Hi Paul.
Thank you for that explanation. I was already familiar with and had not forgotten the notion of periodicity (and so would not have argued for expressing all of the infinity of possible multiples) but I did not know of the convention regarding the Principal Argument excluding the particular value of -pi radians. That makes sense given that +pi radians would suffice. However, from now onwards whenever I see e^i.pi I will also see, in my mind’s eye, e^-i.pi at the same time as well; just to remind myself that -1 can be reached by rotation in either direction. Little things please little minds 
Thanks again.
Thanks Isaac, it’s really gratifying to hear when the material is helping. I have a book on Amazon (Math, Better Explained) but I should do more to market it :). Understanding the nature of e^ix was one of the best aha moments ever.
It seems to me that Leonhard Euler himself was not really aware of the intuitive meaning of the very equation that bears his name.
I say this after reading Euler’s book “Elements of Algebra”, where in chapter 3 he tries to justify why multiplying by two negative quantities gives a positive quantity. For Euler, this is merely applied as a rule of algebra, and he makes no attempt to relate this to complex numbers.
Euler’s writing never gave the impression that he truly understand that a negative number was just a positive number rotated 180 degrees (pi radians) around the origin of the complex plane - he seemed to lack the intuition that underlies the famous equation that bears his name. My guess is that Euler only understood that equation as a consequence of the proof by infinite summation - the Taylor series expansions of sine, cosine and the exponential function - which is not very intuitive to me.
But I am being more than a little unfair to Euler here, because he died decades before before Caspar Wessel and Jean-Robert Argand published their discoveries about complex numbers. Their astounding insight was that multiplying complex numbers is akin to rotating them geometrically - the very insight that Kalid has expressed for us here.
Only after Wessel and Argand published their description of complex numbers, representing them as arrows on a plane, was the world able to intuit the concept of multiplying numbers meant rotating arrows.
Euler’s old way way of thinking about multiplying negative numbers was markedly different to Wessel and Argand. The new way of thinking of a negative number was an arrow with an angle of 180 degrees, and therefore multiplying two negatives meant adding 180 + 180, to get 360 - thus bringing the arrow back to the positive real axis.
Further, multiplying by three or more negative numbers would just keep rotating the resulting arrow between the negative and positive real lines, which is the intuitive result from thinking of complex multiplication as rotation. This is a much more satisfying, more intuitive way of thinking about multiplying by negative numbers compared to Euler’s simplistic algebra rule from his book “Elements of Algebra”.
Perhaps what is saddest is that modern educators are also not making the crucial link between the simplistic, primary-school rule for multiplying negative numbers and the true intuition that complex multiplication is rotation. There should be a magical Ah-Ha! moment when students are first introduced to complex numbers - when they finally get to see the actual reason why multiplying two negative numbers gives a positive number.
Thanks for describing such an arcane topic in so simple terms, now i don’t see the complex no as something very mysterious and such formulas only as some tool to solve other questions.
…and, at the age of 68, I can now say that I am officially the slowest student from the class of '72.
Hi Paul
I hope I didn’t give the impression that I was criticising Kalid’s approach in any way. Kalid’s explanation helped me to visualise in a step by step manner. Without that I would still be taking e^ix at face value as unit vector rotation (which was the case in 1972 when I covered this in engineering maths) without really understanding it. Even taking a derivative would not really have given me the visualisation in those days. Now, however, I can use the i brought down from the exponent - in the derivative - as a short cut (or icon) to memorising Kalid’s visualisation. AK (after Kalid) I am happy at last with Euler’s formula and identity.
For the second example (clockwise rotation) that should read “Similarly, e^-i.x has a derivative of -i times
e^-i.x giving…”
Hi Don,
I guess everyone’s idea of intuitive would be different. You and I are engineers, so what you have written is certainly correct, and makes sense to both of us.
What I am really looking for is something that makes sense as an introductory material for complex numbers and Euler’s identity.
My feeling is that it may be best to start students who are new to complex numbers with a definition of multiplication as a two-part process - scaling and rotating, before getting into any algebra or calculus.
Scaling and rotating are geometric - and thus they can be visualised better than any algebraic or calculus-based description of complex numbers.
What Euler’s identity actually says is that minus-one is exactly as a half-rotation (pi radians) around the origin. An explaination of this without algebra or calculus is what I think of as intuitive.
Hi Kalid and Paul
A simple method for Visualising?
Would it be wrong to encapsulate the idea of e^i.x giving rise to anticlockwise circular motion by simply remembering that its RATE OF CHANGE, ie its derivative, is i times e^ix and that i.e^i.x, always points at 90 degrees to e^i.x? Similarly, e^-i.x has a derivative of -i times e^i.x giving the opposite rate of change - and therefore clockwise circular motion. Would that be complete enough for intuitively visualising the circular motion? Forgive me if this has already been covered somewhere above: I haven’t checked.
Hello Kalid,
I’m posting again to once again remark on the excellent clarity of your explanations. You might consider writing a book, or something. I believe that the general public would benefit from these ideas being brought to light, as to really demystify maths in general and perhaps advance our society forward if more people were to excel in those areas that deal with maths rather than be scared off from even considering the possibilities that understanding maths opens up in life. Seeing e^ix explained as the base for continuous circular growth makes perfect sense and I doubt I will ever “unsee” that insight when dealing with that formula. The veil is being lifted for me on the underlying mechanisms that build the tools of mathematics. Kudos.
I have come across a short article entitled “How Euler Did It - e, pi and i: Why is Euler in the Euler Identity?” by Ed Sandifer, which shows the mathematical lead-up to his Identity. http://eulerarchive.maa.org/hedi/HEDI-2007-08.pdf
Sandifer uses p for pi, which I found confusing at first when the expected exponent pi.i appears as pi. There is also an n missing from the last term at the bottom of the first page (although it appears again in the last term of the first equation on p. 2).
Hi Don Webber,
Thanks for that link to the article. It’s a good read concerning the history of the development of thinking about complex numbers. It’s very mathematical and still unintuitive, unlike Kalid’s explanations.
It highlights that back in the early days, the first mathematicians who tried to grapple with complex numbers were floundering around in the dark. They were chasing mathematical symbols around the page without really understanding what it all meant.
My perspective is that complex numbers should not be taught in the same way they they were discovered. Instead, if teachers begin with the Wessel / Argand concept that multiplying causes numbers to rotate, and then work backwards in time to Euler’s identity, then go further back to the complex solutions of algebraic equations, the subject becomes really quite sensible and intuitive.
But today, complex numbers are still taught in high school and universities by starting with the algebra, and eventually ending up with Argand’s insight about rotation. By the time students get to Argand’s rotation, the whole subject seems labyrinthine and counter-intuitive, and too many students have gotten lost along the journey.
Complex numbers are still taught by moving forward through the history of their discovery, but unfortunately this is intuitively backwards.
Hi Paul
I am one of those students who, back in the '70s (as I explained in post 207, above), got lost and had, in my case, a somewhat hazy grasp of the subject. Yes, I agree: start with the idea of rotation.
I think that not enough credit is given to Jean-Robert Argand for his discovery of the underlying geometry of complex numbers. He did much more that merely understand what Euler wrote - he explained it better for everyone.
Argand’s geometric ideas neatly tied together the rather obscure theorems and equations of those who came before him: Cotes, Bernoulli, DeMoivre, Euler etc.
But Argand worked outside the framework of those established mathematicians - he was an amateur, and obviously a gifted one too. He explicitly said that he wanted to create clarity in the way complex numbers were being discussed, so he invented the breakthrough idea of drawing numbers as arrows rotating around zero.
Perhaps it is because he was an amateur, an outsider, that he made such an important breakthrough. Sometimes professionals can get so caught up in the minutiae of their field that they express their ideas in such obscure ways that they actually create artificial barriers to newcomers.
And at times I would swear that some mathematicians are deliberately making their subject matter more obscure just to try to impress people.
It’s certainly refreshing to read these clear explanations from Kalid, who’s goal is just like Argand’s - to introduce clarity to help others understand.
Maths is actually much easier than most people think it is, it just takes a good teacher to realise that.
Then take a bow, Kalid.
Hi,
there must be a typo here, just want to confirm:
" Grid system: Go 3 units east and 4 units north
Polar coordinates: Go 5 units at an angle of 71.56 degrees"
*polar coordinates is supposed to go 5 units then angle of 53.13 degrees??
I’m just a little bothered about this.
Other than that, the intuition of rotating was the only explanation in which I understood e^i… thank you so much !
Hi Tors, thanks for the correction! Yep, that was a typo, it should be atan(4/3) = 53.13 degrees. Article is updated :). Glad the article helped with e^i, thinking through imaginary exponents is a great test of intuition.
The statement, remember to set your calculator in radians, ruins the mathematical statements, for me. Making it worse, further mathematical statements are made without adding that instruction yet still implying it. Are the mathematical statements true only with that note added? Or is it more like an instruction for how to use a device? The note, itself, seems to lack mathematical rigor and thus seems to be outside the formal mathematical text.