Message #1 Posted by Palmer O. Hanson, Jr. on 16 June 2005, 10:26 p.m.
In an earlier thread on results obtained with analysis of the Albillo matrices with the CC-40 Mathematics module I noted that as a result of the radix-100 mechanization of the CC-40 some numbers will have 13 decimal digit mantissas and others will have 14 decimal digit mantissas. Marcus von Cube asked for more information about radix-100. Valentin Albillo suggested that Marcus should read the paper "Mathematics Written in Sand" to see what Kahan thinks about radix-100 arithmetic and demonstrates some of its side effects. I can't second that recommendation. Much of what Kahan has to say about radix-100 is based on an example from a Visicalc program running on an IBM personal computer which yields incorrect results. My tests show that the radix-100 mechanization in the CC-40 (and in the TI-74) does NOT yield those incorrect results.
There is a small amount of material on radix-100 in the Compact Computer 40 User's Guide. The discussion of Internal Numeric Representation on Page F-2 begins with the paragraph
"The CC-40 uses radix-100 format for internal calculations. A single radix-100 digit ranges in value from 0 to 99 in base 10. The computer uses a 7-digit mantissa which results in 13 to 14 digits of decimal precision. A radix-100 exponent ranges in value from -64 to + 63 which yields decimal exponents from 10^-128 to 10^+126. The expnent and the seven digit mantissa combine to provide a decmal range from -9.9999999999999E+127 through -1.E-128; zero; and then +1.E-128 through +9.999999999999E+127. ..."
The remainder of the page continues on with more discussion of the internal representation including examples of some specific internal representations. It's more material than I want to retype. If you can't find a manual and want a copy of the page let me know. I could not find any mention of Radix 100 in the TI-74 User's Guide. Page A-33 of the TI-74 Programming Reference Guide contains a statement similar to that quoted above from the CC-40 manual.
Pages 6 and 7 of the Volume 9 Number 5 issue of TI PPC Notes provide some additional discussion of the CC-40 arithmetic as written by the CC-40 user community. To see the pages on-line go to Viktor Toth's Programmable Calculator web site at www.rskey.org/. Then go to the Library, to Texas Instruments, to PPC Notes and to Volume 9 Number 5.
Although I do not think the Kahan paper is very helpful to an understanding of radix-100 mechanizations I do recommend reading the paper. My only caveat is that the reader should recognize that the material in the paper is somewhat biased. Readers from the LOL community will see the bias as justifiable pride in a superior product -- the HP-15c. Readers from the dark side will see the paper as containing some useful material together with a lot of blatant salesmanship.
Finally, running the small version of Kahan's Paranoia analysis confirms that for the CC-40 the radix is 100, the precision (the number of radix positions) is seven and the machine has a guard digit for add/subtract.