An Imaginary Tale: The Story of √-1

by Paul J. Nahin

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.

Genres: Mathematics, Science, Nonfiction, History, Stem, Physics, Philosophy

296 pages, Paperback

First published January 1, 1998

About the author

Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire and the author of many best-selling popular math books, including The Logician and the Engineer and Will You Be Alive 10 Years from Now? (both Princeton).

Reviews

[WarpDrive · Rating 5 out of 5]

This is a great book about one of the most beautiful and fascinating subjects of maths: the world of imaginary/complex numbers and of complex analysis.

This is a book unapologetically mathematical in contents and approach, a real math book with real math, sometimes dense and sometimes beyond the “freshman calculus” level, but it is never too difficult if you have the patience and perseverance to go through the many fascinating and intriguing examples and theorems. And if you persevere, the intellectual rewards to the reader are well worth the effort.

The author manages to effectively convey, with concrete and fascinating examples, the intellectual adventure of discovery and unveiling of the beautiful world of complex numbers.
Several very intriguing results are demonstrated by the author throughout the book, many of which could not be possibly be demonstrated without the usage of imaginary numbers. The author, in other words, brilliantly and consistently demonstrates what the mathematician Hadamard once said: "The shortest path between two truths in the real domain passes through the complex domain".

The book presents fascinating examples about the physical signature for the complex roots in the plot of quadratic and cubic functions. The relationship between complex numbers and geometry, and physical solutions to real problems, is also treated quite well.
The author then effectively demonstrates the sheer beauty and simplicity of the geometric interpretation of the complex numbers and of the corresponding definition of "i" as the rotation operator; he also shows how such interpretation, supported by the famous Le Moivre theorem, can be used to generate countless trig identities.

Other beautiful results and identities are also very nicely explained by the author: examples are the famous Euler's identity, the result that "i" to the "i" gives you real numbers, the quite astonishing result that 1^pi (number "1" to the power of pi) - which after all is just a real number by a real power - has an infinity of distinct complex values.

There is also a quite interesting chapter about the utilization of complex numbers in areas of physics such as special relativity, the derivation of Kepler's first law of elliptical orbit (and his other laws) from Newton's physics, and in electrical engineering problems. Disappointingly (and surprisingly) the fundamental character of complex numbers in quantum mechanics is not treated by the author. I was also disappointed by the fact that fractals are not treated in this book either.

In chapter 6 (aptly titled "wizard mathematics"), things get mathematically serious: the book gets into more intense but also very intriguing mathematical territory. This is very rewarding albeit somewhat slow read. The Euler's constant and the zeta function are explained in a nice and clear manner. Some of Euler's derivations, so beautifully presented by the author, can be clearly seen as the product of pure, raw genius. And the beauty of higher mathematics can be seen in its power, when we start digging into things such as the derivation of the value of pi from i, the Fresnel integrals, gamma functions extended to complex values and their relationship to the zeta functions, the Riemann hypothesis etc.
These are all beautiful derivations and examples almost perfectly executed by the author, with only the very occasional minor typo or missing step or partial demonstration (for example, only a trivial case of the reflection formula is actually proved), and occasionally peculiar notational choices. However it is not a textbook and it does not pretend to be a textbook, so the occasional lack of mathematical rigor is totally forgivable, in my opinion.

We finally get, in the last chapter, to the dessert of this rich and rewarding intellectual buffet: complex function theory.
I strongly agree with the author when he states: "it wasn't until my first course in complex function theory that I experienced a totally new emotion - the pure pleasure of learning mathematics that was, in and of itself, pretty."
The analysis of complex functions requires by itself at least a separate book, but the author does an egregious job in conveying the beauty of some fundamental and fascinating results within the space of the last chapter of his book. He focuses on Cauchy's first and second integral theorems. The latter theorem is particularly beautiful: the intimate connection between the value of a complex analytical function f(z) at an internal point inside a region delimited by C, and its contour integral on C, is another illustration of the very special nature of complex functions. The way the author explain Cauchy's contour integrals is just great.

It is clear from the book that the author loves mathematics, appreciates its sheer beauty, and simply loves showing off beautiful equations, graphical tricks, awesome solutions and great intellectual challenges bringing out counter-intuitive and astonishing results.

Overall, this is a hugely rewarding book, highly recommended to anybody who loves mathematics. A joy to read. 4.5 stars, rounded up to 5 stars given the fascinating subject of this book and the energy and enthusiasm of the author.

[Ghada · Rating 5 out of 5]

I've been bored with reading novels lately, so I was looking for something a bit more inspiring and challenging. This book really hit the spot!

I wouldn't call it a non-fiction book per se, but something more of a supplementary book for those interested in digging deeper into a subject. Here the subject under discussion was complex numbers (specifically the imaginary number i).

In the preface, the author claims that no book has ever been written on this subject alone in a non-text book form, so he took it upon himself to do so (he is an electrical engineer and was fascinated with complex numbers while growing up). Well hats off to Eng. Nahin, because he did an amazing job!

The author began with a historical summary of i and the mathematical problems that it surfaced from. He then moved on to some important applications. I really enjoyed chapters 4-6, which contained some problems and uses of complex numbers. There was a chapter titled "Wizard Mathematics"!! I had a field day with that one. The title alone was so exciting!

I loved going through the proof of how according to complex number theory, the shortest distance between two points is NOT a straight line (shortcuts through hyperspace?!?... I know right!), Schellbach's method of using i^i to calculate pi, and more on the genius of Professor Euler. Even the Appendixes were rich with more problems and proofs, it makes you want to savor every page!

I just wish that I had read this book while I was studying complex numbers and Cauchy's Theorems last year. It would've been easier to work out all the derivations while everything was still fresh. I think that the reader would appreciate this book more if she/he had at least some basic knowledge of complex numbers and calculus. It is definitely worth going through the mathematics yourself using pencil and paper, because let me tell you The Story of i is epic!

[Vicki Cline · Rating 3 out of 5]

I was hoping to really like this book, as it involves my favorite equation, Euler's identity,

e^(i * pi) + 1 = 0.

Such an elegant way to connect the five most important constants in math, along with fundamental mathematical operations. Unfortunately, the understanding of the math involved in the book, which I'm sure I used to have 50 years ago when I got my BA in math, has left me. I had to skip over most of the equations in the book (and there are a lot of them), so I don't even know if I can count this book as "read." But what I was able to read was interesting, especially the early history, where the concept of the square root of minus one helped solve otherwise intractable problems, but the men who figured out the methods were so reluctant to believe in it as a number (hence the designation "imaginary").

[OxyMoron · Rating 5 out of 5]

Of the various books about the history and “life and times” of the major mathematical constants, I like this the best. “i” is probably the most difficult of those constants to deal with, given its abstract nature. Nahin well covers the conceptual difficulties associated with the square root of negative one, and how its occurrence in the algorithm for finding the real roots of a generalized cubic equation essentially forced the issue of finding the “meaning of i”. That starting point, followed as it was by the geometry of i struck me as an ideal way to approach its introduction and initial teaching at the high school. For anyone who is curious about i, but does not want to venture too far into the heavier mathematics, then the first three chapters will serve very well.

In respect of the utility of i, Nahin mentions Steinmetz’ 1893 paper, “Complex Quantities and Their Use in Electrical Engineering”. That was one of the first applied uses of i, and Steinmetz facilitated an in-depth analysis of three-phase systems that allowed their general use for distribution and not just transmission. Hitherto polyphase systems had mostly been of the less efficient but much easier to analyse two-phase type. One could say that i, via Steinmetz’ work, is responsible for our three-phase world (recognizing that Ferraris had first described polyphase systems.) "i" deserves a “life and works” treatment, and Nahin has provided just that.

[Matt · Rating 3 out of 5]

Very technical for a popular book, though admittedly most of my reference points are pop science rather than pop maths. (It's certainly no The Road to Reality: A Complete Guide to the Laws of the Universe, but it's far more mathematical than any other pop science book I can think of, and it's much more demanding than, say, Ellenberg's excellent How Not to Be Wrong: The Power of Mathematical Thinking, which is the only pop maths book I can think of off the top of my head.) In any case, I found it heavy going, and had to accept that a fair portion would go over my head unless I was willing to spend a huge amount of time and effort. So I'm not in a position to judge whether it would be enjoyable for readers with the background & intelligence to follow it closely (though I suspect that it would be).

For me it was worthwhile in parts & frustrating in others. Mostly that's simply the result of my own ignorance/laziness/stupidity, but I did sometimes feel that Nahin wasn't quite sure who he was writing for: he would occasionally pause to explain a very basic concept, then in the next breath launch into a torrent of formal mathematics with little in the way of verbal guidance. (Mostly, though, he was clearly aiming at people with a fairly solid mathematical background.) There were some sections that I could have grasped a lot more quickly & easily with just slightly more hand-holding; sometimes a logical leap that would be obvious to a mathematician took me an embarassingly long time to understand. I assume something similar is true of some of the proofs I gave up on following, though others were genuinely too hard for me, and by the final chapter I was doing a lot of skim-reading.

Anyway, I suspect I might have loved this book had I been a bit smarter or better educated. In reality, it was probably worth reading, but only just.

[Jason · Rating 4 out of 5]

Great history and math book. You definitely need a lot of math to fully appreciate this book (if you don't have background up to trig, and preferably calculus, you'll find all but the first couple of chapters impenetrable). If you fit the pre-reqs though, it is very interesting. I found how much my math has degenerated as every now and then I just had to shrug and just move on. (I do look forward to going back and with pencil and paper trying out some of the more hairy calculations.) Now, after all that, I would say even if you don't have a strong math background, the history part is very interesting and some of the simple math concepts as they evolved are keen, so it would be worth your while to crack the book open and read the first couple chapters.

[Jessica · Rating 1 out of 5]

Yeah, yeah, I should know better than to expect much from a book on math, but I've actually read some decent ones. This one was supposed to be on layman's terms, but it was so technical that it might as well have been written in French. Nothing was explained in plain English--it was all equations and made me feel rather stupid. I'm going to be teaching math, after all, but man, it was way over my head. Not that I expect anyone to actually read it, but just in case you were tempted, don't.

[William Schram · Rating 5 out of 5]

Imaginary numbers deal with the square root of negative one. For centuries, people dealt with imaginary numbers by ignoring them or calling them impossible. We can thank the Greeks for this since they had a literal interpretation of numbers and what they represent. I suppose that isn't entirely fair since all civilizations capable of taking roots ignored imaginary numbers, but the Greeks did set a precedent with Geometry.

Author Paul J Nahin discusses the history and applications of imaginary numbers in this book. It contains the equations and ideas that led to the development of the complex number system. Any electrical engineer can tell you that imaginary numbers are anything but imaginary.

The wonderful thing about this book is that there is far more mathematics in it than I was expecting. Trigonometric identities, complex numbers, famous figures, general relativity, and more populate the pages. Nahin covers a lot of imaginary numbers in seven chapters and six appendices.

Thanks for reading my review, and see you next time.

[Alastair · Rating 3 out of 5]

An Imaginary Tale: The Story of the Square Root of Minus One cannot decide whether it wants to be a history book, a textbook or a puzzle book. As a result, it ends up fully being none of them. For those willing to read through some rather tedious sections, and who have a strong stomach for detailed mathematics, this book offers a wealth of information about the mysterious imaginary numbers.

The beginning of the book is superbly written mathematical history. We witness the earliest inklings mathematicians had of imaginary numbers – the square routes of negative numbers – in trying to solve cubic equations. Rafael Bombelli’s work made it clear “that manipulating i using the ordinary rules of arithmetic leads to perfectly correct results” (the roots of the cubic equation).

We step from a pragmatic-arithmetical view of imaginary numbers to a geometric one (what could the geometry of such numbers be?). After a few false starts, such as what even the author admits is an “exercise in splitting hairs” by John Wallis, we arrive at the key move, courtesy of the unknown-to-me Caspar Wessel. Wessel was considering how to make sense of multiplying arrows together. Having found a way to do this, Wessel realised that complex numbers (combinations of the real and imaginary, like 1 + 3i) are arithmetically the same: multiplying them is just like multiplying arrows. In other words complex numbers are the same as arrows in 2D space. Thus we arrive at the central take-home of the whole work. Pure imaginary numbers are rotations: i rotates a real number exactly 90 degrees.

These early chapters are bursting with fascinating historical detail and mathematical insight. The big issue – the book not quite deciding what it wanted to be - first became clear to me in the two subsequent chapters covering applications. The utility of imaginary numbers might well be illustrated here but not much is added theoretically, while the technical knowledge required is a serious obstacle. Those who know about relativity (like me) will zip through that section, while those who don’t know about a particular area (like electrical engineering) may flounder. These sections are decidedly ahistorical and seemed an unnecessary detour from the ‘plot’.

Even in the rest of the book there is a frustrating oscillation between deep ideas and ingenious but overwrought 'virtuoso' displays of calculation. These displays – often begun with a puzzle-book suggestion to ‘try for yourself first’ – certainly offer a lot to get stuck into for those so inclined; I did a few of the calculations myself before realising I’d never finish the book at the rate I was going. Yet, ultimately, seeing pi written in umpteen different ways using incredibly involved calculations gets a little stale. Does anyone really care that pi/4 = 4 arctan(1/5) – arctan(1/239)? Some people will of course care and for them this book will be that bit better, though even as a puzzle book I think a more consistent approach to offering solutions would be an improvement

The book settles back to mathematical history once more in the final chapter, focussing on later developments – notably Cauchy’s powerful integral theorems. This chapter epitomises what the book could have been: mathematically detailed with a good sprinkling of get-your-hands-dirty examples, yet fundamentally focussed on the meaning of the imaginaries and the truly trail blazing discoveries associated with them. That the book didn’t always successfully tread this fine line between detail and the bigger picture is a pity. Yet as perhaps the only focussed history of imaginary numbers, this is still a great read for those with the interest and mathematical acumen to see it through.

[Kyle · Rating 5 out of 5]

Another excellent book by Nahin. This covers the history of using i, or the square root of negative one. It covers the subject without apologies for using mathematics, and I personally enjoy this. Nahin gives explanations of what he is doing, but he does not shy away from just showing steps. The history is also well-done and with one exception, I found it to be comprehensive and well-researched. In other words, it is another book I would recommend if you like math/physics/engineering history.

The exception is the coverage of Copernicus to Kepler. Nahin gives Copernicus a lot of credit for a heliocentric model (and questions Tycho Brahe's model). My experience shows that people that do so have never really investigated or used the Ptolemaic or Copernican models. Kepler's model is the one that I think makes a better case for planets orbiting the sun being a more useful perspective. Other than this, (and Nahin barely comments on this history as it does not have much to do with the history of i), it was quite good.

If you're curious about Copernican vs Ptolemaic models, they are very similar in complexity and so neither is at a computational advantage in my opinion. What the Copernican model has is an explanation for why a particular parameter that relates to an orbital radius increases as we go outward from Mercury to Neptune when you put the sun at the center. [The Ptolemaic model can put the sun at the center from a geometric perspective and the calculations are the same. In addition, there are no preferred frames in physics, so long as you take into account any extra forces, you can use non-inertial frames. So saying heliocentric is better always seemed odd to me. It's just usually simpler to use inertial frames.] In addition, Brahe had strong evidence against the heliocentric model, because they were unaware of optical effects on the apparent size of the stars through the atmosphere, so that if you calculated how big distant stars were, they were hundreds of times larger than our entire solar system, at the smallest. Brahe thought that should count against the heliocentric model, since such gigantic stars would be nothing like our sun. You might not agree with our sun being average, but this is not a crazy argument.

[Stanley Xue · Rating 5 out of 5]

First math book I read that actually had a lot of satisfying math. Only had a high school understanding of complex numbers and calculus beforehand. It was very rewarding to go through this learning experience.
Nahin takes you on a journey through history from the origins of the imaginary number - surprisingly popping up when solving cubic equations, and then to Euler's contributions, case studies in using complex numbers to prove important scientific results such as Kepler's equations, and up to basic complex function theory.
LOTS OF MATHS. Great for people sick of arm-wavy popular science/maths books (and don't have enough energy to wade through a university textbook). Structuring the book through historical discoveries and building up just enough of a picture of who these great mathematicians were (BUT STILL KEEPING A LOT OF MATHS AND CLEAR EXPLANATION TO NOT SOUND TOO WAFFLY) kept me engaged through the whole book.

[Julia Marrinan · Rating 3 out of 5]

2.7/5. Had its moments but I was a little bored. Not my kind of book I guess.

[Lloyd Downey · Rating 4 out of 5]

This was not an easy book to read. But, as maths books go, It was certainly not the most difficult that I’ve attempted. And I did learn, along the way, a lot about how mathematicians approach maths problems.
It is basically a history of the developments in understanding and interpreting the square root of minus one. And, in this respect, I think Nahin does a pretty good job. But I’ve been back over the basic story a number of times and realise that I’m still struggling with the basic ideas behind it. I get it that when you multiply by √-1 then you basically rotate the point in space, counter-clockwise, by 90 degrees. But that is about the limit of my understanding.
Nahin, clearly delights in the ability of complex numbers to deal with complex mathematical problems and a large section of the book is devoted to real world problems. I kind of followed the logic but when it comes to some complex equation or a lot of messy numbers, I just take it on faith that Nahin is right when he says stuff like “ And this reduces quite simply, with a bit of algebraic manipulation, to x”. I just don't have the time or patience to work through it.
At one point he employs an equation and justifies it on the grounds that he knows it actually works, rather than deriving the solution from first principles. An, I found it interesting that he says this is a common procedure among mathematicians. Somewhere I’ve read an explanation of the complex plane that involves a Riemann sphere and I think that explains things a bit better than Nahin manages to do. (I don’t think he mentions this at all).
I was also fascinated to read that the problem of √-1 was solved by Caspar Wessel, a surveyor in 1797 though his paper in Danish was overlooked by the mainstream mathematicians. And, I’ve wondered whether he developed his ideas based around the fact that , with a dumpy level, used in surveying that you are constantly looking a numbers above and below a standard level....so constantly dealing with negative numbers in your charts and having to do trigonometry with them.
Anyway, t he following are a few extracts that I’ve taken from the book, that I though would be helpful to me in any future revision of the book or it’s observations. I’ve tried to use Cambria Math as the font and I’m hopeful that I can actually publish it with the fonts intact because it’s rather difficult without access to math symbols.
"We find the square root of a negative quantity appearing for the first time in the Stereometria of Heron of Alexandria... After having given a correct formula for the determination of the volume of a frustum of a pyramid with square base and applied it successfully to the case where the side of the lower base is 10, of the upper 2, and the edge 9, the author endeavours to solve the problem where the side of the lower base is 28, of the upper 4, and the edge 15. Instead of the square root of 81 - 144 required by the formula, he takes the square root of 144 - 81..., i.e., he replaces √-1 by 1, and fails to observe that the problem as stated is impossible. Whether this mistake was due to Heron or to the ignorance of some copyist cannot be determined.
While Heron almost surely had to be aware of the appearance of the square root of a negative number in the frustum problem, his fellow Alexandrian two centuries later, Diophantus, seems to have completely missed a similar event when he chanced upon it. Diophantus is honoured today as having played the same role in algebra that Euclid did in geometry. Euclid gave us his Elements, and Diophantus presented posterity with the Arithmetica. In both of these cases, the information contained was almost certainly the results of many previous, anonymous mathematicians whose identities are now lost forever to history. It was Euclid and Diophantus, however, who collected and organized this mathematical heritage in coherent form in their great works.
In my opinion, Euclid did the better job because Elements is a logical theory of plane geometry. Arithmetica, or at least the several chapters or books that have survived of the original thirteen, is, on the other hand, a collection of specific numerical solutions to certain problems, with no generalized, theoretical development of methods.
Six hundred years later (circa 850 A.) the Hindu mathematician Mahaviacarya wrote on this issue, but then only to declare what Heron and Diophantus had practiced so long before: "The square of a positive as well as a negative (quantity) is positive; and the square roots of those (square quantities) are positive and negative in order. As in the nature of things a negative (quantity) is not a square (quantity), it has therefore no square root. More centuries would pass before opinion would change.
Bombelli's insight into the nature of the Cardan formula in the irreducible case broke the mental logjam concerning √-1. With his work, it became clear that manipulating √-1 using the ordinary rules of arithmetic leads to perfectly correct results. Much of the mystery, the near-mystical aura, of √-1 was cleared away with Bombelli's analyses. There did remain one last intellectual hurdle to leap, however, that of determining the physical meaning of √-1 (and that will be the topic of the next two chapters), but Bombell's work had unlocked what had seemed to be an unpassable barrier.
When the early mathematicians ran into x2+ 1 = 0 and other such quadratics they simply shut their eyes and called them "impossible." They certainly did not invent a solution for them. The breakthrough for √ -1 came not from quadratic equations, but rather from cubics which clearly had real solutions but for which the Cardan formula produced formal answers with imaginary components. The basis for the breakthrough was in a clearer-than-before understanding of the idea of the conjugate of a complex number.

The big difference now is that the points B and B' do not lie on the base AD, but rather above it. Wallis had stumbled on the idea that, in some sense, the geometrical manifestation of imaginary numbers is vertical movement in the plane. Wallis himself made no such statement, however, and this is really a retrospective comment made with the benefit of three centuries of hindsight.
It would be another century before the now "obvious" representation of complex numbers as points in the plane, with the horizontal and vertical directions being the real and imaginary directions, respectively, would be put forth, but Wallis came very close.
There is, however, one caveat concerning the polar form of representing complex numbers that is most important to keep in mind. A common error made by students who are first learning about the polar form is a failure to appreciate that the tangent function is periodic with period 180°, not 360°. That is, the tangent function goes through its complete interval of values (-∞ to ∞) as the polar angle 0 varies from - 90° to 90°. Or, if we express angles in units of radians (one radian = 180°/ π = 57.296o), then the tangent function goes through its complete interval of values as the polar angle varies from -π/2 to π/2 radians. This means that blindly plugging values of a and b into 0 = tan- (b/a) may lead to mistakes.

Wessel began his paper by describing what today is called vector addition. That is, if we have two directed line segments both lying along the X-axis (but perhaps in opposite directions), then we add them by positioning the starting point of one at the terminal point of the other, and the sum is the net resulting directed line segment extending from the initial point of the first segment to the terminal point of the second. Wessel said the sum of two nonparallel segments should obey the same rule..... So far there is nothing new here, as Wallis had expressed quite similar ideas on how to add directed line segments. Wessel's original contribution was to see how to multiply such segments.
Wessel discovered how to multiply line segments by making a clever generalization from the behavior of real numbers. He observed that the product of two numbers (say, 3 and -2, with a product of - 6) has the same ratio to each
factor as the other factor has to 1. That is, - 6/3 = -2 = - 2/1, and -6/-2 =3 = 3/1. So, assuming there exists a unit directed line segment, Wessel argued that the product of two directed line segments should have two properties.
First, and immediately analogous to real numbers, the length of the product should be the product of the lengths of the individual line segments.
But what of the direction of the product? This second property is Wessel's seminal contribution: by analogy with all that has gone before, he said the line segment product should differ in direction from each line segment factor by the same angular amount as the other line segment factor differs in direction When compared to the unit directed line segment.
Ever since Wessel, then, multiplying two directed line segments together has meant the two-step operation of multiplying the two lengths, with length always taken to be a positive value, and adding the two direction angles....These two operations determine the length and direction angle of the product, and it is this definition of a product that gives us the explanation for what √-1 means geometrically. That is, suppose that there is a directed line segment that represents √-1, and that its length is l and its direction angle θ.
Mathematically, then, we have √-1 = l ∠θ. Multiplying this statement by itself, i.e., squaring both sides, we have -1 = l ∠2 θ or, as -1 = 1 ∠180°, then l2 ∠2θ = 1 ∠180°. Thus, l2 = 1 and 2 θ = 180°, and so l = 1 and θ = 90°. This says √-1 is the directed line segment of length one pointing straight up along the vertical axis or, at long last,.......i = √-1 = 1 ∠90°
This is so important a statement that it is the only mathematical expression in the entire book that I have enclosed.

An imaginary number to an imaginary power can be real! Who could even have made up such an astonishing conclusion? As you will see in chapter 6 this isn't the end of the story, either — in fact, ii has infinitely many real values, of which e^-π/2- is only one.

If, Kasner wrote, one allows y(x) to be a complex-valued function then the limit of arc length to chord length can be less than one! The old adage that "a straight line is the shortest path between two points" is not necessarily true for complex-valued curves. I can't draw a complex-valued curve for you on a piece of paper, of course-how would you draw y(x) = x2 + ix, for example? —but we can still do the formal calculations.

There are, of course, more than just two distinct masses in the universe. The problem of calculating the motion of N gravitationally interacting masses became known as the N-body problem of celestial mechanics, and the myth has spread among most physicists that it remains unsolved for N ≥ 3. This is true only if one demands closed-form, exact equations. In fact, the Finnish mathematical astronomer Karl F. Sundman (1873-1949) solved the three-body problem during the period 1907-19, and in 1991 a Chinese student, Quidong Wang, solved the N-body problem for any N. These solutions are in the form of infinite convergent series, however, which unfortunately converge far too Slowly to be of any practical use. use. Of course, with the development of super-computers, physicists can now directly calculate the future motions of hundreds, even thousands, of interacting masses, as far into the future as one would like, using Newton's equations of motion and gravity. Solving the N-body problem analytically is no longer of any practical importance.

The ancient astronomers could "explain" these mysterious retrograde motions with Ptolemy's crystal spheres, but in fact these motions are simply illusions caused by watching one moving thing (a planet) from another moving thing (the Earth). Kepler knew this, and was the first to explain the illusion using diagrams to illustrate his qualitative reasoning. Complex exponentials, however, make the mathematics of what is happening easy to understand as well.

Amazingly, the quite formal and "mysterious formula" of π = (2/i) In(i), as
Benjamin Peirce called it, can be used to calculate the numerical value of .
That might seem like getting something out of thin air, but this astonishing fact was pointed out long ago by the German mathematician and educator Karl Heinrich Schellbach (1809-90) in 1832...... The Leibniz-Gregory series is, while beautifully elegant in appearance, utterly worthless for numerical calculations since it converges very slowly. Using the first fifty-three terms, for example, is not sufficient to give even just two correct, stable decimal digits....... Now, what Schellbach went on to show was how his method gives other series for π that converge much faster than does the Leibniz-Gregory series.

So what’s my overall take on the book? Actually, it’s quite fascinating. Rather difficult for a non mathematician but there is enough there that a non-mathematician (such as myself) can find it interesting and learn from it. Four stars from me.

[Kcl · Rating 5 out of 5]

Wish I had read this before taking complex analysis. Everything would have been less imaginary.

[Mdaly · Rating 2 out of 5]

This is an extraordinary book. It has to be the craziest book I've ever read, it brooks no doubt and doesn't hold the hand of the reader. A fairly good understanding of calculus is required to keep up with Nahin. The author mingles the history of mathematical ideas with complex mathematical problems. The passion the author has for mathematics is incomparable. He is an engineer and professor of electrical engineering however I found the arrangement of the material quite frustrating and hard to follow.
While outlining the development of mathematical thought around complex numbers he derives some of these key equations in his own way which seems a bit showy if one isn't going to detail how they were originally derived. I would have liked a short primer as an appendix in calculus. Nahin assumes a lot on the part of a casual reader.

[David Doel · Rating 3 out of 5]

The short answer -- way too much algebra and too little history.

The preface to this book is a touching remembrance of the author's father and his impact on the author's decision to study electrical engineering.

If you stop reading there, I believe what you will have missed is a horrific amount of algebra and a little bit of history. There are moments of interest, but too few and too many equations in between. There is no clear goal to this book unless it is to display the author's mathematical prowess.

Chapter 5 includes an interesting discussion of the transition from Copernicus through Kepler and Newton moving from an earth-centered universe to a sun-centered solar system. Nahin uses imaginary numbers to show consistency between Kepler's laws and Newton's physics. That is not how Newton worked, but the section is interesting and does present the history. Most of the advances described in the book were not from using imaginary numbers; they are able to be reproduced using imaginary mathematics.

If you want to brush up on your algebra and trigonometry, this book might be for you. You can just fill in the blanks the author leaves in his many derivations and I guarantee you will get lots of practice.

[Nathan Glenn]

I'm going to hold off on rating this for now, since the Kindle edition is so messed up that I could not read a lot of the formulas.
This book is a comprehensive history of the number i. It explains the history of the idea of the number itself, its geometric and interpretation, and then its applications.
He says that a high school graduate who studied calculus should be fine reading it, but I wouldn't quite agree. You have to have calculus pretty fresh on your mind to just dive into this. It has some really heavy math in it, and it is good to keep in mind that it is meant as a readable history for mathematicians. The theory starts small, showing that "imaginary" numbers are in a number plane instead of a number line, and goes through several aspects of complex analysis including harmonic functions and contour integrals, describing interesting characters along the way. It was fun to read, but I was completely lost on the math. Quite a heavy read.

[Kaylee · Rating 3 out of 5]

It's only been three and a half years since I've been in an upper-level math class, and yet, I felt like a dunce at many points in this book.

Granted, that may have been due to Nahin's decidedly engineer-fascinated-by-math style of writing (that style does exist, I swear; I grew up with my dad teaching me math in a way that can only be described as filtered through an engineer's mind); aside from my dad, the people I spoke math with were all mathematicians.

I should have read this when I first received it as a gift if I wanted to fully grasp all of the equations. As it was, despite my degree in math and the insanely slow pace I took reading this, the lack of constant use of many equations and theorems shown in the book meant I recognized the name and the general idea, but was totally lost on some of his executions.

[Bill W · Rating 3 out of 5]

2.5. While reading this, I couldn't help wondering who the intended audience was. It's marketed as a "popular mathematics" book (an oxymoron, really) but it isn't one. Cauchy's theorem is college-level math. I think that people who have studied mathematics long enough to understand this book also know enough to be bored by it.
I did like the history in the book, which is why I read it. I'm rounding up my rating for Goodreads because I admire the author for his attempt to portray the human side -- the struggles and strokes of genius that lie behind the "dry" analysis we take for granted. This information is very important. I learned a lot from the historical anecdotes in the book.

[Erik · Rating 4 out of 5]

I now realize why I took all that #@$%#$ math in University: it was to be able to read this book. Why can't school math be presented like this? Anyway, if you remember any trig or calc read this and enjoy the part where Einstein's contribution to general relativity gets explained it a way that makes sense.

Edit: This is a real math book, with real math. Like, solving differential equations math. But there's a story you can follow without following every step of the calculations as long as you can intuit why the result makes sense. So it's pretty perfect.

[ajp3 · Rating 5 out of 5]

one of the best mathematics books ever written. the last two chapters have significant mathematical formalism (mostly complex analysis), but up until that point almost any calculus student will understand the arguments presented. some of the most elegant and beautiful ideas are covered in this surprisingly short book. I love it and try to read it often.

[mirela Darau · Rating 5 out of 5]

I simply loved reading this book, and I was thrilled to see how mathematics worked some centuries ago! Another good thing about this book is that the author's arguments and explanations are mostly simple and can be followed...

[Gabriel · Rating 4 out of 5]

While best read by those with a pad of paper, a pencil and their fare share of mathematical knowledge, this has been a very cool read about the history of the imaginary number and how mathematicians think.

[David · Rating 5 out of 5]

Best to have some familiarity with calculus; given that, this is a wonderful book about the historical development of mathematics involving imaginary numbers. The final chapters display some powerful ideas that lead to non-intuitive results.

[Rupesh Kumar · Rating 4 out of 5]

Adding an imaginary number imaginary times may yield a real number. After reading this book you'll come to know what is imaginary is not actually imaginary.
It is complex, isn't it?:-)
A great book for those who love mathematics.

[Mark Kloha · Rating 1 out of 5]

The book’s preface stated that anyone with a knowledge of high school algebra and some calculus can understand this. That is false. This is an advanced mathematics book that requires advanced degrees in mathematics, calculus and electrical engineering.

[Lynn · Rating 1 out of 5]

I had no idea that the few short lessons I had on the imaginary number had such a complicated background. It was an interesting but tedious read.

[Clark Lyons · Rating 5 out of 5]

The author gets his message across very well but is too practical for me.

[Dave · Rating 2 out of 5]

Not overly challenging, but not particularly well written in my opinion.

[Zach]

An excellent book about the history of one of the most important developments in mathematics!