John Allen Paulos is professor of mathematics at Temple University in the US and author of several books and papers on numeracy and probability. But even he, a prime candidate for the description "rational man", was swept along by the stock market frenzy of the late 1990s.

A tip from friends persuaded him to invest in the shares of WorldCom, the telecommunications company. Worse still, as the share price plummeted and the company headed for collapse he bought further shares, some on margin, and even speculated on a WorldCom revival through the options market.

During this disastrous foray, he displayed many of the traits that behavioural finance theorists have identified in investors. He showed "loss aversion", an unwillingness to sell his position and realise his loss; "anchoring", the fixation with a price target that the shares would (he hoped) regain; and "confirmation bias", a willingness to believe all favourable comments about WorldCom and to ignore all negative evidence.

At one level, therefore, A Mathematician Plays the Market is a useful warning for all those amateurs who believe they can trade and beat the professionals from their own living rooms. The stock market has made fools of some very smart men.

But Paulos also uses his mathematical expertise to dissect the way the market operates and the potential traps for those who do not understand the numbers.

Take, for example, the difference between the average and the expected return. Imagine a world in which half of all new flotations rise 80 per cent in the first week and the other half lose 60 per cent. The plucky investor decides to buy every issue and sell after a week, for what he calculates is an average gain of 10 per cent a week.

Indeed, Paulos suggests the average worth of an initial $10,000 stake using this strategy would be $1.4m. The problem is that the most likely outcome (experienced by half of investors) would be a sum of less than $1.95.

This discrepancy is because a few investors would experience massive gains, pushing up the arithmetic average, but most investors would be severely disappointed. This problem helps explain why many investors experience "below average" performance.

Paulos proceeds to skewer many popular investor and business fads. He robustly dismisses technical analysis, the study of chart patterns to predict market movements. "The quasi-mathematical jargon of technical analysis seldom hangs together as a coherent theory. One of its less plausible manifestations [is] the so-called Elliott wave theory" whose followers believe that all market movements, ranging from intra-day fluctuations to patterns spanning centuries, are in waves of five and three.

And of the "Six Sigma" approach to management, made popular by General Electric, which aims to achieve results six standard deviations above the norm, Paulos points out that "a Six Sigma performance is so unlikely that the tables in most statistics texts don't even include values for it".

But the fundamental problem in analysing markets is that humans are involved. Unlike the study of physics, our psychology can affect the results. Take, for example, the efficient market theory, which states that prices already reflect all available information, and thus it is no use trying to beat the market by using publicly available data.

Paulos points out that if most people believed the theory, they would not bother to comb the market for new information. And if that were the case, the market would not react quickly to reflect new information in prices. It would thus become inefficient.

In other words, the efficient market theory can be true only if most investors believe it to be false.

Paulos provides much in this book that is thought-provoking and informative, about both mathematics and the stock market. Those hoping a maths professor might have discovered a way of beating the market may be disappointed by the end of this book. But if they read carefully, they will understand why it is an impossible dream.