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"Certain numbers have magical properties. E, pi and the Fibonacci series come quickly to mind - if you are a mathematician, that is. For the rest of us, the magic numbers are the familiar ones that have something to do with the way we keep track of time (7, say, and 24) or something to do with the way we count (namely, on 10 fingers). The "time numbers" and the "10 numbers" hold remarkable sway over our lives. We think in these numbers (if you ask people to produce a random number between one and a hundred, their guesses will cluster around the handful that end in zero or five) and we talk in these numbers (we say we will be there in five or 10 minutes, not six or 11). "

"Melvyn Bragg and his guests discuss imaginary numbers. In the sixteenth century, a group of mathematicians in Bologna found a solution to a problem that had puzzled generations before them: a completely new kind of number. For more than a century this discovery was greeted with such scepticism that the great French thinker Rene Descartes dismissed it as an "imaginary" number. The name stuck - but so did the numbers. Long dismissed as useless or even fictitious, the imaginary number i and its properties were first explored seriously in the eighteenth century. Today the imaginary numbers are in daily use by engineers, and are vital to our understanding of phenomena including electricity and radio waves."

"SPEAKING two languages rather than just one has obvious practical benefits in an increasingly globalized world. But in recent years, scientists have begun to show that the advantages of bilingualism are even more fundamental than being able to converse with a wider range of people. Being bilingual, it turns out, makes you smarter. It can have a profound effect on your brain, improving cognitive skills not related to language and even shielding against dementia in old age. This view of bilingualism is remarkably different from the understanding of bilingualism through much of the 20th century. Researchers, educators and policy makers long considered a second language to be an interference, cognitively speaking, that hindered a child's academic and intellectual development. They were not wrong about the interference: there is ample evidence that in a bilingual's brain both language systems are active even when he is using only one language, thus creating situations in which one system obstructs the other. But this interference, researchers are finding out, isn't so much a handicap as a blessing in disguise. It forces the brain to resolve internal conflict, giving the mind a workout that strengthens its cognitive muscles. Bilinguals, for instance, seem to be more adept than monolinguals at solving certain kinds of mental puzzles. In a 2004 study by the psychologists Ellen Bialystok and Michelle Martin-Rhee, bilingual and monolingual preschoolers were asked to sort blue circles and red squares presented on a computer screen into two digital bins - one marked with a blue square and the other marked with a red circle. In the first task, the children had to sort the shapes by color, placing blue circles in the bin marked with the blue square and red squares in the bin marked with the red circle. Both groups did this with comparable ease. Next, the children were asked to sort by shape, which was more challenging because it required placing the images in a bin m

"True art would never be mistaken for a crude, paint-by-the-numbers copy. But a researcher has developed a statistical tool for determining whether a purported masterpiece is only a skilled imitation, suggesting that art may be a numbers game after all. Using high-resolution digital images and complex mathematical formulas, associate professor Hany Farid of Dartmouth College analyzed works by Renaissance artists to determine their authenticity. His computer program was able to accurately separate eight drawings by 16th century Flemish artist Pieter Bruegel the Elder from five drawings by imitators. It also found that portions of a painting by Italian artist Pietro di Cristoforo Vannucci, known as Perugino, were probably done by Perugino's apprentices. Farid described his work, presented Monday in the Proceedings of the National Academy of Sciences, as "simply another tool that is contributing to the dialogue of art authenticating" and said more work is needed before digital analysis of art could be done on a wider scale. Art experts reacted warily to the prospect that a masterpiece could be reduced to the sum of its digital parts. "

"What is mathematics? It's neither physical nor mental, it's social. It's part of culture, it's part of history. It's like law, like religion, like money, like all those other things which are very real, but only as part of collective human consciousness. That's what math is. For mathematician Reuben Hersh, mathematics has existence or reality only as part of human culture. Despite its seeming timelessness and infallibility, it is a social-cultural- historic phenomenon. He takes the long view. He thinks a lot about the ancient problems. What are numbers? What are triangles, squares and circles? What are infinite sets? What is the fourth dimension? What is the meaning and nature of mathematics? In so doing he explains and criticizes current and past theories of the nature of mathematics. His main purpose is to confront philosophical problems: In what sense do mathematical objects exist? How can we have knowledge of them? Why do mathematicians think mathematical entities exist forever, independent of human action and knowledge? "

"Female physicists, astronomers and mathematicians are up against more than 2,000 years of convention that has long portrayed these fields as inherently male. Though women are no longer barred from university laboratories and scientific societies, the idea that they are innately less suited to mathematical science is deeply ingrained in our cultural genes. The problem goes back to the ancient Greeks, particularly to Pythagoras, the philosophical giant who dreamed the dream that became modern physics. Pythagoras almost certainly learned his famous theorem about right-angled triangles from the Babylonians, but we owe to him a far greater idea: "All is number," he declared, becoming the first person to say that the physical world could be described by the language of mathematics. Pythagoras also gave us the idea of the "music of the spheres," a set of mathematical relationships that would describe the structure of the universe itself. His vision would eventually give rise to the scientific revolution led by Copernicus, Kepler, Galileo and Newton. The search for a theory of everything today is the latest version of the ancient Pythagorean quest for divine "cosmic harmonies.""

"IT has been more than three years since the beginning of the Wall Street financial crisis, yet we continue to hear about new evidence of glaring errors and widespread misdoings. Even the smartest minds in finance are left scratching their heads: how did we not catch any of this sooner? When I hear this refrain, I am reminded of Boris Goldovsky. Goldovsky, who died in 2001, was a legend in opera circles, best remembered for his commentary during the Saturday matinee radio broadcasts of the Metropolitan Opera. But he was also a piano teacher. And it is as a teacher that he made a lasting - albeit unintentional - contribution to our understanding of why seemingly obvious errors go undetected for so long. One day, a student of his was practicing a piece by Brahms when Goldovsky heard something wrong. He stopped her and told her to fix her mistake. The student looked confused; she said she had played the notes as they were written. Goldovsky looked at the music and, to his surprise, the girl had indeed played the printed notes correctly - but there was an apparent misprint in the music. At first, the student and the teacher thought this misprint was confined to their edition of the sheet music alone. But further checking revealed that all other editions contained the same incorrect note. Why, wondered Goldovsky, had no one - the composer, the publisher, the proofreader, scores of accomplished pianists - noticed the error? How could so many experts have missed something that was so obvious to a novice? This paradox intrigued Goldovsky. So over the years he gave the piece to a number of musicians who were skilled sight readers of music - which is to say they had the ability to play from a printed score for the first time without practicing. He told them there was a misprint somewhere in the score, and asked them to find it. He allowed them to play the piece as many times as they liked and in any way that they liked. But not one musician ever found the err

Computers have become an extension of us: that is a commonplace now. But in an important way we may be becoming an extension of them, in turn. Computers are digital - that is, they turn everything into numbers; that is their way of seeing. And in the computer age we may be living through the digitization of our minds, even when they are offline: a slow-burning quantification of human affairs that promises or threatens, depending on your outlook, to crowd out other categories of the imagination, other ways of perceiving.

"Debunking the social stigma around late risers, or what Einstein has to do with teens' risk for smoking. "Six hours' sleep for a man, seven for a woman, and eight for a fool," Napoleon famously prescribed. (He would have scoffed at Einstein, then, who was known to require ten hours of sleep for optimal performance.) This perceived superiority of those who can get by on less sleep isn't just something Napoleon shared with dictators like Hitler and Stalin, it's an enduring attitude woven into our social norms and expectations, from proverbs about early birds to the basic scheduling structure of education and the workplace. But in Internal Time: Chronotypes, Social Jet Lag, and Why You're So Tired, a fine addition to these 7 essential books on time, German chronobiologist Till Roenneberg demonstrates through a wealth of research that our sleep patterns have little to do with laziness and other such scorned character flaws, and everything to do with biology. In fact, each of us possesses a different chronotype - an internal timing type best defined by your midpoint of sleep, or midsleep, which you can calculate by dividing your average sleep duration by two and adding the resulting number to your average bedtime on free days, meaning days when your sleep and waking times are not dictated by the demands of your work or school schedule. For instance, if you go to bed at 11 P.M. and wake up at 7 A.M., add four hours to 11pm and you get 3 A.M. as your midsleep."

"Which of the following people would you say is the most admirable: Mother Teresa, Bill Gates or Norman Borlaug? And which do you think is the least admirable? For most people, it's an easy question. Mother Teresa, famous for ministering to the poor in Calcutta, has been beatified by the Vatican, awarded the Nobel Peace Prize and ranked in an American poll as the most admired person of the 20th century. Bill Gates, infamous for giving us the Microsoft dancing paper clip and the blue screen of death, has been decapitated in effigy in "I Hate Gates" Web sites and hit with a pie in the face. As for Norman Borlaug . . . who the heck is Norman Borlaug? Yet a deeper look might lead you to rethink your answers. Borlaug, father of the "Green Revolution" that used agricultural science to reduce world hunger, has been credited with saving a billion lives, more than anyone else in history. Gates, in deciding what to do with his fortune, crunched the numbers and determined that he could alleviate the most misery by fighting everyday scourges in the developing world like malaria, diarrhea and parasites. Mother Teresa, for her part, extolled the virtue of suffering and ran her well-financed missions accordingly: their sick patrons were offered plenty of prayer but harsh conditions, few analgesics and dangerously primitive medical care. It's not hard to see why the moral reputations of this trio should be so out of line with the good they have done. Mother Teresa was the very embodiment of saintliness: white-clad, sad-eyed, ascetic and often photographed with the wretched of the earth. Gates is a nerd's nerd and the world's richest man, as likely to enter heaven as the proverbial camel squeezing through the needle's eye. And Borlaug, now 93, is an agronomist who has spent his life in labs and nonprofits, seldom walking onto the media stage, and hence into our consciousness, at all. "

"Andrew Pole had just started working as a statistician for Target in 2002, when two colleagues from the marketing department stopped by his desk to ask an odd question: "If we wanted to figure out if a customer is pregnant, even if she didn't want us to know, can you do that? " Multimedia How to Break the Cookie Habit This article is adapted from "The Power of Habit: Why We Do What We Do in Life and Business," which will be published on Feb. 28. More in the Magazine » Readers' Comments Share your thoughts. Post a Comment » Read All Comments (35) » Pole has a master's degree in statistics and another in economics, and has been obsessed with the intersection of data and human behavior most of his life. His parents were teachers in North Dakota, and while other kids were going to 4-H, Pole was doing algebra and writing computer programs. "The stereotype of a math nerd is true," he told me when I spoke with him last year. "I kind of like going out and evangelizing analytics." As the marketers explained to Pole - and as Pole later explained to me, back when we were still speaking and before Target told him to stop - new parents are a retailer's holy grail. Most shoppers don't buy everything they need at one store. Instead, they buy groceries at the grocery store and toys at the toy store, and they visit Target only when they need certain items they associate with Target - cleaning supplies, say, or new socks or a six-month supply of toilet paper. But Target sells everything from milk to stuffed animals to lawn furniture to electronics, so one of the company's primary goals is convincing customers that the only store they need is Target. But it's a tough message to get across, even with the most ingenious ad campaigns, because once consumers' shopping habits are ingrained, it's incredibly difficult to change them. There are, however, some brief periods in a person's life when old routines fall apart and

"Any love-hate relationship must have its share of pain, so the academic world, in its obsession with college rankings, is suitably dismayed by news that an elite college, Claremont McKenna, fudged its numbers in an apparent bid to climb the charts. Claremont McKenna in California is the latest but not the only college to have admitted submitting false information in an effort to win a high rating. Dismayed, but not quite surprised. In fact, several colleges in recent years have been caught gaming the system - in particular, the avidly watched U.S. News & World Report rankings - by twisting the meanings of rules, cherry-picking data or just lying. In one recent example, Iona College in New Rochelle, north of New York City, acknowledged last fall that its employees had lied for years not only about test scores, but also about graduation rates, freshman retention, student-faculty ratio, acceptance rates and alumni giving. Other institutions have found ways to manipulate the data without outright dishonesty. In 2008, Baylor University offered financial rewards to admitted students to retake the SAT in hopes of increasing its average score. Admissions directors say that some colleges delay admission of low-scoring students until January, excluding them from averages for the class admitted in September, while other colleges seek more applications to report a lower percentage of students accepted. Claremont McKenna, according to Robert Morse, the director of data research at U.S. News, is "the highest-ranking school to have to go through this publicly and have to admit to misreporting." This year, U.S. News rated it as the nation's ninth-best liberal arts college. There is no reason to think the U.S. News rankings are rife with misinformation, and the publication makes efforts to police the data, adjust its metrics and close loopholes. But repeated revelations of manipulation show the importance of the rankings in the minds of prospective students, thei

To illustrate what a growing number of literary scholars consider the most exciting area of new research, Lisa Zunshine, a professor of English at the University of Kentucky, refers to an episode from the TV series "Friends." (Follow closely now; this is about the science of English.) Phoebe and Rachel plot to play a joke on Monica and Chandler after they learn the two are secretly dating. The couple discover the prank and try to turn the tables, but Phoebe realizes this turnabout and once again tries to outwit them. As Phoebe tells Rachel, "They don't know that we know they know we know." This layered process of figuring out what someone else is thinking - of mind reading - is both a common literary device and an essential survival skill. Why human beings are equipped with this capacity and what particular brain functions enable them to do it are questions that have occupied primarily cognitive psychologists.

"Now, as the United States and other Western nations struggle to recover from a debt and property bubble of their own, a growing number of economists are pointing to Japan as a dark vision of the future. Even as the Federal Reserve chairman, Ben S. Bernanke, prepares a fresh round of unconventional measures to stimulate the economy, there are growing fears that the United States and many European economies could face a prolonged period of slow growth or even, in the worst case, deflation, something not seen on a sustained basis outside Japan since the Great Depression. "

The TOK question here is can you compare Japan of the 1990s with the US and Western Europe of today? To what degree can economists, or any human scientists, suggest that one complex situation is "like" another one?

"Melvyn Bragg and guests John Barrow, Colva Roney-Dougal and Marcus du Sautoy explore the unintended consequences of mathematical discoveries, from the computer to online encryption, to alternating current and predicting the path of asteroids. In his book The Mathematician's Apology (1941), the Cambridge mathematician GH Hardy expressed his reverence for pure maths, and celebrated its uselessness in the real world. Yet one of the branches of pure mathematics in which Hardy excelled was number theory, and it was this field which played a major role in the work of his younger colleague, Alan Turing, as he worked first to crack Nazi codes at Bletchley Park and then on one of the first computers. Melvyn Bragg and guests explore the many surprising and completely unintended uses to which mathematical discoveries have been put."

"The internet, an Infographic. Numbers look so much bigger when they're in color, so please take a moment to comprehend the vast extent to which the internet has grown in the short space of a decade."

"By Monday, Four Loko, the alcohol-and-caffeine-laced energy drink, is scheduled to be removed from store shelves nationwide, following a ruling last month by the Food and Drug Administration that the safety of such beverages is unproven and that they should no longer be manufactured or sold. During the resulting media coverage, surprisingly little attention was focused on a corollary topic. What about nonalcoholic energy drinks, which will remain on sale? Are they safe? Effective? Who should be drinking them? Who shouldn't? With excellent timing, a number of new scientific studies and reviews have just been published that address those and related questions about energy drinks, particularly for athletes. Their findings and conclusions are thought-provoking."

December 2005 Over the millennia, many mathematicians have hoped that mathematics would one day produce a Theory of Everything (TOE); a finite set of axioms and rules from which every mathematical truth could be derived. But in 1931 this hope received a serious blow: Kurt Gödel published his famous Incompleteness Theorem, which states that in every mathematical theory, no matter how extensive, there will always be statements which can't be proven to be true or false. Gregory Chaitin has been fascinated by this theorem ever since he was a child, and now, in time for the centenary of Gödel's birth in 2006, he has published his own book, called Meta Math! on the subject (you can read a review in this issue of Plus). It describes his journey, which, from the work of Gödel via that of Leibniz and Turing, led him to the number Omega, which is so complex that no mathematical theory can ever describe it. In this article he explains what Omega is all about, why maths can have no Theory of Everything, and what this means for mathematicians."

"For most of its history, psychology had concerned itself with all that ails the human mind: anxiety, depression, neurosis, obsessions, paranoia, delusions. The goal of practitioners was to bring patients from a negative, ailing state to a neutral normal, or, as University of Pennsylvania psychologist Martin Seligman puts it, "from a minus five to a zero." It was Seligman who had summoned the others to Akumal that New Year's Day in 1998-his first day as president of the American Psychological Association (A.P.A.)-to share a vision of a new goal for psychology. "I realized that my profession was half-baked. It wasn't enough for us to nullify disabling conditions and get to zero. We needed to ask, What are the enabling conditions that make human beings flourish? How do we get from zero to plus five?" Every incoming A.P.A. president is asked to choose a theme for his or her yearlong term in office. Seligman was thinking big. He wanted to persuade substantial numbers in the profession to explore the region north of zero, to look at what actively made people feel fulfilled, engaged and meaningfully happy. Mental health, he reasoned, should be more than the absence of mental illness. It should be something akin to a vibrant and muscular fitness of the human mind and spirit. Over the decades, a few psychological researchers had ventured out of the dark realm of mental illness into the sunny land of the mentally hale and hearty. Some of Seligman's own research, for instance, had focused on optimism, a trait shown to be associated with good physical health, less depression and mental illness, longer life and, yes, greater happiness. Perhaps the most eager explorer of this terrain was University of Illinois psychologist Edward Diener, a.k.a. Dr. Happiness. For more than two decades, basically ever since he got tenure and could risk entering an unfashionable field, Diener had been examining what does and does not make people feel satisfied with life. Seligman's goal was

"When reports emerged 30 years ago that young gay men were suffering from rare forms of pneumonia and cancer, public health investigators scrambled to understand what appeared to be a deadly immune disorder: What were the symptoms? Who was most susceptible? What kinds of infections were markers of the disease? They were seeking the epidemiologist's most essential tool - an accurate case definition, a set of criteria that simultaneously include people with the illness and exclude those without it. With AIDS, investigators soon recognized that injection-drug users, hemophiliacs and other demographic groups were also at risk, and the case definition evolved over time to incorporate lab evidence of immune dysfunction and other refinements based on scientific advances. "If you recognize something is happening, you need a case definition so you can count it," said Andrew Moss, an emeritus professor of epidemiology at the University of California, San Francisco, and an early AIDS investigator. "You need to know whether the numbers are going up or down, or whether treatment and prevention work. And if you have a bad case definition, then it's very difficult to figure out what's going on." Once a disease can be diagnosed reliably through lab tests, creating an accurate case definition becomes easier. But when an ailment has no known cause and its symptoms are subjective - as with chronic fatigue syndrome, fibromyalgia and other diseases whose characteristics and even existence have been contested - competing case definitions are almost inevitable. Now a new study of chronic fatigue syndrome has highlighted how competing case definitions can lead to an epidemiologic "Rashomon" - what you see depends on who's doing the looking - and has stoked a fierce debate among researchers and patient advocates on both sides of the Atlantic. "