As an aside to "compound interest", there has been a vigorous debate recently among economists about _negative_ interest rates, which could potentially be used to combat deflation & unemployment. The Fed's current target interest rate is _minus 5%_ (= 2.1% + 1.3%*inflationrate - 2.0%*unemploymentgap) [Fed Reserve Bank of San Francisco Econ. Letter 6/14/2010]. One proposal from Harvard economist Greg Mankiw: "I can now state the proposed solution: Reduce the return to holding money below zero. Imagine that the Fed were to announce that it would pick a digit from 0 to 9 out of a hat. All currency with a serial number ending in that digit would no longer be legal tender. Suddenly, the expected return to holding currency would become negative 10 percent." http://seekingalpha.com/article/211590-what-s-ben-gonna-do-now Other proposals for "hot potato(e) money": http://www.clevelandfed.org/Research/commentary/2008/0408.cfm Keynes semi-seriously proposed that the govt fill up bottles with currency & bury them in mine shafts to be dug up by the public (by analogy with digging/panning for gold). -- I haven't yet heard of any economists proposing _imaginary_ or _complex_ interest rates, but the night is still young. At 09:56 PM 6/21/2010, 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