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generalheff's review against another edition
3.0
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.
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.
radioactve_piano's review against another edition
3.0
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.
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.
peacefixation's review against another edition
3.0
I held on for the first few chapters but this ended up being quite a difficult read and I had to skim over most of the math. Despite that, it was a very interesting read and I learned a lot about sqrt(-1). I hope to return in a month or so for another go, with a pen and paper close at hand!
trask0730's review against another edition
A pretty good book! I did myself a favor and avoided doing lots of hard math myself haha (I figured I will have plenty of that in the years to come). That being said, I loved learning the historical context of one of my favorite subjects within mathematics!
mathstalio's review against another edition
2.0
I read this with the thought that it would be a history of imaginary numbers (it was) and their applications (ehhhh). The author starts out with an intro that says any student who has taken high school calculus should be able to follow the math of the book. Definitely not. I mean, I could have followed the included calculations but I chose not to because it was simply unnecessary to show the entire derivation of formulas. Tell me ABOUT them and where they came from. That's what I was hoping this book would be. Maybe it's unfair to judge a book on what you wanted it to be, but. Here we are. Too calculation heavy, I wanted more history and DESCRIPTIONS of applications. Although I did learn enough to be able to give my students a bit more background on complex numbers when we get to them.
brettt's review against another edition
2.0
Of all the people who probably hide their heads at the goofs they have made, the person who named "imaginary" numbers is probably among them as far as the field of mathematics is concerned.
So-called "imaginary" numbers describe the square roots of negative numbers, which are impossible to calculate using plain integers. The square root of 1, for example, is 1 because 1x1=1. But the square root of -1 seems impossible to figure, because the only way to get to -1 is to multiply two different numbers together. A negative number multiplied by another negative number leaves a positive number, not a negative one. At some point, mathematicians decided that there would be a square root of -1, and it would just be a 1 that was on another "axis" than the regular positive-negative line. But since the number didn't seem to have any real-world analogue like positive and negative numbers did, it somehow got hung with the tag, "imaginary." So today we say that the square root of negative 1 is i. The square root of -4 is 2i, and so on.
Retired electrical engineering professor Paul Nahin outlines some of the development of i through the history of mathematics in An Imaginary Tale. Some early cultures refused to acknowledge the existence of a quantity that could be squared to form a negative number, and even into the Renaissance and enlightenment years the so-called "imaginary" numbers were considered at best unimportant. They were not useful except in specialized cases and it seemed even mathematicians had reservations about dealing with numbers that didn't represent any real quantity.
Today, i and its counterparts find widespread use in many areas of math, and the only reservations that seem to continue deal mostly with the use of the word "imaginary." Nahin explores how important i is in many fields of engineering, especially his own. This part of the book -- about the latter two-thirds -- is heavily laden with equations and formulas and is going to be beyond most non-mathematician or non-engineer readers. He probably would have had to lengthen the book considerably to bring that subject matter within the grasp of the lay reader, but that doesn't make the string of equations and engineering language any easier to navigate.
Original available here.
So-called "imaginary" numbers describe the square roots of negative numbers, which are impossible to calculate using plain integers. The square root of 1, for example, is 1 because 1x1=1. But the square root of -1 seems impossible to figure, because the only way to get to -1 is to multiply two different numbers together. A negative number multiplied by another negative number leaves a positive number, not a negative one. At some point, mathematicians decided that there would be a square root of -1, and it would just be a 1 that was on another "axis" than the regular positive-negative line. But since the number didn't seem to have any real-world analogue like positive and negative numbers did, it somehow got hung with the tag, "imaginary." So today we say that the square root of negative 1 is i. The square root of -4 is 2i, and so on.
Retired electrical engineering professor Paul Nahin outlines some of the development of i through the history of mathematics in An Imaginary Tale. Some early cultures refused to acknowledge the existence of a quantity that could be squared to form a negative number, and even into the Renaissance and enlightenment years the so-called "imaginary" numbers were considered at best unimportant. They were not useful except in specialized cases and it seemed even mathematicians had reservations about dealing with numbers that didn't represent any real quantity.
Today, i and its counterparts find widespread use in many areas of math, and the only reservations that seem to continue deal mostly with the use of the word "imaginary." Nahin explores how important i is in many fields of engineering, especially his own. This part of the book -- about the latter two-thirds -- is heavily laden with equations and formulas and is going to be beyond most non-mathematician or non-engineer readers. He probably would have had to lengthen the book considerably to bring that subject matter within the grasp of the lay reader, but that doesn't make the string of equations and engineering language any easier to navigate.
Original available here.
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