by Paul J Nahim
This is a fascinating story of i , popularly known to most of us as square root of -1 . In the first part of the book, author, Paul J Nahin , takes you through stories of various mathematicians who struggled and contributed to the understanding of sqrt(-1) . Subsequent to this historical narrative, the author provides a lens in to viewing things from a complex domain. Like they say, the easiest path from one point to another in the real world is through a complex plane, Complex numbers are used almost in every discipline you can think…
Some of the areas where I have come across usage of complex numbers in my limited exposure to things are
- Econometrics : most of the filtering concepts in time series involves extensive use of complex numbers.
- Algebra : Finding roots of nth order polynomial equation which usually arise from pth order auto regressive processes
- Basic Probability : Characteristic functions extensively use complex domain
- Fourier Transform computations become easy by taking them in to the complex space
Basically , before I read this book, my view was that complex numbers, complex theory etc..were nice tools to do quick integration, verify trigonometric identities, root finding etc. However the book changed the way I think about complex numbers . Complex domain is actually a way to see things . If one restricts its usage as a tool, well, that’s all one gets out of it. But if you use it as a lens to see a problem , as the book powerfully argues, it is going to be of considerable use to most of us. I am yet to use in that sense of the term but the book does motivate me to go in that direction.
Ok, now for the first part of the book where the author gives a superb historical narrative:
CHAPTER 1 : The Puzzles of imaginary numbers
Heron of Alexandria : A Greek mathematician was one of the first persons who came very close to stumbling on square root of a negative number. In his calculations of truncated square pyramid, the slant height of the pyramid was expressed as a formula and it clearly showed that for specific values of edge lengths and volume, the slant height can throw up square root of negative number. However, as history has witnessed, Heron seemed to have fudged his formula to avoid sqrt(-1). He came so close to seeing sqrt(-1) but let it slip by
(c.214 – c.280) Diophantus who is kind of father of algebra, also saw sqrt(-1) in the calculations for roots but dismissed them as impossible solutions.
del Ferro : An Italian mathematician who was the first to solve the depressed cube equation( x^3 + p*x = q). His ingenuity in problem solving made him find a closed form solution for the depressed cube. The formula like Heron’s formula had a possibility of sqrt(-ve number) but del Ferro by imposing constraints on p and q avoided such situations. Also, mathematicians around his era were considered with only the positive root. The unique thing about this story is that delFerro did not reveal his secret until his dying moment when he told to his student Antonio Maria For.
Niccolo’ Fontana(Tartaglia) : A self taught mathematician who solved different versions of cubic equations. An interesting story about Antonio Maria For and Tartaglia is where the former challenges the latter on a mathematical duel ( spectator events of those times) and loses badly. Tartaglia also keeps the whole thing secret until he discloses it to Cardano
Girolamo Cardano(1501-76) : Cardano improvised on the del Ferro’s solution and came up with a solution for the generic cubic equation. However in the process of discovery, Cardano realized that square root of negative numbers were appearing everywhere and by working with them , instead of discarding them,he found that the combination of sqrt(-ve) numbers were actually giving out real roots.
Rafael Bombelli was the first person to see that if the sum of expressions which involve square root of -ve numbers is a REAL number , then each of individual elements should be conjugates of the other. A brilliant insight in the hindsight (a+i*b) + (a-i*b) = real ).
Francois Viète solved the cubic using trigonometry. His solution to the cubics was in arc cosine and cosine form. By the very definition of arc cos involved in the formula, the domain of the possible solutions is restricted to real line.
In the entire history that’s mentioned in this chapter, Bombelli stands out as he provided the critical breakthrough of WORKING with sqrt(-1).
In author’s words ” Bombelli’s insight in to the nature of Cardan formula in the irreducible case broke the mental logjam concerning sqrt(-1). With his work, it became clear that manipulating sqrt(-1) using ordinary rules of arithmetic leads to perfectly correct results”
Yes, it is not the usual way that schools teach us , sqrt(-1) is the solution for x^2 + 1 = 0 …. square equations had nothing to do with the evolution of i. It is the cubic equations that led to the development of complex numbers .
The highlight of the first chapter is a method illustrated to find the complex roots using a ruler. Read this section of the book ..It rocks!
CHAPTER 2 : First try at Understanding the geometry of sqrt(-1)
Rene Descartes was a pioneer in the field of Geometry and other domains. He dismissed sqrt(-1) by saying that it was a geometrical impossibility
John Wallis came close to stumbling to the aspect that i could be related to a movement in the vertical direction but he did not actually develop that specific thought and explored.
Thus the geometrical world was still skeptical about sqrt(-1) as they did not any real world interpretation of the imaginary number.
CHAPTER 3 : The Puzzles Start to Clear
Caspar Wessel : A cartographer by profession, his work got him involved in math and finally he discovered something which stumbled eminent mathematicians prior to his life time. What was Wessel’s contribution. He represented numbers in polar form and introduced the concept of rotation operator. He was the first person to actually organize things relating to the geometric behavior of i. However he never publicized it.
Jean-Robert Argand : Another person who was not trained formally in math, but gave the world a geometric representation of A+ iB
Hamilton : He was a mathematician who was not interested in the geometric narrative and instead developed couples (a,b) representing a+ib and formulated rules and methods revolving couples
Gauss : He gave the stamp of authority to Hamilton’s method and thus i became an acceptable notion in the field of mathematics.
CHAPTER 4&5 : Using Complex Numbers
The next 2 chapters give an array of applications of complex numbers, some of which are
- Using to solve recurrence relations
- Applications in vector algebra by treating complex numbers in the argand plane
- Usage in Electrical engineering
- Astronomy , more specifically using Kepler’s laws as example
- Transforming problems in to argand space to solve optimization problems
- Story of sqrt(-1) in an electrical circuit was the first design for a multi billion dollar company today, HP
CHAPTER 6 : Wizard Mathematics
This chapter is about Euler .
I never knew that Euler , one of the best mathematicians that world has seen till date, was blind in the last seventeen years of his life. It is said that Euler did monstrous calculations in his head.
We all come across Euler when we see the equation exp(ix) = cos(x)+ i sin(x) . Euler’s brilliance was in connecting exponentials and complex numbers. He didn’t stop there. He published Euler’s constant gamma(nth partial sum of S1 – ln(n) as n approaches infinity) which equals 0.577215664…. , one of the universally popular constants, besides the popular pi , e, g etc. What’s the connection between this fact and complex numbers. Euler used his identity to prove the convergence of Sp where p is even. Till this date no one has ever figure out the solution for p = odd.
You can’t read this part of the book with out pen and paper as it has a lot of identities and history behind those identities that you would actually want to work them out yourselves to see the beauty of complex numbers.
CHAPTER 7 : Complex Function Theory
The last chapter obviously takes the complex numbers to the next level of complexity 🙂 , functions of complex numbers. Author gives a very basic introduction to contour integration , cuchy’s theorems , greene’s theorem etc. The last chapter is very technical but contains interesting bits of historical narrative. So, one can gloss over the theorems etc to get an overall picture about complex function theory.
The book starts off with a good narrative and becomes progressively more technical. So, one really can’t describe it as a non-fiction book nor can one describe this book as a textbook. I guess its a good primer for people who want to know about complex numbers. Yes, this definitely falls under “reading around the subject” category which will motivate a person to dig deeper in to the concepts , workings and the application of complex number theory in their respective domains. Personally I loved the narrative part described in the first part of the book. Applications of Complex numbers in finance which is immediately relevant to my job, well , it is a looooong way to go , before I can connect things from different domains and implement something useful out of it.
Another takeaway from reading the historical narrative is that, one can’t miss the point that most path breaking discoveries and techniques came from people who were not formally trained in mathematics ,but by people who had a passion for math. It will be interesting to compare these developments in today’s context , where formally trained Math PhDs are spreading their wings and working on areas in various disciplines, finance, biotech, nanotech, internet search technologies, analytics etc. What developments will we see in the next 25 years ?