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Paul Nahin

An imaginary tale : the story of the square root of -1

Imaginary and Complex numbers have the reputation of being difficult. Maybe it's their name - calling them 'Bombelli numbers' might not sound so bad. This work takes a non-textbook approach to the subject, but if you find complex numbers scary then I wouldn't necessarily recommend it to you - there are quite a lot of equations. On the other hand the mathematics isn't particularly difficult. I would say that it is aimed at the keen high-school students, who will get a foretaste of more advanced mathematics. Those with a little more mathematical knowledge should enjoy it as a bit of light reading.

The book starts with the history of complex numbers, which were accepted as legitimate mathematics by Rafael Bombelli in the 16th century. We also find out about Euler's famous equation e^iπ+1 = 0. Nahin goes on to look at practical uses complex numbers, for example in electrical engineering. This is followed by a selection of algebraic tricks and techniques where such numbers are useful, including a discussion of the Riemann Zeta function. The final chapter gives an introduction to complex function theory. If you would like to get a taste of these subjects, but don't want to study them in detail then I would say that this is the book for you.

Amazon.com info

Hardcover 257 pages
ISBN: 0691027951
Salesrank: 540269
Weight:1.2 lbs

Published: 1998 Princeton University Press

Amazon price $31.96

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Amazon.co.uk info

Hardcover 257 pages
ISBN: 0691027951
Salesrank: 296012
Weight:1.2 lbs

Published: 1998 Princeton University Press

Amazon price £18.49

Marketplace:New from £13.85:Used from £10.69

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Amazon.ca info

Hardcover 257 pages
ISBN: 0691027951
Salesrank: 213357
Weight:1.2 lbs

Published: 1998 Princeton University Press

Amazon price CDN$ 44.42

Marketplace:New from CDN$ 39.71:Used from CDN$ 17.05

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Product Description
Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.

Product Description

Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.

In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.

Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.

First-Rate Introduction to complex numbers *****
This is a first-rate introduction to complex numbers. Little background is required to start reading, though a decent high-school background will help. If you ever wondered what complex numbers were for, if you have forgotten why the number i was created, if you want to learn the art of equation solving, if you are curious about Gauss and Euler, this is the book. I've read several other books by Paul Nahin, and this one is among his finest.

First-Rate Introduction to complex numbers *****

This is a first-rate introduction to complex numbers. Little background is required to start reading, though a decent high-school background will help.

If you ever wondered what complex numbers were for, if you have forgotten why the number i was created, if you want to learn the art of equation solving, if you are curious about Gauss and Euler, this is the book.

I've read several other books by Paul Nahin, and this one is among his finest.

Perfect enrichment for math students & teachers *****
Teaching mathematics is often an uphill battle against the forces of abstraction and dullness. This delightful book is a perfect antidote, weaving as it does the history, applications and actual mathematics surrounding the concept of "imaginary" and "complex" numbers. But don't get the wrong expectation -- it's a real math book, with equations, proofs, etc, varying in level from high-school algebra and geometry to college calculus and physics. I myself bought it in a search for material to motivate a bright 11-year-old that I am tutoring. I introduced imaginary and complex numbers to him, but all of the actual applications seemed far out of his reach. So now when I mention imaginary numbers he screws up his face and asks for more boolean algebra instead. But with this book, I now have a number of examples and historical anecdotes to motivate and fascinate him, particularly geometric interpretations and applications. Here, for example, is one extremely elementary application that I did not know about. Prove: the product of two sums of squares is itself the sum of two squares in two different ways. Symbolically, given any integers a, b, c, d, there are integers p, q, r, s with... (a^2 + b^2)(c^2 + d^2) = p^2 + q^2 = r^2 + s^2 This was demonstrated by mathematicians a long time ago, but not particularly easily. Using complex numbers, it's almost trivial to see, however, certainly within reach of a student of Algebra I. (There's an even simpler version of the proof that Nahin presents, but it's a bit messy to write without properly typeset mathematics.) This also makes the important point that complex numbers are very useful to help understand non-complex mathematical phenomena, a point Nahin makes throughout the book. This also illustrates that this is a real math book, not simply a popularization piece ~about~ mathematics and mathematicians. It's really too bad that reviewers who expected the latter are downgrading their ratings of the book, because if you understand and accept what it is trying to be, it's a gem! Much of this material is, of course, available by searching the internet. But it's not easy to find, and of highly variable quality. So Nahin's book is a real service to teachers and students at all levels.

Perfect enrichment for math students & teachers *****

Teaching mathematics is often an uphill battle against the forces of abstraction and dullness. This delightful book is a perfect antidote, weaving as it does the history, applications and actual mathematics surrounding the concept of "imaginary" and "complex" numbers. But don't get the wrong expectation -- it's a real math book, with equations, proofs, etc, varying in level from high-school algebra and geometry to college calculus and physics.

I myself bought it in a search for material to motivate a bright 11-year-old that I am tutoring. I introduced imaginary and complex numbers to him, but all of the actual applications seemed far out of his reach. So now when I mention imaginary numbers he screws up his face and asks for more boolean algebra instead. But with this book, I now have a number of examples and historical anecdotes to motivate and fascinate him, particularly geometric interpretations and applications.

Here, for example, is one extremely elementary application that I did not know about. Prove: the product of two sums of squares is itself the sum of two squares in two different ways. Symbolically, given any integers a, b, c, d, there are integers p, q, r, s with...

(a^2 + b^2)(c^2 + d^2) = p^2 + q^2 = r^2 + s^2

This was demonstrated by mathematicians a long time ago, but not particularly easily. Using complex numbers, it's almost trivial to see, however, certainly within reach of a student of Algebra I. (There's an even simpler version of the proof that Nahin presents, but it's a bit messy to write without properly typeset mathematics.) This also makes the important point that complex numbers are very useful to help understand non-complex mathematical phenomena, a point Nahin makes throughout the book.

This also illustrates that this is a real math book, not simply a popularization piece ~about~ mathematics and mathematicians. It's really too bad that reviewers who expected the latter are downgrading their ratings of the book, because if you understand and accept what it is trying to be, it's a gem!

Much of this material is, of course, available by searching the internet. But it's not easy to find, and of highly variable quality. So Nahin's book is a real service to teachers and students at all levels.

Wonderful book -- very highly recommend it *****
A fantastic resource for anyone who has an inclination to learn math with history of how it really developed. I truly felt sorry that I didn't have this book when I was learning Trignometry in high school -- would have used De Moivre's theorem to derive the interesting identities without having to resort to painful coordinate geometry proofs. ps: this book is not bedtime reading

Wonderful book -- very highly recommend it *****

A fantastic resource for anyone who has an inclination to learn math with history of how it really developed. I truly felt sorry that I didn't have this book when I was learning Trignometry in high school -- would have used De Moivre's theorem to derive the interesting identities without having to resort to painful coordinate geometry proofs.

ps: this book is not bedtime reading

An Imaginary Tale: The Story of "i" [the square root of minus one] *****
A good readable review of i. Begins with a discussion of Cardan(o)'s solution of the cubic equation with its unexpected side effects, and ends with contour integration.

An Imaginary Tale: The Story of "i" [the square root of minus one] *****

A good readable review of i. Begins with a discussion of Cardan(o)'s solution of the cubic equation with its unexpected side effects, and ends with contour integration.

calculus required **
I thought this book would touch on the philosophical implications of the imaginary number. I was quite surprised to see the calculus level equations on pretty much every page of the book. If you are an accomplished math genius and want to know about the history of imaginary numbers, then this might be a great book. However, if you are a mere mortal and looking for an interesting read about math, this one might be a bit much.

calculus required **

I thought this book would touch on the philosophical implications of the imaginary number. I was quite surprised to see the calculus level equations on pretty much every page of the book. If you are an accomplished math genius and want to know about the history of imaginary numbers, then this might be a great book. However, if you are a mere mortal and looking for an interesting read about math, this one might be a bit much.

Good, but you have to work at reading it ****
If I had never read any of Eli Maor's excellent books I would have scored this book as 5 stars. It is a very good book that guides you through a series of difficult mathematical concepts without being a textbook. It is very readable, but it is peppered with 'roadblocks' where you suddenly have to pay a lot more attention, and possibly re-read sections, before you can proceed. It also, despite being a new 'bugs removed' edition, has at least one grammatical error which makes a paragraph hard to follow. Having said all that, it really is a very good book. It is just that I have been spoiled by Eli Maor's books, which cover similar ground (trigonometry, e) in a similar way (history, characters, mathematical ideas, related concepts), but manage to make it an effortless joy for the reader. This book somehow never became a joy to read.

Good, but you have to work at reading it ****

If I had never read any of Eli Maor's excellent books I would have scored this book as 5 stars. It is a very good book that guides you through a series of difficult mathematical concepts without being a textbook. It is very readable, but it is peppered with 'roadblocks' where you suddenly have to pay a lot more attention, and possibly re-read sections, before you can proceed. It also, despite being a new 'bugs removed' edition, has at least one grammatical error which makes a paragraph hard to follow.

Having said all that, it really is a very good book. It is just that I have been spoiled by Eli Maor's books, which cover similar ground (trigonometry, e) in a similar way (history, characters, mathematical ideas, related concepts), but manage to make it an effortless joy for the reader. This book somehow never became a joy to read.

Disappointing presentation of the material **
I read this book on the back of having just finished Eli Maor's excellent "To infinity and beyond". Unlike Maor's book, "An imaginary tale" is poorly written and presented. While Maor has a fluid and engrossing writing style, Nahin is much less convincing. The material is all there, but it's the presentation with which I have a problem. It's not all bad -- the chapter on the geometry of i is well done, for example, but that's the exception rather than the rule. Another problem is the poor quality of the diagrams. Cubic curves are hastily drawn freehand. Right angled triangles don't always have right angles, and so on. On the whole, I came away with an impression of a book with lots of potential, but most of it left unrealised.

Disappointing presentation of the material **

I read this book on the back of having just finished Eli Maor's excellent "To infinity and beyond". Unlike Maor's book, "An imaginary tale" is poorly written and presented. While Maor has a fluid and engrossing writing style, Nahin is much less convincing. The material is all there, but it's the presentation with which I have a problem. It's not all bad -- the chapter on the geometry of i is well done, for example, but that's the exception rather than the rule. Another problem is the poor quality of the diagrams. Cubic curves are hastily drawn freehand. Right angled triangles don't always have right angles, and so on. On the whole, I came away with an impression of a book with lots of potential, but most of it left unrealised.

Eulogy *****
I rate this book as one of the three best general mathematical books that I have ever bought. Its style is clear and light and the scope of the mathematics is breathtaking; I learnt a great deal from it and saw explained some hard ideas in a very readable way. Not every question is answered but as the author says it isn't a text book. If you want to get into complex analysis and learn about its development and the geniuses who have been involved in it I can think of no better path to take-but you will need to work at some bits! The author avoids actually defining complex numbers in a rigorous way and I would have liked to have seen them defined somewhere as ordered pairs of reals with a reasonable definition of addition and a funny definition of multiplication, with i simply a change of notation. Not easy to fit into the historical development but worth an appendix. Buy the book. If you don't like it I reckon the problem's with you!

Eulogy *****

I rate this book as one of the three best general mathematical books that I have ever bought. Its style is clear and light and the scope of the mathematics is breathtaking; I learnt a great deal from it and saw explained some hard ideas in a very readable way. Not every question is answered but as the author says it isn't a text book. If you want to get into complex analysis and learn about its development and the geniuses who have been involved in it I can think of no better path to take-but you will need to work at some bits! The author avoids actually defining complex numbers in a rigorous way and I would have liked to have seen them defined somewhere as ordered pairs of reals with a reasonable definition of addition and a funny definition of multiplication, with i simply a change of notation. Not easy to fit into the historical development but worth an appendix.

Buy the book. If you don't like it I reckon the problem's with you!

Fantastic! Thorough, scholarly, interesting! *****
This is an excellent, beautiful book! Just the section on Kepler's laws is worth the price of the book (hardcover to boot!) If you like math, if you are willing to spend a bit of time understanding the wonderful results -- get it! Some calculus background needed -- nothing beyond high school. The book goes well beyond providing a narrative on the history of "square root of -1". It actually shows in complete detail how to use "i" to do wonderful things. Along the way the author provides the important historical events and plenty of notes and references for anyone interested in getting some more. It is clear the author took his time to research and study the subject. He has presented it well, thouroghly, and in an interesting way -- without sacrificing detail!

Fantastic! Thorough, scholarly, interesting! *****

This is an excellent, beautiful book! Just the section on Kepler's laws is worth the price of the book (hardcover to boot!)

If you like math, if you are willing to spend a bit of time understanding the wonderful results -- get it! Some calculus background needed -- nothing beyond high school.

The book goes well beyond providing a narrative on the history of "square root of -1". It actually shows in complete detail how to use "i" to do wonderful things. Along the way the author provides the important historical events and plenty of notes and references for anyone interested in getting some more. It is clear the author took his time to research and study the subject. He has presented it well, thouroghly, and in an interesting way -- without sacrificing detail!

Lots of fun if you enjoy math *****
Nahin's book requires some effort to work through, but it is well worth the time to discover, or rediscover, the beauty of complex numbers. The book is understandable for undergraduates with a math background. I look forward to reading more of Nahin's books.

Lots of fun if you enjoy math *****

Nahin's book requires some effort to work through, but it is well worth the time to discover, or rediscover, the beauty of complex numbers. The book is understandable for undergraduates with a math background. I look forward to reading more of Nahin's books.

Spectacular Failure *
Nahin states that only high school mathematics is required to read his book. That may be true, but the book is still a disaster. Nothing more than page upon page of mind-numbingly boring equations; it's worse than a textbook. There is no attempt to actually tell a coherent story, unlike many other mathematics books I've read. Unless you are already a master at mathematics or you suffer desperately from insomnia, don't even consider buying this book. Hopefully someone who actually knows how to write will tell the story of complex numbers in an insightful, interesting way. Paul Nahin is not that person.

Spectacular Failure *

Nahin states that only high school mathematics is required to read his book. That may be true, but the book is still a disaster. Nothing more than page upon page of mind-numbingly boring equations; it's worse than a textbook. There is no attempt to actually tell a coherent story, unlike many other mathematics books I've read.

Unless you are already a master at mathematics or you suffer desperately from insomnia, don't even consider buying this book. Hopefully someone who actually knows how to write will tell the story of complex numbers in an insightful, interesting way. Paul Nahin is not that person.

A fun read of not only math but history ****
this book is just really intresting because brings math to life with the history of the subject of root negitive one.

A fun read of not only math but history ****

this book is just really intresting because brings math to life with the history of the subject of root negitive one.

Wish more books like this *****
Inspiring! Explaining the true physical meaning of an imaginary real quantity and showing its real imaginary applications.

Wish more books like this *****

Inspiring!
Explaining the true physical meaning of an imaginary real quantity and showing its real imaginary applications.

somewhat dense and no problems to solve by the reader ***
This book is well written, but, it does feel like the venerable professor took his lecture notes and strung them together, but dear me, he left out problems for the reader; this to me is a cardinal sin when it comes to expository math. Maybe the professor could create a website with problems + solutions related to the subject matter - give us puzzle people a chance at solving at least a few problems on our own.

somewhat dense and no problems to solve by the reader ***

This book is well written, but, it does feel like the venerable professor took his lecture notes and strung them together, but dear me, he left out problems for the reader; this to me is a cardinal sin when it comes to expository math.

Maybe the professor could create a website with problems + solutions related to the subject matter - give us puzzle people a chance at solving at least a few problems on our own.