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.
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).
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.
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.
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.
Hi Stephen, thanks for the comment. Great suggestion, I’m planning on moving to a more forum-based discussion, wordpress comments aren’t setup for these types of long-running conversations. I hope to have it available in the near future!
Nice article(s). This one was referenced on one of the Python dev lists.
I don’t have a real answer to Brandon (comment #2), but I’m sure that somewhere in my Mathematical travels (probably either Mac Lane’s Form and Function in Mathematics or Penrose’s Road to Reality) I came across a description of the use of complex analysis in number theory, specifically about the prime numbers. Another (now ancient) resource for those who want to understand math (without teaching you to do it, which is one of the real pluses to your work, Kalid, it does teach the “how” as well as the wonder!) is Newman’s 4-volume The World of Mathematics.
BTW, as many comments as you get, it would be useful to use a threading comment system. A couple of people in the 200s commented that they hadn’t checked if their question was already asked and answered – and who can blame them?
What an excellent job you’ve done! Really, good on ya! Can I send you some questions regarding Hamilton
’ s hodographs or other maths? I would love to learn more from you!
Intuition in mathematics is both the art and science behind the formulas. Btw are you a Gōdel fan? If so you should have a look at Nicolas of Cusa and Proclus on the philosophical side of things. Also, the previous email was wrong. This is the right one.
I have mixed feelings about your article:
It is outstanding, and it spoiled my weekend pet project
It is also fair to say that I have poor mathematical training and I would not reach even close to your explanations.
I had the same opinion as your: Formulas are not magical spells.
Unfortunately part of their beauty vanishes with their magic.
I am always skeptical about icons. I prefer to think of Euler, Gauss, and others as hard worker rather than magicians and I like to think they are close friends rather than icons. I guess I am a iconoclast
Whether the celebrities in Mathematics are close to us or not, this way of thinking motivates us to investigate their work harder and deeper in order to fully understand what appear to be magical.
Thanks Khalid for your wonderful explanation. I so much agree with you that intuition is not optional!
I have recently run into a problem/paradox that I cannot resolve.
Using these polar expressions -1 can be written in many forms:
e^(pii), e^(-pii), e^(3pi*i) etc, are all equal to -1.
The problem happens when we raise -1 to the power of i. The different representations of -1 give different results when raised to the i.
(-1)^i = (e^(pii))^i = e^(piii) = e^(-pi)
(-1)^i = (e^(-pii))^i = e^(-piii) = e^(pi)
(-1)^i = (e^(3pii))^i = e^(3pii*i) = e^(-3pi)
The correct value is supposed to be only the first one (at least Wolfram Alpha gives only the first value as a result). But how do we reconcile the rest?
Great question. Just like a square root can have multiple values ($\sqrt{9} = \pm 3$), the natural logarithm of a number like -1 can have several solutions. (I.e., what power should we use in $e^x$ to get to the value?)
