On cyclic numbers and an extension of Midy's theorem

In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic.Comment: 6 pages, aimed at undergraduate student

Detection of H-alpha emission from the Magellanic Stream: evidence for an extended gaseous Galactic halo

We have detected faint, diffuse H$\alpha emission from several points along the Magellanic Stream, using the Rutgers Fabry--Perot Interferometer at the CTIO 1.5-m telescope. At points on the leading edges of the H I clouds MS II, MS III, and MS IV, we detect H$\alpha emission of surface brightness $0.37 \pm 0.02$ Rayleighs, $0.21 \pm 0.04$ R, and $0.20 \pm 0.02$ R respectively, corresponding to emission measures of 1.0 to 0.5 \cmsixpc. We have observed several positions near the MS IV concentration, and find that the strongest emission is on the sharp leading-edge density gradient. There is less emission at points away from the gradient, and halfway between MS III and MS IV the H$\alpha surface brightness is$< 0.04$R. We attribute the H$\alpha emission at cloud leading edges to heating of the Stream clouds by ram pressure from ionized gas in the halo of the Galaxy. These observations suggest that ram pressure from halo gas plays a large role in stripping the Stream out of the Magellanic Clouds. They also suggest the presence of a relatively large density of gas, $n_{\rm H} \sim 10^{-4} cm^{-3}$, in the Galactic halo at $\sim 50$ kpc radius, and far above the Galactic plane, $b \sim -80\deg$. This implies that the Galaxy has a very large baryonic, gaseous extent, and supports models of Lyman-$\alpha and metal-line QSO absorption lines in which the absorption systems reside in extended galactic halos.Comment: 15 pages, aaspp latex, + 1 table & 3 figures. Accepted in A.J. Also available from http://www.physics.rutgers.edu/~bweiner/astro/papers