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I am asking this question as a response to reading two different questions:

Is it ever Pi time? and Are complex number real?

So I ask, is it ever $i$ time? Could we arbitrarily define time as following the imaginary line instead of the real one?

(NOTE: I have NO experience with complex numbers, so I apologize if this is a truly dumb question. It just followed from my reading, and I want to test my understanding)

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OctaviaQ
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    If you take the Argand viewpoint, you're then asking if "time" can have more than one dimension... if it does have two dimensions, then there's a way to interpret $i$... – J. M. ain't a mathematician Oct 14 '11 at 02:21
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    "following a complex line" - it's a "real line", but a "complex plane"... ;) – J. M. ain't a mathematician Oct 14 '11 at 02:27
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    If you think of the complex plane, at 12:00 both hands point to $i$. – Ross Millikan Oct 14 '11 at 02:33
  • @J.M -- I thought that complex numbers formed a plane, but that imaginary numbers were on a line (and the plane came from combining the imaginary with the real)... Oh, maybe I should have said "imaginary line?" – OctaviaQ Oct 14 '11 at 02:33
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    If time were imaginary some things would get weird. For example, if you accelerate at $50 \frac{mi}{hr^2}$ and the hours were imaginary numbers, then the $hr^2$ would end up flipping the sign of your acceleration. We know this doesn't happen physically, so imaginary time in this sense won't work. – tomcuchta Oct 14 '11 at 02:51

2 Answers2

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Yes; if you'll refer to the Wikipedia page on Imaginary Time, you'll see that imaginary time is a useful concept in quantum mechanics.

EDIT: As an aside, your question is very far from dumb. The desire to generalize anything and everything to complex numbers (and, for the number theorists out there, to $p$-adic numbers) has shown, historically, to be a natural and very fruitful instinct.

Robert Haraway
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In the Wikipedia article titled Paul Émile Appell, we read that "He discovered a physical interpretation of the imaginary period of the doubly periodic function whose restriction to real arguments describes the motion of an ideal pendulum."

The interpretation is this: The real period is the real period. The maximum deviation from vertical is $\theta$. The imaginary period is what the period would be if that maximum deviation were $\pi - \theta$.

Michael Hardy
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