Type: Seminar

Distribution: World

Expiry: 18 Mar 2016

Auth: seangard@cpe-58-173-78-158.cqzc1.ken.bigpond.net.au (sgar9702) in SMS-WASM

This week’s SUMS talk is by Mark Greenaway. Lunch in the form of pizza will be available afterwards. Abstract: In this talk we will construct the natural numbers, the integers and the rational numbers. We will then discuss the fact that the rational numbers are not complete, but have "holes", motivating the construction of the real numbers. It will be shown that while the rational numbers are countable, the real numbers are not. This forces us to approximate if we wish to perform calculations on a finite memory computer in finite time. The three major approaches to this problem will be introduced - fixed point, floating point and arbitrary precision. Of these, floating point is the one most programmers are likely to first encounter, and the one implemented in hardware that you can actually buy today. We will discuss the floating point numbers, along with the implications of their definition using some elementary results from numerical analysis. We will also discuss alternative representations, particularly arbitrary precision and continued fractions, along with their benefits and drawbacks. Time permitting, there will also be a discussion of unums, which break with previous ideas of floating point representation completely. As mathematics is not a spectator sport, all major results will be proven, assuming only the ZFC axioms. This talk will be accessible to anyone with a basic knowledge of high school mathematics and an inquiring mind.