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Am I right to say that $e^{ix}$, where $x$ is the angle in a unit circle, is just the name of a point on the unit circle corresponding with some angle?

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Blue
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LuckyHomer
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    Well, it's not just "a" point corresponding to "some" angle ... it's the point corresponding to that angle, $x$. – Blue May 30 '19 at 02:38
  • Personally, I'd say that $e^{ix}$ is "just the name" of the complex number $\lim_{n\to\infty} 1+\frac{ix}{1!}+\frac{(ix)^2}{2!}+\cdots+ \frac{(ix)^n}{n!}$. –  May 30 '19 at 02:45
  • What do you mean by "just the name"? In my mind, $e^{ix}$ is what you get when you plug $ix$ into the Taylor series for $e^z$. It is true that this turns out to be a point on the unit circle, but that is not obvious in advance. – littleO May 30 '19 at 02:47
  • I tried mapping the unit circle a cartesian coordinate system and found that really the imagination term was invented to identify a section of base 10 number system and the the real term identifies the term within that section – LuckyHomer May 30 '19 at 03:30
  • It is just a name i think, but the object is mapped to 2 properties, angular frequency and time – LuckyHomer May 30 '19 at 03:32
  • So really the term is just a symbol which identifies a number within a section – LuckyHomer May 30 '19 at 03:34
  • @littleO and do you know how the Taylor series was invented? Or – LuckyHomer May 30 '19 at 03:37
  • Or do you only know whats its for and how to use it? – LuckyHomer May 30 '19 at 03:37

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You're correct in saying that, in the Argand plane, the point corresponding to the complex number $e^{i\theta}$ certainly does lie on the unit circle (centered at the origin) at angle $\theta$ from the $x$-axis, measured anticlockwise.

However, it's not the name of any point or anything in any plane other than the Argand one. And even though it is the name of the point on the Argand plane, it's not "just" that point. It also happens to be a complex number, and as such you can do a lot of cool mathematics with it.

auscrypt
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  • A.k.a. the Gauss-Argand plane. – DanielWainfleet May 30 '19 at 03:52
  • Why were complex number invented may i ask? And what was the process on invention? – LuckyHomer May 30 '19 at 04:00
  • e is the name given to some quantity, like that of 1 or 2 or 3 – LuckyHomer May 30 '19 at 04:01
  • Things like complex number are just names given to something. How was this something invented or discovered? – LuckyHomer May 30 '19 at 04:02
  • I can assign delboy = epi^2 But what is the significance of epi^2 – LuckyHomer May 30 '19 at 04:03
  • Thats what im trying to say – LuckyHomer May 30 '19 at 04:03
  • arguably mathematics wasn't "invented" but discovered; it's just a natural consequence of mathematics i guess. i mean i can see why people would disagree with the 'natural' part. unfortunately this delboy stuff isn't very natural. It's fairly obviously you made that one up on the spot without much foresight, without practical considerations, or anything like that. Feel free to google around for practical and theoretical applications of complex numbers if you want – auscrypt May 30 '19 at 04:05
  • The symbols were invented to create order, since the tendency of the universe to increase in disorder. Though i agree that mathematics can be discovered – LuckyHomer May 30 '19 at 04:07
  • Because symbols were initially representative of the naturally occurring relations from observation – LuckyHomer May 30 '19 at 04:09