Amir Aczel is a professor of statistics whom the IRS treated badly during an unnecessary and unjustified audit. For example, the IRS auditor repeatedly telephoned Aczel at his home before dawn, depriving Aczel, his wife and small baby of sleep. Wanting to avoid such a nasty experience in the future, Aczel used his statistical skills to detect the rules that the IRS uses to choose taxpayers to audit.

He used a super computer to compare thousands of audited income tax returns with thousands of other returns which were not audited. The result is this fascinating book. It explains 14 rules for avoiding an IRS audit.

This book is invaluable...I won my audit because of the tips I learned from this book.

Thorough textbook that can be used by professionals to refresh knowledge. Comes with CD of examples and presentations. Lots of real life examples.

This book places heavy emphasis on the technical and mathematical aspect of investing. It is quite excellent but I found the math to be very advanced and therefore it is probably not suitable to most individual investors. Like the previous reviewer, I also found the book to be too brief.

This book definitely has merit, but it isn't right for beginners or the mainstream investment community. The best audience would be investors with highly specialized investing methods.

This is a very good introduction to the basic mathematics used in Wall Street. The author is a professor of mathematics, which explains the ease in which he explains all of the mathematical concepts for even the average reader. The downside to this book is that it does not go very far. Given its small size, only 100 pages, I feel that he could have gone on much further into more advanced topics. He starts with means and variances, and works his way through correlation, risk measures, performance measures, investment planning, and indexes. He wraps up with some ultra-brief comments on ARCH models and neural networks. I would love for him to come out with a much larger, much more detailed, much more advanced book.

Savvy investors in the stock market need knowledge of the math behind the market, and The Math Behind Wall Street provides it: a slim book masks a wealth of information covering statistics, probability, and other practical applications of business math concepts. From risk factors to annual rates of return, The Math Behind Wall Street will prove invaluable.

Aczel, whose book about Fermat's last theorem was an enjoyable romp through the history of mathematics, now turns his attention to Einstein's theory of general relativity and its implications for cosmology. Based on his work with some historians who are taking a fresh look at Einstein's life and work through recently discovered notebooks and correspondence (Renn, Stachel, et.al), Aczel is able to reveal some previously unknown factoids about the 20th century's greatest scientist. For example, a previously unknown notebook from about 1912 reveals that Einstein had produced his field equation for gravitation nearly 3 years earlier than its final publication in 1915. Apparently Einstein was not convinced of the accuracy of this equation, for he abandoned it, only to rederive it 3 years later with apparently no recollection that he'd been there before. Aczel also spends some effort refuting the popular myth that Einstein was no good at mathematics. He was a superb mathematician, says Aczel, and largely self-taught, which speaks to his agile intellect and intuitive sense for fruitful areas of research.

Unlike any other biographies of Einstein or expositions of relativity that I've read, Aczel takes a "mathematician's eye view" of general relativity, and spends considerable time tracing the development of the geometry of curved space through Gauss, Reimann, and several other lessor known contributors. He also reveals, which I had not known previously, that Einstein kept up an ongoing correspondence with the legendary British mathematician David Hilbert, and that Hilbert published some work of his own based on early copies of Einstein's field equations. This incident has apparently been fodder for considerable historiagraphical debate, and was only recently settled that there was no plagarism or other funny business occurring on the part of either man.

God's Equation is not all Einstein, however. Aczel also introduces us to many of the nagging questions in modern cosmology, and astronomers' attempts to reconcile the recently discovered accelerating expansion of the universe with current theories. Astronomer Saul Perlmutter is central to the story's recent developments, whose supernova observing program lent considerable weight to the accelerating expansion scenario. Taking center stage for this discussion is the resurrection of the cosmological constant, Einstein's famous "blunder," which Aczel argues, has never really left cosmology. As modern astronomers have looked further and further into the universe and back in time, the cosmological constant seems more and more necessary to some theorists, as a repulsive force to counteract the attractive force of gravity (which is itself a brute simplification, since anybody familiar with general relativity knows that gravity is not a force at all, but rather a result of curved spacetime).

Overall, I do recommend this book, though I'm frustrated that Aczel didn't do much more with this opportunity. This book could have easily been twice as long. I get the sense that he was hurried to get it to print for some reason, passing over stories that begged for further clarification (more, for instance, on the eclipse expeditions so central to providing proof for general relativity, and less on the roots of World War I, which delayed the expeditions). All in all, it's an excellent addition to the existing material on Einstein's life and work, and a teaser for more detail on what's really going on in modern cosmology (in the last two or three years, particularly). It makes me hunger for some publications based on Renn and Stachel's work on Einstein. I found a few typographical errors (in a discussion about the effect of Minkowski's lectures on Einstein while at the ETH, he gives a date for Minkowski's birth four years after Einstein published his paper on special relativity).

Inspired by the fact that the universe is ever expanding,
Aczel wrote the history from Einstein to the present
of the thoughts around Einsteins cosmic constant. The main
part deals with Einsteins struggles with his main equation
and the discovery of the first proof for general relativity,
the bending of star light around the sun. This history part
is presented in kind of zooming in at those times and people,
so that one temporarely becomes part of the times of the
process of verificaton and recognition of general relativity.
From the statements about the cosmic constant the author
then leads the reader into modern times, but this time rather
zoomed out, mentioning many people an theories.
It's all gripping to read, but one does not get answers about
the phenomena which introduced the book, namely, how the
universe could possibly accelerate it's expansion.

In this book, Aczel proposes that Einstein's Cosmological Constant, discarded and by the genius himself considered his greatest blunder, is in fact an integral part of the equation that defines the nature of the universe, its past and its future. Some mysterious force is accelerating the expansion of the universe, pushing out on space, countering gravity and making the universe accelerate towards infinity. Aczel argues that in addition to the four known forces: gravity, electromagnetism, the weak and the strong nuclear forces, there is a fifth: the cosmological constant which is the quintessence of the universe. He spoke to many experts in the fields of mathematics, physics, astronomy and cosmology and integrated the ideas of prominent scientists like Eddington, Penrose and Grossman. The chapters deal with stuff like Euclid's Riddle, Riemann's Metric, the expansion of space, the nature of matter and the geometry of the universe but it also serves as a type of biography of Einstein and a history of the development of his theories. There are quotes from Einstein' work and the text is enlivened by portraits, photographs and illustrations. Although an engaging and thought provoking text, it is sometimes difficult to grasp all of the intricacies as there are many formulas that a non-mathematician would not understand. Nevertheless a uniquely stimulating work that concludes with a helpful bibliography and thorough index. I also recommend Marcus Chown's The Universe Next Door, Mark Ward's Universality: Beyond Chaos and Martin J. Rees' Before The Beginning: Our Universe And Others.

While this book is at times entertaining, it did disappoint me. This is really too bad, since the author is well versed in the content matter of this book.

The books picks up some players in mankind's quest to come to grips with the concept of infinity. It focuses on the ancient Greeks, Jewish mysticism and modern mathematics. To juice the narrative up, the author chose to portray two of his characters, Cantor and Goedel as the real live counterparts of Adrian Leverkuhn in Mann's Doctor Faustus.

It is obvious that some considered this sloppily edited book a success. It may be that mathematicians who are fully versed in the mathematical subject matter, were entertained by the cute inclusion of Kaballah and small biographies. Unfortunately, I have only the basic knowledge of undergraduate coursework in real analysis and set theory. Based on this and a similar level of knowledge of Jewish mysticism, I can not help but consider this book an utter failure.

The history of science has been one of disagreement and often feud. The same is true regarding the development of a great many of the cornerstones of current mathematics. The line of reasoning followed here regarding the fate of Cantor and Goedel is on the same level as that of recent publications on the already dispelled magic of "Bible codes".

In addition, the clear but casual treatment of many of the important mathematical milestones in the book, simply does not give credit to and insight into the struggles that these mathematicians went through to find order into what previously appeared to be chaos.

Well too bad. Let's hope that Simon Singh will one day revisit the subject of this book.

Is infinity a number? Who invented or discovered infinity? Is one kind of infinity larger than the other? Is there anything yet unknown about infinity? If you have such questions, this is a good book for you.

Amir D. Aczel is a mathematician, and wrote the much acclaimed book "God's Equation: Einstein, Relativity and the Expanding Universe." He produced here another interesting book by telling stories about the study of infinity. In the introductory chapter numbered aleph-0 the author writes briefly about the life of the mathematician Georg Cantor and his "continuum hypothesis" together with its equation using the Hebrew letter aleph. The symbol aleph with a subscript 0, 1, 2, etc., invented by Cantor, denotes the order or the different level of infinity.

In the following chapters Aczel describes the history of searching infinity. The story starts from its discovery by Greeks in the fifth or sixth century B.C. and includes the concept of infinity in the Kabbalah (a system of Jewish mystical philosophy), discoveries about the nature of infinity by Galileo and Bolzano, and studies by mathematicians in Berlin during the period from 1860 to the start of World War I. Next, the author explains about irrational numbers. Reading about these, the reader might almost forget about Cantor told at the beginning of the book.

After reading almost 40% of the text, however, the reader is brought back to the central story about Cantor. Cantor got strong opposition against his research from his former teacher Leopold Kronecker. This fact and the difficulty of proving his continuum hypothesis seem to have been responsible for Cantor's mental problems. The story of Kurt Gödel, who further developed the work of Cantor, comes next, and the reader learns that Gödel also got mental illness. The author writes in one of the endnotes that Cantor and Gödel were not the only mathematicians working in the field of the foundations of mathematics to suffer from mental illness and that it is interesting to contemplate the reasons for this. Some readers might wish to read Aczel's another book on this contemplation in the near future.

In the final chapter we find an interesting question, "Do numbers actually exist?" The author writes the answer he believes to be the case, and gives a beautiful ending by citing the sentence from a commemorative plaque of Cantor in Halle. The sentence, Aczel thinks, captures Cantor's deepest conviction about mathematics.

Even the reader without much knowledge of mathematics can enjoy this book, if she or he is a little patient to read technical passages. In those passages the author understandably explains the proofs of some natures of infinity discovered by Cantor.

The author does a very good job at showing how Cantor was intrigued by the infinite. He gives a chapter on Kabbalah to give the reader information on the concept of infinity. He does this to give the reader background on how awesome such a concept is. The book tells how Cantor went insane trying to understand the infinite. He was hospitalized with depression many times when he came to close to the infinite. What intrigued me most about the book is Cantors mathematics. Cantor was trying to show by this discipline what it means to be infinite. He writes about lower and higher sets of infinity. To make the reader understand what this means, we turn to philosophy. What does it mean for instance, when we say that God has infinite knowledge? Most people would probably say that God has knowledge of endless things. This is true. In a deeper sense though it means that God has infinite knowledge of infinite things. In other words, if we took a subject like mathematics, God would know infinite knowledge of algebra, infinite knowledge of calculus, and so on. Such knowlege truly boggles the mind. Cantor went insane because he was trying to go above his head. He was trying to limit the infinite to his understanding, and this is impossible. What is limited can never take in the infinite. All we know is that God cannot make a contradiction, but trying to understand the infinite can never be done. All we can do is take in small doses at a time. This book is one math book I highly recommend. It makes for the most intresting reading. It is far more fascinating then any fiction.

This was a great read! It was as pleasurable as sailing on a smooth blue sea. Aczel did a very nice job presenting the history of the compass, in perfect chronological order, from ancient China, through the Middle East, to Italy and then northern Europe.
The book takes the reader on a magical journey from the emperors of ancient China with their infatuation with the magnetic needle, through the travels of Marco Polo, to the medieval city of Amalfi in southern Italy, where the European invention of the boxed compass used in navigation took place.
This is a fast-paced and very informative page-turner that after a satisfying and complete discussion of how the compass was invented takes the reader to Venice to show how the invention was exploited to improve trade and increase wealth. Finally, there is a wonderful concluding description of the Great Age of Exploration--and how the compass brought it about.

This is a very good introduction to the history of the magnetic compass--and to the story of navigation in general--at a level of a popular readership. The author does a good job arguing that early navigators did not need the compass--they used the stars and the currents and habits of migrating birds. But, once the compass was developed, navigation expanded dramatically, and that's where the power of the invention really comes in. He traces the development from China in the first century through to the Mediterranean, Amalfi, northern Europe, Venice, and the great age of exploration. He places the story of the rise of Venice in the context of navigation and the compass. Then he shows how new lands could not have been discovered without the invention. This book is on my top 5 books for this summer.

Amir Aczel presents us with the story of the origins of the compass, in as much detail as scholars have been able to discover. Aczel covers the use of the compass with ancient mariners and how these mariners had to rely on other navigational aids in the days before the compass, such as wind, plants, sounding lines, sea life, geography, currents, etc.. Also mentioned as well is the use of stars in determining latitude, longitude was much more difficult to determine due to the lack of accurate chronometers in early times.

Much of this volume deals with the origin of the 16 point wind rose and how it became incorporated into the modern compass, documented with events and ancient documents in China, and Italy, up to medival times and beyond. This includes discussions of the Etruscans, the cities of Amalfi and Venice, the explorer Marco Polo, all relating to the development of the compass. The second to last chapter sketches the voyages in the Great Age Of Exploration which were vastly aided by the compass, in addition to the astrolabe, a precursor of the sextant.

I believe that Amir Aczel made a very good case here that the compass is one of the pivitol inventions of humanity. Ask yourself this: if the compass had never been invented (which would have slowed down trade and the exchange of information and ideas) how many years of progress would have been lost? My wild guess is 50-100 years of lost progress, a lot.

This book described the history of Fermat's last theorem. It also described the history of some of the theories that Andrew Wiles built upon to solve Fermat's last theorem. For example, it gives an in-depth account of how Wiles proved that all elliptic curves are modular, thus drawing the world closer to solving Fermat's last theorem. It is a must-read for anyone interested in high-level mathematics.

I very much enjoyed the way that the author brought details of the history of mathematics into the story of the final proving of Fermat's Last Theorem. I enjoyed the intrigue and the suspense. I especially enjoyed the little tidbits, such as the story of happenings when the Pythagoreans discovered irrational numbers. If my teachers back in junior high school had shared such interesting sidelights of history, I am sure that I would have come to appreciate math at a much earlier age. I bought copies of this book as Christmas presents for those of my friends who enjoy trivia of the world of mathematics.

This is an exceptional book because the author does a great job in laying out to the layman how the incredibly difficult solution to this simple looking equation was achieved in terms most could understand.

The bulk of the book is kind of a history of number theory in which the author gives mention to many great feats in mathematics that had to be achieved before the solution to Fermat's Last Theorem could be devised. It really amplifies the the words of Sir Isaac Newton, "If I have seen further, it is because I have stood on the shoulders of giants".

Before tackling this book, I had read some of Dr. Aczel's other works, such as The Mystery of the Aleph and Fermat's Last Theorem. I found those books quite enjoyable, so when I heard that he had published a new book on quantum entanglement, I could hardly wait to grab a copy.

After reading it through, I must say I am sorely disappointed to the work. I am still somewhat unclear as to what exactly entanglement is in any depth. His explanations, which are repeated at least a dozen times to no effect, are poor and left me confused. Diagrams are peppered throughout the book with no explanation or captions, leaving one to think that they're there just for eye-candy. Typos abound in the book and some pages are misnumbered (pages 232-4 especially). The book goes on and on about John Bell's famous theorem, but I am still confused as to what this theorem was and why we should care about it. In fact, the practical effects of entanglement aren't even gotten to until practically the last chapter -- the entire rest of the book is simply a biography on those who are or were working on entanglement.

Overall a terribly written book from an author whom I know can do much better. I am left confused as to the very subject matter the book purports to explain. Save your money on this one.

Einstein's did not believe that Universe may be changing or expanding, despite the fact, that his equations of GR were telling him opposite. He introduced Cosmological Constant in order to boost his believes. He was not right, but since 1998 his constant has become a Lambda force, the most important constant of the Nature and in cosmology science.
Einstein's skepticism surfaced again in 1935, when he questioned quantum theory, the one he contributed to immensely by describing the "photoelectric effect".
Einstein was calling quantum theory "incomplete" according to his notions of realism and locality. What experiments and knowledge has been developed later, we can learn reading Amir Aczel book. Will "entanglement" phenomena become as important for modern quantum science as Lambda force for today's cosmology? We do not know it at this time. Essentially Amir Aczel's book describes CERN theorist John Bell's theorem (1966) as a tool for probing certain unknown quantum properties. I believe this book could have been written better. First 122 pages elegantly presents history of classic quantum physics from Young to famous Einstein-Podolsky-Rosen's paper. Classic quantum formulas are introduced with a sense of measure. However the Copenhagen Interpretation of the quantum theory is barely mentioned and explained. Later book gets less clear. Many experiments have been performed around the world in order to prove that Einstein was wrong, but pictures of difficult instrumentations often lack of proper connectivity with text and adequate explanations.
More about entangled states and quantum information can be found in Scientific American magazine (November 2002).

This book had a wonderful topic: quantum theory and especially the idea of entanglement (hence the title). This book isn't for people who are new to quantum theory; one should be familiar with some of the mathematics involved to understand this book. The reliance on many equations is a bit boring, so I'd recommend some other book like "Timeline" by Michael Crichton to introduce the idea of quantum theory.

One of the major problems in any study of SETI is the Fermi paradox. If they are SETI why do we not know about them.

To the credit of this book he does make an attempt to deal with the problem and not like so many other writers try to brush off the problem. His solution is that we are among the most advanced civilisations. At present this is the best response that I have heard to this question. The other is that more advanced civilizations go somewhere else - where is the question.

Unfortunately his arguments do not appear to be very convincing an example would he puts the question that a pre-Columbian Indian sage would reason that life existed on Earth only in the Americas because if there were other continents other then America - life would have developed intelligence, and one of these civilizations would have built large canoes and come here.

My answer back is that he would not pose this question as not pre-Columbian knew of America as a continent. All they knew was their local area. Around which they would have known were inhabited by unknown tribes. If one assumes that this sage knew all that the US Indians knew, then he would have known that tribes in Alaska had contact with another continent.

Also that the relationship
life-> intelligence -> civilization -> SETI
has hardly been proven.

His final arguements is a rather simple probablity theory that in mathematical terms states if the odds for life is very small on a planets. If you have countless planets, then life must exist on some of them. A point that tells us nothing new.

The best books on this subject, I suggest is
Barrow and Tipler - The Anthropic Cosmological principle
Frank White - The SETI factor

"Probability 1, Why There Must Be Intelligent Life in the Universe" was disappointing. The title hints at a mathematical analysis that is largely absent, making this book unsatisfactory for a reader with some mathematics and science training. What I found most unsettling, however, was Aczel overly confident, dogmatic conclusion that a rather simplistic probability analysis fully resolved the "intelligent life" issue and that there is no room for further argument.

Fewer than 25 pages (a portion of the first chapter and some of the final chapter) directly look at the probability of extraterrestrial intelligence, and this examination largely avoids mathematics. We meet a simple equation containing nine multiplicative factors that claims to indicate the abundance of intelligent life in the universe. The last chapter concludes that the number of stars is so great that it overwhelms the low probability of the other critical factors. Fiddle with the other components, but the answer is always the same: intelligent life must indeed exist.

The intervening nine chapters were largely a digression, bland tutorials on astronomy, chemistry, DNA, and evolution, accompanied by non-mathematical discussions of deterministic physical processes, statistical physical processes, chaos theory, sampling theory (inspection paradox), and probability analysis (birthday problem). Under this flood of topics, the lay reader could easily be forgiven for neglecting to question the structure of Aczel's title, not an unbiased "Is There Intelligent Life ...", but "Why there Must Be Intelligent Life... ".

The question is not whether intelligent life does exist (humans are direct evidence), but whether intelligent life is sufficiently common to offer a reasonable likelihood that we humans might achieve some form of communication with another intelligent life form.

"Sufficiently common" does not automatically arise from the observation that there are many galaxies, each composed of many stars, each with some likelihood of having planets, some of which might be expected to orbit within a favorable temperature window that might allow chemical and biological processes underlying evolution to occur.

Let's return to Enrico Fermi's simple question, "Where is everyone?" The answer may well be answered by Aczel's bus stop analogy. Wait long enough and a bus will come by. The SETI search should continue to look for a bus.

Meanwhile, I would have liked discussion about a missing factor that may offset the numerical advantage of the nearly unlimited stars. Recent advances in astrophysics suggest that we live in a most unfriendly universe populated by supermassive accreting black holes located at the core of most large galaxies, gamma ray bursts that for a few seconds emit power that exceeds the total output of all galaxies in the universe, colliding galaxies, and locally, disruptive planetary orbital events. For example, none of the newly observed planetary systems exhibit well-behaved, almost circular, non-overlapping planetary orbits that characterize our solar system.

Catastrophic astrophysical events appear to be common, possibly too common to permit life. It is like waiting for a bus in an unending war zone. There may be little hope that a bus will ever arrive.

Our star, the sun, appears to be derived from element recycling in two or three prior supernova events. After a rather violent early history that probably extinguished early life on earth, our solar system has been relatively well-behaved for about 4.5 billion years. Life has evolved steadily, with a few setbacks like the asteroid impact responsible for the dinosaur extinction, culminating to-date with man and his radio telescopes.

What if, viewed on a larger scale, our particular history was exceptionally well-behaved, exceptionally placid, characterized by only minor life threatening astrophysical events? What if the extended periods of relative quiet required for evolution simply do not occur, or occur only rarely? Living in the suburbs of a galaxy may reduce the likelihood of astrophysical events that terrorize the more dense, central core, but even on the rim of the galaxy the requisite extended period of quiescence may seldom occur.

The fundamental question may be how often (or more pessimistically, how rarely) the long, slow process of evolution is allowed to enjoy an extended period without catastrophic events. Is evolution ever allowed to run its course? The answer must be yes, as we are direct proof. But it may be that intelligent life is quite rare, and lives in distant isolation, each searching the skies in vain.

Until we better understand catastrophic astrophysical phenomena and know whether long periods of quiescence can be expected to occur, I reserve my opinion on the issue, and I question the confident conclusion of "Why There Must Be Intelligent Life".

I conclude with a thought more eloquently stated by others: "Which would be more remarkable and more unsettling - to discover that other intelligent life exists, or to conclude that we are alone?"

This book has one quality: it is reasonably well-researched and talks about recent (mid-90's) discoveries. With that said, the book is overall quite disappointing. The biggest problem about this book is that if the reader has a scientific background, or even is a serious amateur, he or she will be turned off by the many mistakes and inaccuracies in this book. I will mention just one example, when he writes: "Then there is the motion of all stars (detectable in observations of distant galaxies) away from each other due to the expansion of our universe." This is very misleading, since it implies that the stars in our own galaxy are moving away from each other due to the expansion of the universe, a gross mistake. This could be a typo, but it's hard to give the author the benefit of the doubt, given the large number of other inaccuracies in the book. This book looks like it was not reviewed. It's hard to believe that Martin Gardner, one of my favorite authors, gave a positive review of this book, as suggested by the quote in my edition.