Oh, I wasn't suggesting all irrationals could be defined as a limit. I meant "an irrational" -- meaning that there are one or more irrationals that could be defined in this way. Then it struck me that there was an ancient method for finding an approximation to pi that can be extended to an...
OK. This is becoming clearer. I was tricked by the seemingly illogical impossibility of interleaving two continuous series, but I guess "common sense" breaks down when you start to consider infinite numbers and infinitesimals.
Could an irrational be considered to be defined as the limit of...
Thanks guys.
So, between the rationals a and b, there exists an infinite number of other rationals, as well as infinity of irrationals. Two series interlaced within the series of the reals. Yes?
Hmmm... I thought that that was the definition of a continuum. No?
Between *any* two rationals is another, third, rational. So between the third rational and one of the first two is yet another, fourth, rational. Etc. Ad infinitum.
Rational numbers are those that can be represented as a/b.
It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them.
\frac{X+Y}{2} = \frac{ad+bc}{2bd}...
Ah! :)
The equation I wrote down was as follows:
m\theta'' = -mg\frac{3}{\pi}R\theta
But perhaps that is incorrect? Perhaps the m on the left hand side should be mR^2k^2 (where R.k is the distance between the centre of mass and the point of contact with the floor)?
This would imply...
Hmmm... that's what I thought.
I started by calculating the position of the centre of mass for an arch, and then using the limiting case when the inner radius approaches the outer radius. I calculated the limiting value for this as (3R)/π (i.e. approximately 95% of the full radius).
When you...
Hi all,
I'm trying to calculate the rocking frequency of a half-cylindrical arch. That is, a half-cylinder, that has had a smaller half-cylinder "bitten" out of it. If placed with the curved surface on the floor, it can be made to rock from side to side (sort of like an inverted pendulum).
In...
Thanks for your reply.
I think the problem is that I didn't really use the right initial conditions. When I tested the code with a sinusoidal initial condition, I got a standing wave as the result -- which means that the code is correct.
Then, when I expanded it to two spatial dimensions, I...
1D wave equation -- bizarre problem!
I am trying to write a solver for a 1D wave equation in python, and I have run into a bizarre problem that I just can't find a way out of.
I start with the wave equation, and then discretise it, to arrive at the following,
phi(i,j+1) = deltat2/deltax2...