Caveat Reador: You may question your understanding of numbers after you read this.
When a visitor asked the Smithsonian docent how old was that dinosaur, she was told three million years, four months, five weeks and six days. How can you be so certain, sez she, as she flashed her puzzled eyes? Well, sez he, when I joined the Museum the bones were dated as three million years, and I’ve been here for four months, five weeks, and six days.
Thus emerges a major quirk in human gullibility: we are taken in by numbers and stats that have no grounding in measurable reality. The highly paid forecasters in many domains have discovered the advantage of avoiding bland integers. Predicted sales of, say, aluminum riding boots in, say, 2003, are $2,567.93. Gainsay me not! Who would pay you to estimate sales of $3,000?
Chuck Allison’s editorial (December 2001) reminded us of the complexities of mapping the real numbers to our pathetic finite floating-point registers. Chuck also asked how much real, pure mathematics must be “known” by computer scientists. In the applied fields tackled by the latter, many/most of our sums require only six decimal places to ensure that the bridge or building survives, absent a terrorist discontinuity.
Elsewhere it is Other. We all know about deterministic systems where infinitesimal errors in measuring/recording the initial parameters can generate Chaos.
The weird, ineluctable power of the real numbers, beyond your latest Intel:
There’s a real number in the interval [0,1] or indeed in any non-empty subset thereof that can usefully represent the whole history of this cosmos (even throw in an infinite number of other cosmoi) to any degree of accuracy allowed by Heisenberg.
Won’t bore you: for each particle, real or virtual, at each “moment” in time, imagine a huge but finite number describing its “state” warts and all.
Concatenate all these...voila!
I happen to know the magic number, but my lips and disks are sealed.
As I’m going to press, pressed beyond belief (at my age, deadlines are “winged chariots” hovering near), I picked up the fascinating exchanges between Tim Roberts and Mike Ring (“We Have Mail,” February 2002) on how to “beat” the bit/byte limitations.
Did I note a slight semantic slur? Numerology is not to be confused with Number Theory!
Well, astrology remains more amusing than astronomy?
There’s no doubt we can represent and manipulate numbers as strings to any degree of precision limited only by our finite storage devices. BCD (Binary Coded Decimal) strings and Mathematica give us the power to sum away without any wisdom filter.
Hate to recall that way-back in my EDSAC-1 days, we had to pre-run our algorithms on the old reliable double-barreled Brunsviga before risking expensive validation.
The amazing fact (against which you cannot fight) is that there are 1070 particles in the known universe each of which has a given state at t=after n big-bang fermo-seconds.
Seems we need but 137 bits to define each measurable time slice plus say 1012 bits for each particle state given the usual Feynman summations. Give or take a few, the real numbers can cope.
We holy-pure mathematicians can delve even deeper into Hyper-real numbers where non-zero infinitesimals fill the gaps between zero and epsilon! See, for example, Mathematics Magazine, MAA, December 2001.
Bridge builders and tomato genome-twisters daily use Euler’s fuzzy equations.
Cavet Comptor?
Stan Kelly-Bootle has been computing on and off since 1953 when he graduated from Cambridge University in Pure Mathematics and hacked on EDSAC I (the first true stored-program computer). He is a contributing editor for Linux Journal and a Jolt Judge for Software Development Magazine. With the demise of UNIX Review/Performance Computing, his 16-year-old Devil’s Advocate column has moved online to <www.sarcheck.com>. His many books include 680x0 Programming by Example, Mastering Turbo C, Lern Yerself Scouse, The Devil’s DP Dictionary, The Computer Contradictory, and Unix Complete. Under his nom-de-folk, Stan Kelly, his songs have been recorded by Cilla Black, Judy Collins, the Dubliners, and himself. Stan welcomes email via skb@atdial.net and his website <http://www.feniks.com/skb/>. Stan’s ramblings can also be found at <www.unixreview.com>.