SUMS: Greenaway -- How I learned to stop worrying and love floating point numbers

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.