Simon G. Stacey

• Consultant Anaesthetist & Intensivist, Bart's Heart Centre, Bart's and The London NHS Trust, London, UK

The principles that Sun Tzu listed as most important in winning the confict are realistic and practical assessment of the situation insomnia articles order generic unisom on-line, initiative insomnia x macbook generic 25 mg unisom with amex, fexibility and surprise faithless insomnia buy discount unisom on line. Tese four are eternal and can still be found in the principles of war that are listed in the latest strategic doctrines of most of the military forces in the world sleep aid zzzquil reviews safe unisom 25 mg. Sun Tzu attached great importance to ‘knowing the enemy and yourself’ to win a war. A commander has to be realistic and practical regarding the emerging situation and must have carried out sufcient analysis of the variables before making the fnal decision to fght. Sun Tzu maintained that the military action in war must be adroitly guided to ensure that one’s own initiative is complete. In cases where the adversary is also manoeuvring ofensively, it is necessary to contest the initiative and win, failing which one is likely to be defeated or even annihilated. He stressed the importance of employing the force fexibly taking into account the position and manoeuvring of the adversary, but within the broader capabilities on one’s own forces. The concept of retaining the element of surprise in all the actions that are initiated is one of the central themes of the operational and tactical discussions in the treatise. The explanations that Sun Tzu provides to reinforce the four major principles remain valid in contemporary conficts today. The outcome of war is predicated not only by the national security strategy, but also by the concepts of warfare propounded by a nation. Emerging technology has always energised the conduct of war and military forces that failed to convert state-of-the-art technology to bold ofensive strategies have always found themselves at the losers’ end in any confict. Today, more than ever before, technologies are efecting changes in the conduct of warfare at a much faster pace. Tere is a now an acknowledged need to formulate a capstone strategic concept based on knowledge superiority to take full advantage of emerging technologies—in other words understanding ends, ways, and means in a complex security environment. It has been accepted that the strategic advice it ofers concerns much more than the conduct of war and that it is a book of proverbial wisdom. However, it is within the ambit of human confict that it conclusively proves to be a timeless classic because the axioms that Sun Tzu put forward in relation to the command and direction of a land campaign are very easily translated to the conduct of a contemporary air campaign or a joint campaign. Air power, as we know it today or for that matter in any form, did not exist during Sun Tzu’s time. The fact that the strategic and operational advice that is contained in the book covers all the contingencies that an air commander conducting a sophisticated air campaign could possibly encounter should establish the credentials of the author as one of the greatest strategists of all times and the book as the ultimate strategic treatise. This book is an attempt at interpreting the ancient classic within the constraints of the employment of air power in a contemporary confict scenario. The interpretations and extrapolations have been consciously kept at the strategic level in order to reduce the number of variations that could be attempted. The verses that go down to the operational level in the original, for example the one dealing with tactical superiority in Chapter 6, have been explained in as much of a strategic context as possible. However, these explanations provide only one perspective when superimposing them onto a contemporary scenario. It is hoped that the reader will be sufciently challenged to broaden the thinking and interpretations from this foundation. The chapter headings have been given subtitles that point in a generic manner to what each one means when explained in terms of an air campaign. The subtitles also indicate the relevance of Sun Tzu’s axioms to modern-day warfare with particular reference to the air campaign. It has to be mentioned here that although the primary intention has been to try and gain as much input as possible from the ancient text in understanding the employment of modern air power, a conscious efort has been made to situate this employment within the bounds of a joint campaign. This is particularly so when dealing 11 John Minford, The Art of War, Penguin Books, New York, 2003, p. Machell-Cox, The Principles of War by Sun Tzu, A New Translation from a Revised Text, Royal Air Force, Colombo, Ceylon, 1943, Introduction p. The reader will also notice fairly early in the reading itself that control of the air is one of the major themes in this interpretation of the classic. This has been necessary for two reasons other than the often repeated fact that control of the air is the raison d’etre for the existence of an air force. First, a number of concepts that Sun Tzu elaborates on hinge on control of the air for success, either directly or indirectly, when translated to contemporary situations. Second, the four major principles of war that Sun Tzu expounded—analysis, initiative, fexibility and surprise—are all completely dependent on the military force’s freedom to manoeuvre. However, in contemporary warfare freedom of manoeuvre is a direct by-product of control of the air, thereby entwining the two inextricably. The greater the general, the more he contributes in manoeuvre, the less he demands in slaughter … Nearly all the battles that are regarded as masterpieces of the military art … have been battles of manoeuvre in which very often the enemy has found himself defeated by some novel expedient or device, some queer, swift, unexpected thrust or stratagem. Winston Churchill, 1874−1965 End Toughts The concept of security has undergone a drastic revision in the last two decades. It used to be clearly associated with the political process that provided for negotiations rather than belligerence aimed at avoiding wars and ensuring peace as the normal condition amongst states. However, the end of the Cold War saw the beginning of an evolutionary change in the concept of security. The conscious withdrawal of the superpowers from the poorer developing world led to international development agencies stepping into the vacuum with their emphasis on aid and reconstruction, rather than combat and warlike activities. The stress on territorial security that was the mainstay of the Cold War slowly 12 Introduction gave way to the concept of human security—security through armament became security through sustainable human development. He extols the virtue of a commander who could win a confict without having to carry out combat operations. This is in keeping with the contemporary strategy of negotiation before any combat operations are undertaken. The current trend towards irregular warfare is also encompassed in the treatise in terms of the use of deception and surprise and the ‘indirect’ ways to wage war. Troughout history, the desired end-state of every war that has been fought has been peace and survival of the State. Sun Tzu made this very clear at the start of his book stating, ‘War is a matter of vital importance to the state, a matter of life and death, a road to survival or extinction. Let us see if by moderation we can win all hearts and secure a lasting victory, since by cruelty others have been unable to escape from hatred and maintain their victory for any length of time … this is a new way of conquering, to strengthen one’s position by kindness and generosity. Sun Tzu advocated the use of strategic assessments to objectively analyse one’s own position vis-à-vis that of the potential adversary in a methodical manner, thereby identifying a winning situation within its framework. Even before considering the possibility of a confrontation leading to confict, the complete environment should be analysed to compare the strengths and weaknesses of one’s position, the relationship between grand strategic objectives and military goals, and the military capabilities that can be brought to bear in any given situation. Such an objective assessment of one’s own strategic position in comparison to that of the adversary will assist in developing a deeper understanding of relative strengths and weaknesses. This understanding is an essential part The Art of Air Power of planning any military action with the framework focused on the unique aspects of the possible confict. Sun Tzu also emphasised the use of tactical paradoxes—the ability to project to the adversary a contradictory view of one’s own plans—as a winning strategy, when employed by skilled military leaders. This chapter very clearly indicates the indelible connection at the highest levels of national security between the grand strategic intent of the nation, its power elements and the role of the military. It also explains the importance of strategy, enumerates and describes the fundamental factors that afect strategy and details their individual characteristics, and then indicates how these can be used to calculate the chances of success in a campaign. Sun Tzu starts on the premise that war is the most serious undertaking of a nation, while also being the most complex of human endeavours. He goes on to lay down two basic premises—that a country must have a comprehensive security strategy and that it should also have adequate military capability to support the strategy. Much later in history, Carl von Clausewitz propounded similar ideas in his world famous treatise On War (1832). Military forces have a built-in tendency to employ, in their current confict, concepts and techniques that have been successful in the past. It needs clear and contextual analysis of the strategic situation to ensure that outmoded techniques, however successful in previous conficts, are not employed in emerging confict. The art of war is of vital importance to the State—military afairs, therefore, becoming a country’s vital political concern. The survival or destruction of a country and the life or death of its people may depend on military action. Trough the centuries of human history, war has come to be recognised as a visible extension of a nation’s strategic intent. They have to be prosecuted with clear objectives, the political will and military wherewithal to win. No war is a stand-alone military confrontation; it is always an afair of the entire State. Terefore, anything short of complete victory is bound to dilute the status and power of the State, both internally and externally.

The poor economy has been hard on families insomnia event unisom 25mg otc, many of whom struggle for decent meals sleep aid vitamin shoppe unisom 25mg for sale. The Russian Federation is massive insomnia headache purchase unisom online pills, nearly 17 million square kilometres xanax sleep aid dosage cheap 25mg unisom, with varied cultures and geological features, including tundra, forests and mild seaside climates. The Russian Federation would have difficulty feeding its own people without stable trading ties with Ukraine and other members of the Commonwealth of Independent States. In the heyday of the Soviet economy, in the late 1950s and early 1960s, Russian health (defined by infant mortality and life expectancy) was on par with Western Europe and the United States. The Russian Federation has a firmly established, yet now underfunded, system of socialized medicine. Although most of the population has 325 Food at work: Workplace solutions for malnutrition, obesity and chronic diseases access to free and basic medical care, the quality is low compared with Western standards. The system had begun to decline during the waning years of the Soviet Union, nearly bottomed out during the mid-1990s, and is only now recovering. Many factors have contributed to the population’s health decline: inadequate nutrition and occasional food shortages; air and water pollution; smoking; alcoholism; and a lack of medical supplies or medicines to control infectious diseases, such as cholera and tuberculosis. The interrelationships between under-nutrition, stress, low resistance and disease were particularly evident during the influenza epidemic during the winter of 1995–96, in which over one million people in Moscow alone were infected. Thus, just when the country was most economically vulnerable, workforce productivity sagged under the cycle of poor health, poor medical care and poor work performance. The rise in alcoholism, con sidered the country’s most critical health problem after cardiovascular diseases and cancer, further reduced productivity. A government health campaign in the mid-1980s sharply reduced alcohol consumption to around 7 litres of ethanol (pure alcohol) per capita, lower than some Western European countries. Workers’ health was highly valued during the era of the Soviet Union, in principle, to maintain a high level of productivity. Up until the 1980s, Russia had a vast network of neighbourhood and work-site health clinics, with a ratio of medical professionals and clinics to patients far higher than in most countries. Workers were entitled to 18 days’ holiday, a 70 per cent discount on health resorts, and a heavily subsidized factory meal. The All-Union Central Council of Trade Unions, with around 30 branches and an essentially mandatory membership requirement, monitored social benefit schemes. The trade union more or less followed the orders of the Government; its main function was to distribute workers’ benefits. About 85 per cent of the Soviet Union’s 137 million workers in the mid 1980s were on state payrolls; the remaining workers were de facto state workers on farms and in independent activities. Workers were much like foot soldiers, and the army analogy describes workers’ conditions well: hard work, 326 Solutions for families inexpensive meals, discounted lodging, access to factory shops, medical coverage, free education and pensions, all of which came with no frills. Each company had its own large canteen that usually operated 24 hours, in line with shiftworkers. Most workers had access to free mineral water; and those in “unhealthy” professions exposed to industrial hazards received free milk. Most companies also had low-priced shops, although the selection was very limited. Some factories, such as those in remote areas of Siberia and the Sakhalin Oblast (the far southeast), offered better benefits (lodging, commodities) and higher pay as a means to attract workers to such distant and desolate regions. Workers in light industries, such as textile and food processing, had lower pay and fewer benefits, although this was not common knowledge until the 1990s. During the 1970s, the Soviet Union experienced food shortages, particularly meat; and by the end of the decade the government mandated food production at the workplace. It was during this period of food shortages that Soviet productivity began to fall, with some workers drinking vodka during lunch and stealing production instruments and materials. Many workers’ benefits, including meal programmes, were largely cut with the collapse of the Soviet Union in 1991. Around 50 per cent of Russian Federation factory canteens have been closed since 1994. The main reasons for closure have been the lingering economic crisis, company bankruptcy and the reduction of government nutrition subsidies. What communism set up in its creation of a culture of dependency on the Government, capitalism knocked down with a cruel reality of no care for no pay. The trade unions are still too weak and overwhelmed with other issues (pay, working hours, safety) to petition strongly for canteens. That is to say, companies seem willing to offer better benefits to workers once profitability is established, a sign of resentment of runaway capitalism or the sweatshop mentality. The majority of companies, particularly those in hard-hit rural regions, still have few workers’ benefits in comparison with Soviet times. Life is not necessarily harder in modern Russia compared with the Soviet era, but for many workers it is less predictable. The pressing concerns for workers and unions today are consistent and decent pay, paid overtime or a cap on the 40 hour week, and safety. Many workers have taken to packing a filling yet unbalanced meal for lunch, such as bread with a little cured meat. Yet nutrition in general, and workers’ nutrition specifically, needs to be part of the national health agenda if the Russian Federation hopes to compete in the open market economy. The toll of heart disease, cancer and alcoholism will continue to cripple productivity. Fundamental changes, not in government ideologies but rather in Russian diet and lifestyle (sausages and sour cream, little exercise, heavy smoking and heavy drinking, particularly among men), need to take place; and the workplace is a logical location to start. Better nutrition at work leading to better productivity leading to a better economy will break the cycle of poor health and working conditions that many Russians are now experiencing. Neighbouring Finland’s North Karelia Project managed to change lifestyles for the better – with significant reductions in smoking, alcohol consumption and fatty foods, and increased consumption of vegetables, which many thought to be impossible in Finnish culture. To do the same in the Russian Federation will require both time and a similarly well-planned intervention programme suitable to the country’s indomitable spirit. The anti-drinking campaigns, for example, however well intentioned, resulted in rapid bootlegging of homemade, low-quality alcohol. However, the positive changes occurring in Rostov-on-Don, as documented in the following case study of a ball-bearing plant, is a sign that the Russian Federation may be turning the page to a bright, healthy, productive future. The company mostly produces five general varieties of bearings, comprising 70 per cent of the company’s total output. Employees: 3,220 (2,580 industrial employees), down from 6,600 during the Soviet era. Food solution – key point: low-cost grocery shops, subsidized canteens, and cafeterias. Meals were subsidized by 90 per cent, and the foodstuffs in the shops were sold at a minimum mark-up. One of the canteens offered special dietary meals for workers with chronic diseases, such as stomach and liver problems. The plant had two shifts: a day shift from 07:00 to 15:30 hours, and a night shift from 15:30 to 24:00 hours. The canteens served both shifts, and around 80 per cent of the workers used the canteens. One other interesting feature was that workers in “unhealthy” professions – presumably those working with solvents and the like – received a free half litre of milk each day. Food solution Workers have lost a few benefits over the years after the end of the Soviet Union. The food is 329 Food at work: Workplace solutions for malnutrition, obesity and chronic diseases cooked in one kitchen and delivered to the canteens and cafeterias. All these facilities are well stocked and renovated, with air-conditioning and clean lavatories. A company sanitary inspector is hired to maintain a high level of hygiene and food quality. The foodstuffs for the canteens, cafeterias and shops are purchased by the special provision service of the plant. At the three grocery shops, the cost of food is about 5 per cent lower than that in the city. Management had decided to keep this benefit from the Soviet era to supplement the nutrition of workers and their families. The stores sell typical Russian foods, such as dairy products, sausages and baked goods.

It is fun to write programs for it insomniax download purchase 25mg unisom free shipping, play games on it faithless - insomnia order generic unisom from india, and to build new parts for it insomnia movie cast buy discount unisom on line. It is fascinating to try to figure out what part of the program it is in by the way the lights nicker or the radio buzzes insomnia 720p yify buy unisom 25mg cheap. There is no need to buy this software package or that circuitboard; I can design one better. The juicy little news bits, the how-to-fixits for the problem that has been bugging me the last two weeks. He forced himself to take exception to those articles of faith, testifying that he had "been there" and seen the problems with them. Point by point he demonstrated the folly of hacking, and concluded by writing: "By now the computer has moved out of the den and into the rest of your life. His express desire was to allow the club to develop as an anarchist community, a society of non-joiners wed, whether they knew it or not, by the Propaganda of the Deed. If he desired a blueprint for failure, he need only look to the south, at the Southern California Computer Society. Its leaders were not happy with the mere exchange of information: they envisioned group buying plans, a national magazine, and an influence which would allow hobbyists to dictate terms to the growing microcomputer industry. Homebrew had no steering committee to confer on goals and directions; it only incorporated as an afterthought, almost a year after inception; it had no real dues requirements only a suggested contribution of ten dollars a year to get its modest newsletter. He later described the difference between the two groups: "Homebrew was a place where people came together mysteriously, twice a month. The board meetings became incredibly tempestuous, and the bad feelings spread to the general membership meetings. He was perfectly happy dealing with an army of Bob Marshes and Tom Pittmans, some changing the world by dint of useful products manufactured in the spirit of hackerism and others just going their way, being hackers. The eventual goal would be a mass distribution of the wonderment that Lee Felsenstein had experienced in his basement monastery. The user will have to spend some amount of time probing around inside the equipment, and we will have to make this possible and not fatal to either the equipment or the person. That sounded all right to Lee, who had been doing work in documentation and schematics for Processor Technology all along. The company distributed schematics and source code for software, free or at nominal cost. The Dazzler used color, and produced its flashy effects by constantly going back to the memory in the main chip of the Altair (or any of the other new computers that used a similar hardware bus). Its output was black and white, and instead of using dots it actually formed alphanumeric characters. It worked like a mini-time-sharing system, where the two users were the video display and the computer itself. His magazine had followed up on the coup, he had delivered more computer-related cover stories, and now he was hoping to present a complete computer video display terminal a self-contained item which would have the power of the computer as well as a display capability. It would be the next step beyond the Altair, a combination computer-teletype with video. No more goddamn bloody fingers from the flicking switches on the Altair front panel. Bob talked to Lee, who agreed to do most of the design, and as they discussed it they realized that what Les Solomon wanted was not merely a terminal, but a complete computer. All required some sort of terminal, usually a klunky teletype, for the user to do anything with it. Marsh wanted to use an 8080 chip, an idea which Lee at first still opposed for political reasons (why one centralized silicon dictator? Lee figured he would use his junkyard-paranoid style to balance out the rest of the design, so that the brain would not be tempted to run amok. Lee, just about living on orange juice, spent endless hours staring at the Mylar spaghetti of the layout on the fluorescent light table. Two weeks later, a day before the newly scheduled delivery date in late February 1976 in New York, they were racing to get all the workings to fit on an Altair-style bus, along with a kluged-up power supply, a keyboard, and even some preliminary software. Finishing just in time, they had to race to a heliport in order to make the plane. Nothing was open at the airport, even for coffee, so Solomon invited them over to his home in Flushing for breakfast. He would often entertain the young hardware hackers who designed these products, and his wife would always recognize them at a glance. She used to say there was an inside personality, and though they looked like disreputable bums, you looked them in the face, you looked in those eyes and you knew who they were. After a quick day-trip to New Hampshire to meet the folks at the new hobbyist magazine Byte, Lee was able to get to a workbench and find the problem a small wire had come loose. The resulting Popular Electronics article spoke of an intelligent computer terminal. But it was clearly a computer, a computer that, when Processor Technology packaged it in its pretty blue case with walnut sides, looked more like a fancy typewriter without a platen. There were new schematics for the revised kit (under one thousand dollars), which of course were provided to anyone who wanted to see how the thing worked. Marsh later estimated that they got thirty to forty thousand requests for schematics. It looked like the Sol would be the machine that broke the computer out of the hobbyist market and brought hacking into the home. It was an odd affair, the first time the tradesmen of this hobbyist-computer business all got together to show their collective wares. Some indignant elderly retirees living at the hotel almost attacked Steve Dompier in the elevator when they saw his long hair. Homebrew-inspired companies like Processor Tech and Cromemco finally met similar souls from other parts of the country, and everybody stayed up far into the night, swapping technical hints and plotting the future. The hackers all seemed to agree that with its low profile, its typewriter-style built-in keyboard, and its video display, the Sol was the next step. The normally abrasive television personality came face to face with the newest manifestation of the hacker dream a Sol computer running a game program written by Steve Dompier. The game was called "Target," and it consisted of a little cannon on the bottom of the screen by which the user could shoot down a series of alien spaceships, made of alphanumeric characters, sailing across the top of the screen. It was a clever little hack, and Steve Dompier, as he later said, "basically gave it away. Imagine these grungy post-hippies being able to bring a computer over to a television studio, set it up, and have a total technical illiterate like Tom Snyder do something with it. Tom went along, and before you could say "commercial break" he was deeply involved not in the least kidding in shooting down aliens, which would zip across the screen in greater numbers as the game progressed, and even dispatch little parachutists loaded with grenades. A feeling that gave you a small taste of what it must be like to use this machine to actually create. He had no great social scheme, did not incubate plans for a Community Memory-style assault on the foundations of the batch-processed society. Meeting after meeting, Steve Wozniak would be at the back of the room, along with a loose contingent of followers of his digital exploits mostly high school-age computer nuts drawn by the sheer charisma of his hacking. His hair fell hap-hazardly on his shoulders, he had the kind of beard grown more to obviate the time-consuming act of shaving than to enhance appearance, and his clothes jeans and sports shirts, with little variation never seemed to fit quite right. Still, it was Steve Wozniak, known to his friends as "Woz," who would best exemplify the spirit and the synergy of the Homebrew Computer Club. Stephen Wozniak did not reach his views of hackerism through personal struggle and political rumination as Lee Felsenstein did. He grew up in Cupertino, California, amidst the curving streets lined with small single-family homes and the one-story, sparsely windowed buildings that sowed the crop of silicon which would be so central to his existence. Even in grammar school, Wozniak could get so engrossed in mathematical ponderings that his mother had to rap on his head to bring him back to the real world. He won a science contest at thirteen for building a computer-like machine which could add and subtract. His friend Alan Baum later remembered him at Homestead High School: "I saw a guy scribbling these neat diagrams on a piece of paper. Every Wednesday they would leave school and have a friend sneak them into a computer room at the Sylvania company.

Because s sleep aid on morning joe cheapest generic unisom uk, = 2t ands are both integers sleep aid honey purchase 25 mg unisom free shipping, s insomnia 79th and amsterdam order unisom with visa, s = s insomnia x macbook discount unisom 25mg with amex, t, = (s t), must also be an integer. We briefy mention several other types of numbers here, though we do not ret to them until Chapter 12. A number a is algebric if it is the root of a polynomial with integer coeffcients; tat is, a is algebrac if there exist integers a0, a1. The irational number, is algebraic, because it is a root of the polynomial x2 2. This fllows fom the fct that the number a/b, where a and b are integers and b = 0, is the root of bx a. The numbers e and r ae also tanscendental, but the proofs of these fcts (which can be fund in [HaW r08]) ae beyond the scope of this book. The Greatest Integer Function In number theory, a special notaton is used fr the lagest integer that is less than or equal to a particular real number. The greatest integer in a real number x, denoted by [x], is the largest integer less than or equal to x. Instead of using the notation [x] fr this fnction, computer scientists usually use the notation LxJ. Besides being important in number teory, as we will see troughout this book, it plays an important role in the analysis of algorithms, a branch of computer science. Additonal propertes of the geatest integer function a fund in the exercises at the end of this secton ad in [GrKa94]. Thefactional part of a real number x, denoted by , is the diference between x and the largest integer less than or equal to x, namely, [x]. Because [x]::x < [x]+1, it fllows that 0:: = x [x] < 1 fr every real number x. The greatest integer in x is also called the integral part of x because x = [x]+. We have {5/4} = 5/4 [5/4] = 5/4 1 = 1/4 and {-2/3} = -2/3 [-2/3] = -2/3 (-1) = 1/3. Diophantine Approximation We kow that the distance of a real number to the integer closest to it is at most 1/2. But can we show that one of the first k multiples of a real number must be much closer to an integer? A importnt pat of number theory called diophantine approximation G studies questons such as tis. In partcula, it concentates on questions that involve the approximaton of real numbers by rational numbers. The proof will depend on the famous pigeonhole principle, intduced by the Ger mathematcian 1 Dirichlet. Informally, this principle tells us if we have more objects than boxes, when these objects a placed in the boxes, at least two must end up in the same box. Although this seems like a partcularly simple idea, it tus out to be extemely useful in number theory and combinatorics. We now state and prove this importat fact, which is known as the pigeonhole principle, because i you have more pigeons than roosts, two pigeons must end up in the same roost. H k+1 or more objects ae placed into k boxes, then at least one box contains two or more of the objects. This contradiction shows tat one of the boxes contains at least two or more of the objects. I a is a real number and n is a positive integer, then there exist integersa and b wit 1 ::a ::: n: such tatlaa b I < 1/n. Each of tese n + 1 numbers lies in one of the n disjoint intervals 0::x < 1/n, 1/n ::x < 2/n. Be­ cause there ae n + 1 numbers under consideration, but only n intervals, the pigeonhole principle tells us that at least two of these numbers lie in the same interval. Because each of these intervals has length 1/n and does not include its right endpoint, we know tat the distance between two numbers that lie in the same interval is less tan 1/n. I fllows that there exist integers j and k with 0:::j < k:::n such that l I <j 1/n. We will now show that when a= k j, the product aa is within 1/n of an integer, namely, the integer b = [ka] [ a]. It fllows that when: a=, and n = 6, we can take a= 5 and b = 7 to makelaa bl < 1/n. Furtermore, in Exercise 34 we show how to use the Dirichlet approximation theorem to show that, given an irational number a, there ae infnitely 2 many diferent ratonal numbers p/q such tatla p/qI < 1/q, an important result in the theory of diophantine approximation. The sequence , where an= n, begins wit the terms 1, 4, 9, 16, 25, 36, 49, 64. The sequence, where n bn= 2, begins wit the ters 2, 4, 8, 16, 32, 64, 128, 256. The sequence , where cn= 0 if n is odd and cn= 1 if n is even, begins wit the terms 0, 1, 0, 1, 0, 1, 0, 1. For example, each term in the sequence of powers of 2 is 2 times the previous term. Even tough the initial terms ofa sequence do not deterine the sequence, knowing the frst fw terms can lead to a conjecture fr a frula or rule fr the terms. Conjecture a frmula fr an, where the frst eight terms of are 4, 11, 18, 25, 32, 39, 46, 53. We note that each term, staing with the second, is obtained by adding 7 to the previous term. Conjecture a frmula fr an, where the frst eight terms ofte sequence are 5, 11, 29, 83, 245, 731, 2189, 6563. We note tat each term is approximately 3 n n times the previous term, suggesting a frmula for an in terms of 3. Looking at these two sequences n together, we fnd that the frmula an= 3 + 2 produces these ters. Conjecture a frmula fr an, where the fst ten ters of the sequence a 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. After examining this sequence from diferent perspectives, we notce that each term of this sequence, afer the first two terms, is the sum of the two preceding terms. This is an example of a recursive definiton of a sequence, discussed in Section 1. The terms listed in this exaple are the inital ters of the Fibonacci sequence, which is discussed in Section 1. Among the sequences we will study ae the Fibonacci numbers, the prime numbers (covered in Chapter 3), and the perfect numbers (introduced in Section 7. Neil Sloane has amassed a fantastcally diverse collecton of more tan 170,000 integer sequences (as of early 2010) in his On-Line Encyclopedia ofInteger Sequences. You may find this a valuable resource as you contnue your study of number theory (as well as oter subjects). We now define what it means fr a set to be countable, and show that a set is countable i and only if its elements can be listed as the terms of a sequence. A set is countable if it is finite or it is infinite and tere exists a one-to­ one corespondence between the set of positive integers and the set. A infinite set is countable if and only if its elements can be listed as the terms of a sequence indexed by the set of positve integers. To see tis, simply note tat a one-to­ one correspondence f fom the set of positive integes to a set S is exactly the same as a listing of the elements of the set in a sequence a. The set of integers is countable, because the integers can be listed startng with 0, fllowed by 1 and -1, fllowed by 2 and -2, and so on. At first glance, it may seem unlikely tat there would be a one-to-one correspondence between the set of positve integers and the set of alratonal numbers. First, we arge all the rational numbers in a two-dimensional aray, as shown in Figure 1. We arge these by placing the faction with a pacular numeator in the position this numerator occupies in the list of 12 the Integers all integers given in Example 1. Next, we list all factions on successive diagonals, fllowing the order shown in Figure 1. Finally, we delete fom the list all factions that represent rational numbers tat have already been listed.

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