Computer Science Back cover copy "I like this book very much. This amazing oversight by past authors is presumably the result of the topic requiring an author with a pretty sophisticated mathematical personality. Havil clearly has that. Many instructors will surely find the book attractive.

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Blaise Pascal It is tempting to think that there are just three special mathematical constants: [pi], e and i. In fact there are many, each with its own definition, each originating in some natural way in its own area of mathematics, each given a special symbol and a name too.

They need symbols to represent them because they are awkward; that is, they have no convenient, finite numeric representation and no patterned infinite one: the ratio of the circumference to the diameter of any circle is not 3. Compared with these, writing i for [square root of -1] is a small convenience.

Its value is the unprepossessing 0. The mid s brought with it the hand-held, microchip-centred, battery- powered, comparatively cheap calculator, thereby bringing to an end the role of logarithms and the slide rule as calculative aids.

Yet the appearance of them in a piece of mathematics is seldom a cause for surprise. Anyone who has studied calculus would see them materialize time and again, quite probably in the expression for the integral of some function or in their role as the inverse of the exponential function, with e vying with p for constant supremacy.

They can also arise without warning in situations that seem remote from their influence, and when they do so they exercise a surprising control in unexpected places-as we shall see: we will also see that the harmonic series, and others related to it, enjoy an important existence of their own. We start by looking at the peculiar way in which logarithms were initially defined, a way which reveals the immense intellectual effort that must have been invested to turn multiplication into addition, to utilize an idea from the old world that helped to usher in the new.

The harmonic series, with its three peculiar properties, is discussed and then its specializations and generalizations, before looking more closely at that definition of [gamma] and having done that, and having convinced ourselves that the number actually exists, at ways of approximating its value, using both decimal and fractional methods. The finale is really just another application of logarithms, but since the application is the Prime Number Theorem, leading to the Riemann Hypothesis neither of which we prove!

How difficult is the mathematics? That of course is a subjective matter. Certainly, we have not shied away from the use of symbols, since to do so would have condemned us merely to talking about mathematics rather than actually doing it. Yet, there are few really advanced techniques used, it is more that in some places simple ideas have been used in advanced ways.

In these terms we think the content is often elementary but in places not so very simple. The reader should expect to make use of a pen and paper in many places; mathematics is not a spectator sport! The approach is reasonably rigorous but informal, as this is no textbook, it is more a context book of mathematics in which the reader is asked to take time out from studying the mathematics to read a little around it and about the mathematicians who produced it or of the times in which they lived; sometimes in detail but other times just a few lines and then not always, as this is no history of mathematics book either; it merely acknowledges that mathematics comes from mathematicians, not books, and seeks to bring a sometimes shadowy figure forward to share the prominence of his ideas, and to give some sort of feel for the way in which those ideas developed over time.

We hope that the material will appeal to a variety of people who have a little probability and statistics and a good calculus course behind them, and before that a rigorous course in algebra, if such a thing still exists: the motivated senior secondary student, who may well be seeing many of the ideas for the first time, the college student for whom the text may put flesh on what can sometimes be dry bones, the teacher for whom it might be a convenient synthesis of some nice ideas and maybe the makings of a talk or two , and also those who may have left mathematics behind and who wish to remind themselves why they used to find it so fascinating.

The reader will judge to what extent this book achieves its aim: to explain interesting mathematics interestingly. The names of many mathematicians appear, names that should bring wonder to anyone interested in the subject and its history, but it is that name Euler that will force itself onto the page more than any other.

It is not that we happen to pass through the mathematical territory to which he holds title, but more that it would be difficult, if not impossible, to go far in any mathematical direction without feeling his influence.

For example, much of the notation that we now take for granted originates from him; in particular, e, i, f x , [summation], [DELTA], sin x, cos x, etc. It can be hard to appreciate, or easy to forget, just how many important ideas his name is associated with or perhaps even attached to; he invented many vastly important concepts and touched every known area of the subject-and everything he touched he adorned.

According to R. Extravagant use of the word serves only to dilute its meaning or to bring into question the judgement of the author, but we have used it already and will risk employing it on a number of other occasions, no more fittingly than with Euler, safe in the conviction that if he was not a genius and these people were not geniuses then none have yet been born.

Yet, to the majority, his name is probably as mysterious as his constant. He breathed life into [gamma] through his Zeta functions the generalizations of [H.

We hope that the reader will understand if the story is not always complete, and agree that where it is not complete it is at least representative. We hope that the reader will share our enthusiasm as we take brief excursions though countries, centuries, lives and works, unfolding the stories of some remarkable mathematics from some remarkable mathematicians.

Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher. Excerpts are provided by Dial-A-Book Inc.

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