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From what I have been reading about imaginary numbers, I've arrived at this summarization:

  1. Numbers describe the existence and count of objects.
  2. Imaginary numbers describe the rotation of these object(s).

So for example, if we have 2 pies, and both pies are rotated to an angle of i (90 degrees), we can describe both these properties of our pies with the equation -> pies = 2i. 2 pies, at angle i (or 90 degrees).

if I'm right, which I think don't think I am exactly, if we have 2 pies rotated at angle i^2 or 180 degrees, this would be equal to 2i^2 which equals -2. So does the negative in the context of this new equation not describe the existence, count, and rotation but now just describes the count and rotation of our pies?

-> Or does this just depend on whether we substitute i^2 for -1 or not, and whether we declare that, in this context, we are describing existence with a negative sign and or not angle?

I'm trying to get a fundamental, intuitive grasp on this concept, so if I'm harshly wrong, please explain in an intuitive way rather than mathematical?

dandev
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    You can think of numbers in general as describing scaling & rotations. A real number only has $\pm 1$ rotations. – copper.hat Feb 20 '19 at 04:32
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    @Dantebiase How do you define the addition, subtraction, multiplication, and division of complex numbers with your notion of rotations? – Mark Viola Feb 20 '19 at 04:34
  • @copper.hat but isn't that the point of imaginary numbers to describe the rotational axis of numbers in another dimension, because the number 1 can only be in the positive or negative direction but with "i" it can be up or down? – dandev Feb 20 '19 at 04:34
  • @MarkViola I can't yet, I'm looking to know if my intuition is at least on the right path, I just started really trying to understand this concept intuitively. – dandev Feb 20 '19 at 04:35
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    THIS might be useful. – Mark Viola Feb 20 '19 at 04:37
  • I suppose I would say the point of imaginary numbers is that for a suitable rotation we have $r^2 = -1$. – copper.hat Feb 20 '19 at 04:37
  • @MarkViola the notion I am referencing is based of the analogy and ideas posted after reading over: https://math.stackexchange.com/questions/199676/what-are-imaginary-numbers and https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ – dandev Feb 20 '19 at 04:38
  • Hi Mark! ${}{}$ – copper.hat Feb 20 '19 at 04:38
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    Hi Joe! How are you? – Mark Viola Feb 20 '19 at 05:32
  • @DanteBiase Your intuition about imaginary numbers describing "rotation" is analogous to looking at the complex plane. Instead of a coordinate plane composed of two real intersecting number lines, one line is real and the other is imaginary. In other words, lines of the form $z=a+0i$ are perpendicular (90 degrees) to the lines $z=0+bi$ – Andrew Feb 20 '19 at 07:21
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    Possible duplicate of What are imaginary numbers? . I think Clive's answer (the one with 750+ upvotes) addresses your question very nicely. – rschwieb Feb 20 '19 at 15:05
  • @rschwieb I referenced this in my comment above, this question is a continuation of that post and another resource, I’m not asking what is, I’m asking if I’m understanding right. – dandev Feb 20 '19 at 15:09
  • @DanteBiase Clive's answer more than confirms your second intuition point. IMO your first intuition point is not accurate or useful. So unless you want the answer to your question to be "your first point is not useful and your second point is confirmed everywhere else," I think it's a good candidate for duplication. – rschwieb Feb 20 '19 at 15:27
  • @rschwieb my first intuition point of general numbers is almost exactly the same as the Wikipedia definition, and my second intuition points is built upon the first, and furthermore, I have a follow up conflict question about my second point in the case it is right as you confirmed. So, I hear you but I disagree that this is essentially the same question. – dandev Feb 20 '19 at 16:16
  • @DanteBiase Yeah, written for a general audience, the wiki article has a pretty basic scope. Really, there isn't much utility for a precise idea of "number" in mathematics. We talk about more specific things. – rschwieb Feb 20 '19 at 16:27
  • @DanteBiase IMO, one could consider anything in any field a "number." From that perspective, fields are exactly the things that can be used to make coordinate axes for Pappian planes. That is not mentioned at all in the wiki article, but it is nevertheless a good version of what a number can do. – rschwieb Feb 20 '19 at 16:29
  • @rschwieb you gave me a lot to look up, I appreciate that. I just started trying to dig into the math world and redefine everything I think I know which is obviously not a lot given my question, so you might be/probably are right. – dandev Feb 20 '19 at 21:27
  • @DanteBiase It looks like I thought about it before in a solution: you might be interested in this. – rschwieb Feb 21 '19 at 18:34
  • @rschweib yeah I keep seeing the term of rings popping up to describe numbers and also especially the exact comparison to fields is made, again I’m going to have to dig at both answers right now because at first glance, destining rings as numbers does not make any profound sense to me. But again I didn’t start digging yet. If you suggest any intuition based resources to understand even basic things like this please let me know. I will be looking at all this. Thanks. – dandev Feb 21 '19 at 19:15

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