil in the news business -- a system many people wish had been built into the Web's original architecture, along with those standardized page locations.
We all know the story of how the information-wants-to-be-free ethos of the Web threatened the newspapers with extinction. Wouldn't it be ironic if books turned out to be their savior?
The "aero-auto hybrid" made its flight debut on an airstrip in New York.
Wednesday, March 18, 2009
By Brittany Sauser
Credit: Terrafugia
On March 5, at Plattsburgh International Airport in upstate New York, a "roadable aircraft" called Transition took flight for the first time. The flight lasted just 37 seconds (see video below), and according to the pilot, Phil Meteer, a retired U.S. Air Force colonel, it was merely a brief, wheels-up test along the runway. Transition made six more test flights between March 5 and 7. The flights were approved by the Federal Aviation Agency, which inspected Transition prior to takeoff.
The weird-looking vehicle was developed by a startup based in Woburn, MA, called Terrafugia, which confirmed the test flight in a press conference at the Boston Museum of Science held this morning. The "aero-auto hybrid," as the company calls it, was also temporarily on display in the museum's main hall.
The aircraft is designed to be driven on public roads: its wings fold up in 30 seconds, and it operates in front-wheel drive and uses 27 miles per gallon. Transition also fits in a standard garage. In the air, it can reach speeds of 115 miles per hour on flights of 450 miles or less at 30 miles per gallon.
But sadly for budding flying-car pilots, it will not hit the market for at least two years, and even then it'll cost around $194,000.
Building a Denser Chip The more memory and capacity a semiconductor chip has, the more wires and transistors it needs. But packing an ever larger number of wires and transistors onto the same size silicon wafer is an increasingly difficult and expensive proposition. The latest effort to decrease the distance between wires and other features on chips, called extreme ultraviolet lithography, uses a slow-moving machine that costs about $50 million to make. But now, MIT researchers have developed a technique that can refine the patterns of computer chips quickly and cheaply. Physicist Mark Schattenburg and mechanical-engineering graduate student Chih-Hao Chang used a nano ruler, recently invented by Schattenburg, that lets them rapidly print precise lines over a large area. They combined that with a novel process that transferred the pattern to the right layer of the silicon wafer, allowing them to etch one pattern on a chip, shift the chip incrementally, and then etch again. After repeating this four times, the scientists produced an even grid of lines that were only 25 nanometers apart—versus today’s standard 65 nanometers—with relatively minimal fabrication costs. Such a process, Schattenburg says, has potential applications for everything from semiconductors to solar-cell production.
Who discovered zero? Why have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.
The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)
Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.
One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.
The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.
If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.
Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.
What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.
In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.
However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.
Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.
The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.
The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!
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John Weber![[The Journal Report: Technology]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ti2-WTRkYvV9tRzlr-p_0T4vkFoO0iS8EXk8l3tQC-YDdGRdkA8_QXexiJifuXLWOzV79QGrLm4J-wfBCFjATFNdHbRri-ljdxW8ApXQH8YHdZCryvhPtBGVU32Wb9-ZfDfkooGczDgulOl_0SFKE=s0-d)
Geoffrey A. Fowler
