In the 100I% case, with an uncompounded time interval of T, the multiplier is 1 + IT. The complex norm of the multiplier is 1 + T^2. If we divide this into N intervals for compounding, the norm of one interval of T/N is 1 + T^2 / N^2. For N intervals (total time T), the norm is the Nth power of the single-interval value, ( 1 + T^2 / N^2 ) ^ N. For large N, this is roughly 1 + T^2 / N + some O(1/N^2) stuff. The effect of large N, getting close to continuous compounding, is to make the norm, for the full time interval T, approach 1. Some fine print is required to make sure that the O(1/N^2) stuff doesn't mess up the limit being 1. [1] This tends to obscure the main point, that subdividing the time interval T makes the norm closer to 1, by a factor 1/N^2, overpowering the *N effect of having N intervals. This establishes that the norm is 1 for the whole duration. A similar argument establishes that the polygonal path of the compounded values (1 + IT/N)^K, K in [0...N], is roughly the unit circle, and the limiting result is an arc of length T along the circle. My personal opinion is that the rigor doesn't help understanding, but does need to be checked. Once you accept the compound interest viewpoint, the power series for sine & cosine come from imaginary & real parts of using the binomial theorem for (1 + I T / N) ^ N, as N->infinity. I've always been puzzled that the Greeks didn't come up with logs & exp, since they knew about log spirals. Rich [1] For suitable k & x, 1 + k x < (1+x)^k < 1 / (1 - kx). ------- Quoting Joshua Zucker <joshua.zucker@gmail.com>:
On Wed, Jun 23, 2010 at 5:54 PM, <rcs@xmission.com> wrote:
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.
I disagree - you have already set the interest at 100% of i per time unit, compounded continuously. The question is why, after all that compounding, the speed is still 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).
I think you need to do something equivalent to a second derivative somewhere in there to show that the magnitude of the velocity vector is always 1. If there's no need for calculus, I'd like to see some of the math detail you skipped there so I can understand how we know the limit point of this sequence will be 1 unit of arc length away from the starting point.
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.
Definitely: unit speed travel around the unit circle is what's going on here. The question I have is why 100i% interest compounded continuously results in a constant unit speed (since certainly 100% interest compounded continuously only has unit speed when you start!)
Thanks, --Joshua
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