• March 22nd, 2016

Assignment 9

Paper, Order, or Assignment Requirements

Written assignments For each assignment, do the starred exercise and one other exercise of your choice. The answer to the required exercise should be in WP, but calculations in other exercises can be by hand. ——————-

Assignment IX  In this assignment, work any two problems.  The mathematics required may be beyond what you have studied.  If so, please let me know and we can work out an alternative assignment.

 

  1. Show that a prime complex number is irreducible.

 

  1. Show that the Gaussian integers form a Euclidean domain. That is, show that, given two Gaussian integers z, m, there exist two others, q, r, such that z = qm + r and N(r)  < N(m).

 

  1. How do students in high school “understand” negative numbers? Do they understand why a negative times a negative is a positive?  Is such an understanding necessary?

 

  1. Prove the theorem that every bounded increasing sequence of real numbers has a limit number, using Dedekind’s cuts and also using Cantor’s fundamental sequences. Which proof is easier?

 

  1. Explain the differences between Cauchy’s definition of continuity on an interval and the usual modern definitions of continuity at a point.  Does a function that satisfies Cauchy’s definition satisfy the modern one for every point in the interval?  Does a function that satisfies the modern definition for every point in an interval satisfy Cauchy’s definition?

 

  1. Describe the effect of the publication of Nightingale’s pie charts, immediately and in the long term.

 

  1. Is the analytic form of non-Euclidean geometry as presented by Taurinus, Lobachevsky, and Beltrami a better way of presenting the subject than the synthetic form?  How can one make sense of a sphere of imaginary radius?

 

  1. Comment on the oft-repeated statement that Riemann’s work was a precursor of Einstein’s general theory of relativity.

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