An Adventurer's Guide to Number Theory

Book Description

This witty introduction to number theory deals with the properties of numbers and numbers as abstract concepts. Topics include primes, divisibility, quadratic forms, related theorems, explored in imaginative chapters such as "Seven jogged my elbow," "On a clear day you can count forever," and "When the clock strikes thirteen."

Customer Reviews:

An informal introduction to number theory with a historical perspective........2007-01-01

Friedberg's text, which is written in an inviting conversational tone, is an idiosyncratic introduction to number theory which stresses the subject's historical development. The material is introduced through problems that motivate the results that Friedberg discusses. These results include Euclid's theorem that there are infinitely many prime numbers, the use of the sieve of Eratosthenes to find prime numbers less than the square root of a positive integer n, Gauss' Fundamental Theorem of Arithmetic, perfect and amicable numbers, Pythagorean triples, modular arithmetic, factoring numbers of the form x^2 + ny^2, and the Law of Quadratic Reciprocity. Friedberg ably links these topics together and places them in historical perspective. However, there are better introductions to number theory. This text has no formal exercises, so you do not have an opportunity to reinforce what you are learning. It is also a poor reference because definitions, theorems, and proofs are stated within paragraphs, the whimsical chapter titles do not convey what topics are covered, and there is no subject index to help you find the definitions and theorems that are buried within the paragraphs. Also, the scope of coverage is less than that of other introductions to number theory.

In his introduction, Friedberg, a physicist, distinguishes between the common and scientific meanings of the word theory. He also discusses the difference between a scientific theory and a mathematical theorem.

Friedberg uses sequences to introduce proofs by mathematical induction. Friedberg shows how proofs of mathematical induction work and discusses why they are valid. In the text, however, he tends to use Fermat's method of infinite descent to prove assertions indirectly rather than using direct induction proofs.

While discussing these sequences, Friedberg refers to 1 as a prime number, contrary to the usual definition that a prime number is a positive integer larger than 1 whose only factors are 1 and itself. Defining 1 to be prime would violate the assertion of the Fundamental Theorem of Arithmetic that each positive integer has a unique prime factorization. For instance, if you allow 1 to be prime,

6 = 2 x 3 = 1 x 2 x 3 = 1 x 1 x 2 x 3 = ...

This is problematic, so Friedberg disregards his own assertion that 1 is prime when discussing the Fundamental Theorem of Arithmetic.

Friedberg then discusses some results from ancient Greece, including the fact that the square root of 2 is irrational, Euclid's theorem that there are infinitely many primes, the sieve of Eratosthenes, perfect numbers, and amicable numbers. He also proves the Fundamental Theorem of Arithmetic while covering these topics.

During a brief discussion of Diophantine equations, Friedberg discusses how to find and factor Pythagorean triples, that is, triples (x, y, z) of positive integers that satisfy the equation x^2 + y^2 = z^2. A better explanation of how to find Pythagorean triples is given by John Stillwell in his texts Mathematics and its History and Elements of Number Theory. After Friedberg's discussion of the problem, he tackles the more general problem of how to factor numbers of the form x^2 + ny^2, where n is a positive integer. The mathematics used to solve this problem, including modular arithmetic, is quite powerful, which is conveyed by the simple proofs Friedberg provides of results proved with more difficulty earlier in the book and by his proofs of Fermat's Last Theorem for the cases n = 3 and n = 4.

Friedberg concludes the book with a proof of Gauss' proof of the Law of Quadratic Reciprocity. The material on quadratic residues calls upon many of the previous results. However, while there is a table classifying the theorems in the text (albeit without their actual formal statements), the lack of a subject index makes finding the necessary definitions and theorems difficult. Consequently, Friedberg's arguments are more difficult to follow than they need to be.

If you are seeking a basic introduction to the subject, try working through Oyestein Ore's an Invitation to Number Theory (New Mathematical Library), which is accessible to a bright high school student. Ore is also the author of a slightly more advanced text, Number Theory and Its History (Dover Classics of Science and Mathematics), which, like Friedberg's text, introduces number theory through its historical development. There are numerous more advanced treatments of the subject, which serve as good introductions. They include, among others, The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport, Elementary Number Theory by Gareth A. Jones and J. Mary Jones, and An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery.

What a carefully written exploration!.......2002-09-05

I think this book is a masterpiece in mathematical exposition. All you need to know is how to add, subtract, multiply, and divide and maybe a vague memory of algebra. Mr. Friedberg will walk you through a lot of number theory after which (or maybe even during which) you may find a number theory textbook more approachable. If you read carefully you will really internalize what a proof by contradiction is and what an infinite descent is. You'll get a real appreciation for the logic of a proof and you'll see some ingenius tricks used by some great mathematicians ... and you'll understand them!
This book is approachable and doable by anyone with a motivation for what can be understood about numbers. And I can't stress how carefully, thoughtfully, and articulately it is written.

Be carefull.......2001-08-24

You must have a medium understanding of mathematics and algebra.

"Quirky" is exactly the right word........2001-06-25

A lot of us know that you can't double the square. You can't find two square whole numbers, one of which is twice the other. This, of course, is an ancient Greek problem.

If b squared were equal to two times a squared, the right side of the equation would contain an odd factor of two, which is obviously impossible by the fundamental theorem of arithmetic. This is the modern way of proving this assertion.

Richard Friedberg prefers the old way. He uses Fermat's method. On page 45 we read:

"At each point we can prove that the numbers we have reached are even. So we can go on dividing forever. But this is impossible. Eventually we must reach 1, or some other odd number. Since we have proved something that is impossible, we must have assumed something that isn't true. The only thing we assumed was that there are two numbers a and b, such that two times a squared equals b squared. So there can be no such numbers."

He continues in this way all the way to quadratic recipocity, and concludes with a Table of Theorems, all rigorously proved in his own quirky way.

I continue to be frustrated by Friedberg's approach to number theory. It is historically accurate but very difficult to assimilate or combine into present day orthodoxy. I'm not sure whether he is worth my time, but nevertheless I continue to study his book. I've read it on and off now for more than five years. There is no doubt in my mind that he is a genius . . . hence the five stars.

Whether one wants to embark on this slippery slope of classical geometry, historical number theory, the defects in Euler's reasoning and other incredibly obscure topics in number theory, the reader must decide for himself or herself. I don't think I'll ever know as much about the history of number theory as Richard Friedberg does, so I decided to put in my two cents mid-way through the course of my studies.

Not that adventurous.......2000-12-27

I felt like I suited up for space travel and got grounded by equipment malfunction. Perhaps I took the title too literally. Since there are so many books on number theory, surely one with such a title should cover the outer reaches. This is nothing but a basic introduction. More is covered in Albert Beiler's "Recreations in the Theory of Numbers" and it's much more adventurous. Still worth 3 stars, and worth owning - but not worth keeping under your pillow.

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An Adventurer's Guide to Number Theory
Richard Friedberg
Manufacturer: New York: McGraw-Hill Book Company, 1968
ProductGroup: Book
Binding: Hardcover
ASIN: B000NVEH2S

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An Adventurers Guide To Number Theory
Friedberg
Manufacturer: Mcgraw Hill Book Company
ProductGroup: Book
Binding: Hardcover
ASIN: B000JD0XT6

Book Description

Equalling Tarawa, Iwo Jima and Okinawa in scale and ferocity, the battle for Peleliu has long been regarded as the Pacific War’s “forgotten battle”, and perhaps one that should never have been fought. A massive carrier-based attack some weeks before the invasion destroyed all aircraft and shipping in the area and virtually isolated the Japanese garrison. 1st Marine Division commander, General Rupertus, made extravagant claims that the capture of Peleliu would “only take three days – maybe two.” But the Japanese fought a bloody battle of attrition from prepared positions an in a struggle of unprecedented savagery a whole Marine Division was bled white.

Customer Reviews:

Good, but suffers from poor editing........2006-05-16

I agree completely with the other reviewer. Overall, a good addition to the Campaign series, but hopefully the next edition will emerge without the numerous small errors (units, locations, dates) that can make it an occasionally frustrating read.

Good overview, poor editing.......2004-09-30

This is a better Osprey book than most, but it doesn't reach its potential. In spite of the name, the book describes the U.S. Marine and U.S. Army infantry actions on all of the Palau Islands, though the emphasis is on Peleliu. In the typical Osprey format it begins with an overview of the campaign in the context of the Philipine campaign, describes the opposing commanders, and then delves into the details of the campaign itself. The last sections include the aftermath of the battle and a brief description of the islands today.

For the most part the book is pretty solid. World War II is not my specialty, and I'm just starting to read about the war in the Pacific. From what I have read elsewhere the book is reasonably accurate. The writing style is clean. Unlike when reading some of Osprey's other campaign books, I had little trouble following the action. Although the authors don't stick to a strict chronological order, they don't hop around more than is necessary. As always, the maps are gorgeous and excellent. I liked the Orders of Battle, though I wish they included the number of men at each organization level, but that's a minor quibble.

I had a couple of issues with the narrative. I wish they had spent more time explaining what was going on with the U.S. Navy elements during the campaign. The authors mention the initial, pre-invasion bombardment. They describe how Admiral Oldendorf declared that his ships had hit all available targets and ordered some to leave for the Philipines. They don't list the ships that were left, what they did during the rest of the campaign (were they conducting bombardment or just acting as a picket line?), or where they were located. This is important because several barges of Japanese reinforcements made for Peleliu during the battle, with 600 soldiers actually landing on the island. The book doesn't make clear how these soldiers managed to sneak through the navy cordon. There are some other details I'd have liked to have known, too, such as whether carrier airstrikes in support of the U.S. troops were a daily occurance, and when exactly the Marine fighters arrived on the island.

Although the authors did a good job of explaining the Japanese command structure, and the initial Japanese counterattacks, there is little on Japanese deployments, and virtually nothing beyond those counterattacks describing Japanese troop movements. They state that Col. Nakagawa was one of the two best tacticians the Japanese had, but there is no sense of why they believe this. The book doesn't describe any Japanese movements except in terms of their perimeter shrinking and the number of men left. Did Nakagawa place all of his men at the beginning of the battle and pretty much leave them there, or was he constantly shifting troops around? When the Marines first assaulted the Umurbrogol Mountains in the south, were there Japanese troops sitting around waiting in the north, or did Nakagawa move troops from quiet sectors to critical sectors? This isn't explained. I wanted an idea of the number of Japanese casualties per day, but Japanese casualties aren't mentioned except at a few key points in the battle. A couple of paragraphs describing Japanese infiltration techniques and the "limited counter-attacks" would have been most useful, but the book doesn't go into this level of detail for the Japanese (it does go into some of this kind of detail on the Marine side, especially in reference to a couple of the full color plates showing typical combat scenes). I really wanted a better idea of how the Japanese defenses were laid out, even if it was just a small diagram of a typical defensive layout but that wasn't supplied. Perhaps much of the information on the Japanese side of the campaign isn't available. If that's the case, it should have been mentioned.

In spite of these shortcomings I'd still give the book 4 stars for content, considering the size limitations. I only give it 3 stars because of the editing. This is, unfortunately, a common problem with Osprey books. The book was published in October, 2002, yet the text mentions something that happened in January, 2002. Nine months from finished manuscript to printed book is too fast to do a half decent editing job. Editing mistakes, such as sentence fragments, and phrases repeated at the end and the beginning of the same sentence, are found throughout the book, in the text and in picture captions. Editing problems throw some minor details into doubt. In one section of the text we are told that the island of Koror had a seaplane base. In another section we are told it was a submarine base. The map shows it as a seaplane base. Which is correct? Is the text and map right, or was the map based on the erroneous piece of text? A cursory reading of the book should have caught these errors (I certainly caught them easily enough). Osprey is sloppy, which is unfortunately the norm.

The book is a worthwhile overview of the campaign, in spite of its shortcomings. I only wish that it had a little extra detail and that more time had been spent editing it.

Bloody Forgotten Battle.......2003-07-16

A nice very easy access to this very bloody and forgotten battle. There is a very good section on how the battle figured in the overall strategic plans. In final analysis (and no one wants to admit it), like modern day Iraq, Peleliu was a battle that need never have been fought. Monday morning quarterbacking is easy, but there are some real glaring inconsistencies that someone should have taken the blame for at the time. If aerial recce's were done of Peleliu they would have instantly reccognised the central ridge of about 200 metres in height poking out of the jungle (I have flown over it twice at several elevations and can assure you that it is VERY visible).Apparently aerial recce's never saw this feature according to the text. Whay was this info. not conveyed to operational commanders? Bleeding the Marines in this battle gained the Americans nothing. Better to have taken Anguar established their base there and left the rest of the Palaus alone. It would have worked because that is in fact what more or less happened: the landings at Peleliu and Anguar still left the main island and numerous lesser islands occupied but neutralised. Why not merely neutralise and not land on Peleliu as well. That is the central question that will remain unanswered. This Osprey Edition does a good job of the overview of the battle and the hell that both sides faced in this "forgotten battle."Text is good but suffers from inaccurate rendering of Japanese names into Roman Script (eg. Taksgo is not a Japanese name). As do so many books on the Pacific war. The easy Japanese phonetic alphabet make such mistakes inexcusable.

Another Fine Addition to the Campaign Series.......2003-01-03

Peleliu 1944, Osprey Campaign volume number 110, covers one of the more obscure but controversial battles of the Pacific War. Jim Moran and Gordon L. Rottman, two authors well versed in US Marine Corps history, have done a fine job synthesizing the available secondary sources on the battle into a succinct campaign narrative. A detailed, blow-by-blow campaign narrative, combined with excellent maps and photographs, makes Peleliu 1944 a fine addition to the Osprey Campaign series. The authors succeed in laying out the two most enduring questions about the battle: was the invasion of Peleliu necessary and why was it so costly?

In standard Osprey Campaign series format, Peleliu 1944 begins with short sections on the origins of the campaign, a chronology, and sections on opposing plans, commanders and forces. The campaign narrative itself is 48 pages long, followed by a brief aftermath section, comments on the battlefield today and a bibliography. There are a total of three "Birds-Eye View" 3-D maps (D-Day on Peleliu, Securing the North of Peleliu and the Reduction of the Umurbrogol Pocket), six 2-D maps (Strategic situation, the Palau Islands, Peleliu Island, D-Day to D+8, Capture of Angaur Island and the Umurbrogol Mountains), and three battle scenes (Beach Orange on D-Day, the Japanese tank counterattack and Reducing the Umurbrogol Pocket). The authors also provide very detailed orders of battle for the 1st Marine Division, the US Army 81st Infantry Division and the Japanese forces on Peleliu.

The controversy about Peleliu began even before the invasion. Admiral Halsey wanted to neutralize the island by bombardment and then bypass it, as so many other islands had been bypassed. However, Admiral Nimitz disregarded this suggestion and ordered the invasion to proceed in order to remove a potential threat to General Macarthur's imminent invasion of the Philippines. The authors clearly point out that most of the Japanese aircraft and naval vessels in the Palau Island group had been destroyed by US air raids well before the invasion. Thus, the threat to Macarthur was minimal. Securing Peleliu provided the US Navy with a convenient but not critical base (although it did help to rescue the survivors of the sunken USS Indianapolis in 1945). Thus the question really devolves to whether or not the invasion was worth the cost, since it was clearly not critical to winning the war in the Pacific.

On the conduct of the battle itself, Moran and Rottman clearly place the blame for the heavy US casualties on two causes: inadequate intelligence preparation of the battlefield (IPB) and incompetent senior leadership. In modern parlance, IPB focuses on gaining information about the terrain, weather and enemy that will enable the friendly commander to make the best possible decisions. However, the US Marines landing on Peleliu were continuously presented with unpleasant surprises about the terrain, weather and enemy. Instead of the "mostly low and flat terrain" briefed prior to D-Day, the invaders were confronted with undetected anti-tank ditches, coral outcrops and the very nasty Umurbrogol Mountains. How did aerial reconnaissance miss mountains this size? The Marines had only expected a short three-day fight for Peleliu and had not really considered the weather. However, temperatures around 100? F fatigued the Marines and the monsoon rains disrupted the build-up. Finally, the Japanese altered their tactics on Peleliu. The Japanese were beginning to learn from previous battles that wild Banzai charges consumed their troops quickly and led to short battles. On Peleliu the Japanese demonstrated a new preference for protracted, attritional struggles based around interlocking and cunningly hidden defenses. Indeed, it is hard not to read this book and gain great respect for the incredible defense put up by the Japanese troops on Peleliu. All three factors - terrain, weather and new enemy tactics - ruined the American timetable and exacted a heavy toll on the invaders.

Moran and Rottman blame Major General Rupertus, the commander of the 1st Marine Division, for much of the heavy casualties on Peleliu. Rupertus initially claimed that Peleliu would be secured in 3-4 days of hard fighting. It quickly became apparent on D-Day that the Marines were running into unexpectedly tough resistance, but Rupertus did not elect to alter his timetable. Instead, Rupertus continually urged his regimental commanders to make costly frontal assaults until his division was virtually decimated. Rupertus refused to ask for the 81st Division to relieve his Marines and he foolishly released his tank battalion just when his troops most needed armor support. Rupertus kept urging his exhausted marines into futile assaults until there was nothing left to give. At that point, the First Marines were relieved and replaced by the 81st Division, which eventually secured the island after two more months. In the end, it took the Americans over two months to secure Peleliu and cost over 1,500 dead. One big question left unanswered is why Nimitz didn't intervene to stop this senseless bloodletting and why Rupertus wasn't relieved until after the operation. Obviously, there were more than a few reputations to protect from fallout from the Peleliu slugfest. For military professionals, Peleliu 1944 provides an instructive lesson on the perils of over-confident predictions based on inadequate knowledge of the terrain, weather and a resourceful enemy.

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Recent history relevant to today

Belau
Douglas Faulkner
Manufacturer: Xlibris Corporation
ProductGroup: Book
Binding: Paperback

Environmental Science | Earth Sciences | Science | Subjects | Books
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ASIN: 0738821888

Book Description

BELAU The PROMISED LAND is about Douglas Faulkner's efforts to save Belau from imminent destruction. Its islands and reefs are too fragile to be subjected to big development projects; projects that care little for beauty.

Customer Reviews:

Recent history relevant to today.......2004-12-21

Belau, The Promised Land is a collection of verbatim records from seven meetings of the United Nations Trusteeship Council between 1981 and 1987. Five are records of the author's personal petitions to the council. The remaining two records detail the USSR representative's input on the same issue. Like the iconic image of the protester standing in front of a tank in Tianamen Square the author single handedly defied the might of both the U.S. military and the federal government to oppose establishment of a large U.S. military base amid the exceptional natural beauty of the Pacific islands of Belau (or Palau as it is know to the outside world).

Although now history and the base plans abandoned with the end of the Cold War this account is perhaps more relevant than ever for a variety of reasons. It not only documents the duplicity and indifference of powerful forces pursuing their own agendas but also reveals the reservations and the too often hidden humanity of persons of good will on all sides of the issue. It also shows that armed only with reason, facts, and good will an individual can make a difference.

In a time when opposing views rage Faulkner's Belau account is a salutary reminder to not only recognize the deceits of power but also that strength of conviction is no guarantee of correctness and that basic humanity exists on all sides of opinion.

As for Belau, although it dodged the threat of a Superport and was spared the collateral damage of the military base it is now exposed to the ravages of having become a popular tourism destination. Unspoiled natural beauty has come to have high economic value and the battle now is to find a balance between economic and esthetic or spiritual values. How much of each are we willing to sacrifice for the other.? As Faulkner reveals, this is not so much a battle against outside forces as it is one within ourselves and although his particular skirmish is over the war continues with no light apparent at the end of the tunnel. As a tactical account of a successful battle in the ongoing struggle for the human spirit Faulkner is well worth reading. If nothing else a recount of recent history helps us to remember that things are rarely as bad as we fear or as good as we hope and getting too excited over either is unwarranted.

In his classic photo book on Palau, This Living Reef, published in 1974 Faulkner somewhat prophetically wrote:
"For the moment Belau is a "new world", little touched by the occupations and diversions of industrial man. The reefs and islands have survived many geologic and climatic changes, but they may not be able to sustain the degradations of our unthinking carelessness. This is my only fear in bringing to light this world which I love. If we destroy this living reef, a great beauty and richness will go out of the world, and out of our lives."

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Lee Boo Of Belau: A Prince in London
Daniel Peacock
Manufacturer: University of Hawaii Press
ProductGroup: Book
Binding: Paperback

Lonely Planet Palau: Diving & Snorkeling

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worth the price
Very helpful while diving Palau
Excellent, and more than just a diving & snorkeling guide !
Excellent, and more than just a diving & snorkeling guide !

Lonely Planet Palau: Diving & Snorkeling
Tim Rock , and Francis Toribiong
Manufacturer: Lonely Planet Publications
ProductGroup: Book
Binding: Paperback

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Papa Mike's Palau Islands Handbook

Product Features:

dive site depth range and conditions
common and hazardous marine life
topside practicalities
diving services and environmental organizations
10 easy-to-read maps

ASIN: 1864500190

Product Description

The Palau archipelago in Micronesia boasts a huge barrier reef that protects the mushroom-shaped limestone Rock Islands, offering tranquil diving and snorkeling in the lagoon's warm waters. Dramatic drop-offs at the famed Ngemelis Wall and Blue Corner makes Palau an adrenaline-pumping drift-diving paradise. Explore historic WWII shipwrecks, venture into mysterious sea caves, and snorkel ancient Jellyfish Lake. From the southern tip of Peleliu to the northern reaches of Kayangel Atoll, this guide covers 48 of Palau's best dive sites, with full-color photos throughout.

Customer Reviews:

worth the price.......2005-11-24

Well worth the money to purchase this book. There are LOTS of quality places to snorkel and dive in Palau, this book gives you some help picking out the best places for you. Also some (not a ton, but adequate) suggestions for dry-land activities.
And yes, I was in Palau spring 2005- the jellys in jellyfish lake are back to a healthy population after El nino!

Very helpful while diving Palau.......2002-01-29

I spent 10 days diving Palau in June 2001 and this book was incredibly helpful. There are so many dive sites to choose from in this region, and this book did an excellent job detailing each and every one. Since most dive operators in this region take "requests" rather than dictate where you're going, it's nice to have some information to make an informed decision, depending upon whether you want to focus on macro, big animal enounters, drift dives, wrecks, etc. Going to dive Palau without this book is simply a bad idea.

Excellent, and more than just a diving & snorkeling guide !.......2001-07-28

I used by no means to be a fan of diving and snorkeling. Yet, the beautiful seas of the Pacific, combined with this wonderful excellently written guidebook, introduced me to the wonderful world of this activity. The guidebook provides excellent and invaluable information and advice for divers and snorkelers - beginners and experts alike - with special focus on how great this can be in Palau. Meanwhile, the guidebook is also a wonderful travel tool for visiting Palau, whether or not you are primarily there for snorkeling and diving. All in all, it is surely a worthwhile, if not crucial, purchase, before you set off to the Pacific.

Excellent, and more than just a diving & snorkeling guide !.......2001-07-28

Micronesia, the Problem of Palau (Report / Minority Rights Group)

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Micronesia, the Problem of Palau (Report / Minority Rights Group)
Roger Clark
Manufacturer: Minority Rights Group
ProductGroup: Book
Binding: Paperback

Human Rights | Constitutional Law | Law | Subjects | Books

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Palauan Legends (no. 1)
Cheldecheduch er Belau
Manufacturer: Palau Community Action Agency
ProductGroup: Book
Binding: Paperback

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Palauan Legends (no. 2)
Cheldecheduch ed Belau
Manufacturer: Palau Community Action Agency
ProductGroup: Book
Binding: Paperback
ASIN: B000M7S01W

Patterngrams How To Copy Designs at Home

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Patterngrams How To Copy Designs at Home
Nancy Olson
Manufacturer: Fairchild Publications
ProductGroup: Book
Binding: Hardcover
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