# Mathematical Digest

### Short Summaries of Articles about Mathematics

in the Popular Press

"A Lucid Interval," by Brian Hayes. *American Scientist*, November-December 2003, pages 484-488.

As accurate as digital computers are, round-off errors are hard to avoid when performing arithmetic computations with non-integer numbers. Include division, and approximation is almost inevitable. Floating-point arithmetic can't avoid round-off errors either: if a number cannot be exactly represented using this notation, it must be approximated by its closest floating-point neighbor. The author, Brian Hayes, suggests that interval arithmetic---using intervals instead of single numbers---may be a way to determine more accurate answers, or at least a more accurate range of results. For example, a number *x* could be represented by the interval *[a,b]* consisting of the two floating-point numbers *x* falls between. While certain aspects of interval arithmetic present challenges---including possible division by zero, or comparing intervals---Hayes points to people who suggest ways to solve or manage these problems. Currently, the computer hardware needed to support interval arithmetic has yet to be developed, and acceptance of interval-arithmetic standards has not been adopted by any standards-setting organizations. In spite of this, interval methods have been successfully applied to a number of problems, including research relating to Newton's gravitational constant. And while perhaps not the best solution to practical problems requiring a single result, Hayes suggests that interval arithmetic could be at least part of the means to more accurately solve real-world problems.

*--- Claudia Clark*