I'm not sure what you're looking for. From a math viewpoint, you can adjust the numerical speed by adjusting the time unit, so you are free to choose 1. From a pedagogical viewpoint, a speed of 1 is simple: With no compounding, simple interest of 100% for one year, in the four directions +-1 & +-i, gives a diamond with corners at 0, 1+i, 2, 1-i. Compounding at six-month intervals gives a kite, with corners .25, .75+i, 2.25, .75-i. Decreasing the compounding interval ultimately leads to corners 1/e, cos 1 + i sin 1, e, cos 1 - i sin 1. Some amount of math detail is needed to fill in the rigor, but it's plausible that the i direction result is on the unit circle, along an arc of distance 1 from the starting point (1,0). The reason I like this is it explains why pi, defined as the length of a semicircle of radius 1, appears in the equation e^2piI = 1. I haven't seen a "natural" explanation of this. Rich --------------- Quoting Joshua Zucker <joshua.zucker@gmail.com>:
What's the justification for the constant speed of 1 (radian per year, or unit per year) in this approach?
--Joshua
On Mon, Jun 21, 2010 at 9:56 PM, <rcs@xmission.com> wrote:
I've always liked the "compound interest" explanation for the Euler-DeMoivre formula: Imagine continuous compounding of $1 at 100X % interest for one year, giving $e^X. Visualize this as an expansion, beginning at the point (1,0), at the rate 100X% / year. Letting X = I, the expansion becomes a rotation. The number of years' rotation to go around the unit circle is just the length of the circle, which explains why e^2piI = 1.
Rich
---- Quoting Dan Asimov <dasimov@earthlink.net>:
Mike wrote:
<< Is there a geometric way to understand the Taylor series for sin and cos? The closest I've been able to find is a combinatorical explanation (below), but it doesn't seem to help much.
I'd love to know such a way to understand their Taylor series.
I recently gave a talk to a bunch of smart youngsters about complex numbers, and was unable to find a truly graceful way to explain why
exp(ix) = cos(x) + i sin(x),
(without deriving their Taylor series).
So if anyone knows a way to see this, I'd love to know it.
But of course that would require giving exp a meaning on the imaginaries.
Mike continued:
<< The paper "Objects of Categories as Complex Numbers" by Marcelo Fiore and Tom Leinster and says that under certain conditions, objects can behave as though they had complex cardinalities. It turns out that data types of typed programming languages satisfy the conditions; the data structure T = T^2 + T + 1 of trees with 2 (ordered), 1, or 0 children, behaves like "i". It is not the case that T^4 is isomorphic to the one-element set, but T^5 = T. Since it's a countably infinite set, this may seem obvious, but there's a way of constructing the isomorphism using only distributivity and the defining isomorphism above, so the cardinality is also preserved. . . . . . .
I don't really understand this, but it sounds fascinating.
--Dan
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