Archive for June, 2010

Jimaginary

Posted in Uncategorized on June 22, 2010 by Alex

I first became aware of imaginary numbers in high school.  They were presented as a clever way of solving irreducible quadratic equations, and of course they were represented by the letter i, for imaginary.  I quickly figured out that imaginary (or more generally, complex) numbers could be manipulated in the same way as regular old real numbers, and I didn’t give any thought to the notion of how that letter came about in the first place.  In fact, at the time I wasn’t really aware that there was any history behind this, or any other branch of math.  I just assumed that this was something known since antiquity, and now I had to learn about it.  It is an interesting thought that mathematics is taught as if the results are obvious and nobody ever struggled with it, except for stupid students like me.  Over the last few years I have read a lot about the history of math and come to the opposite conclusion; there are tons of stories of ancient mathematicians, many of whom were the smartest men of their time, struggling to solve problems that today are taught as elementary in high school!

At the time, I perceived imaginary numbers as mathematical oddities.  I could add, subtract, multiply and divide complex numbers almost as easily as I could with real numbers, but beyond that I could not see why anyone would use them beyond just fooling around with the arithmetic.  Granted, I wasn’t exactly an expert on the practical applications of math back then, but I had some inkling of the usefulness of other topics.  But by the time I was finished with this preliminary study, it was firmly ingrained in me that the imaginary number was, in fact, i¸ and I never questioned why that symbol was used, if only because that was what everyone else wrote.

A couple of years later, while I was in college, I hadn’t thought much about complex numbers.  They briefly made an appearance when I took my first course in differential equations, but all the imaginary parts neatly canceled out when the final solutions were written down.  Then I took my first circuits class, and they came back with a new letter, j!  This dumbfounded me more than any of the pretty results that can be gained from playing with complex numbers (Euler’s formula being the most notable) and was probably the first time I ever wondered why people write things the way they do.  Why mess with a good thing?  It’s the imaginary number, of course it should be written as i!  Who the hell had the bright idea of changing the symbol that everybody else in every other technical field uses?  I asked my professor about this (he was not an electrical engineer, but a physicist) and he simply said that he did not try to understand the motives of electrical engineers, for that way madness lay.  The flippant explanation that I give my students today is “it’s j, for jimaginary.” 

Over the years of studying electrical engineering I have come up with a few rationalizations for this, and heard a few more.  The most common is that i is the symbol for current and it would be awfully confusing to see the same symbol used for two completely different things in the same equation.  To me this only raises more questions, such as why we use the letter i for current in the first place.  There is no i in current!  The more I learned the more I noticed these odd, seemingly arbitrary choices for letters used in all courses of study.  To list a few:  L for inductance, G for conductance, p for momentum, L again for angular momentum, alpha for about thirty different things, and ξ, a letter I can’t write or pronounce! 

I imagine that there is a story behind the choice of each letter for each thing it represents.  Some of them may be interesting, most of them probably not.  I have speculated on how the decision to use the letter j was arrived at.  I think it may have gone something like this:

During the late nineteenth century the field of electrical engineering was coming into its own.  There was a meeting of all the great minds of electrical engineering to discuss the great problems of the day.  The master of ceremonies got up to speak to set the agenda for the meeting.

MC:  Gentlemen!  We have much to discuss about the future of our field on this day.  We shall hear the latest ideas on trans-Atlantic wireless communication, on the feasibility of powering every home in the world with electricity, and of lighting the streets of every city, thus turning night into day.  We shall debate the relative merits of alternating vs. direct current for the use of power transmission, and also discuss the fundamental nature of electricity itself.  These problems are at the very forefront of our field, and their solutions will help to shape the world for centuries to come.  But I feel there is a dark cloud hanging over our profession, and that is the use of imaginary numbers in our calculations.  The letter i is no longer satisfactory, for it makes our calculations difficult and unwieldy.  I therefore propose, before we get to these other topics which are minor in comparison, that we use a new letter for the square root of negative one!  This decision will mark the beginning of a new era for electrical engineers.  The choice should be made extremely carefully, for the fate of the world just might hang in the balance.  It is up to you, the greatest minds of this age, to come to a consensus and usher in this glorious new era of light, potential and current!

(Stunned silence from the gathered engineers.)

Voice from the crowd:  How about j? It’s the letter right after i.

The meeting then erupts in applause, and the engineer who made the suggestion is carried off into the sunset, the rest of the agenda left to be discussed another day.

Maybe the decision to use j instead of i didn’t occur exactly like that, but the decision itself must have been just as arbitrary and weird.  Besides, as my lawyer always says, never let the truth get in the way of a good story.