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!
Well that was easy. Awesome
Hey, Kalid,
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.
Dear Kalid,
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.
Good job!
Oswaldo
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?)
By convention, we decide on a ‘principal value’, more here: http://mathworld.wolfram.com/PrincipalValue.html
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