Re: Code to translate to negative base...

*To*: mathgroup at smc.vnet.net*Subject*: [mg64591] Re: Code to translate to negative base...*From*: Cca <cca at gregosetroianos.mat.br>*Date*: Thu, 23 Feb 2006 00:34:51 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

>Does anyone have code to translate to a negative base? Surprinsingly enough, this is very simple, although not much known among students. I find the approach below conceptually very intuitive and ellegant: everything depends on a REMAINDER function. Suppose that you already have implemented the traditional algorithm that gives you the digits of a integer number N in base B, for B>0 in Integers. Then, just enter with a negative value for B and you will get what you want. Let me ellaborate on this a little. We can see the traditional algorithm as a Mathematica command, say (although below I´ll not give any formatting instructions) myBaseForm[A, B, r] where r is a REMAINDER FUNCTION, that is, a rule r[A, B] such that A is congruent to r[A,B] mod B. The Euclidean remainder -- the "standard" one -- is given by r[A_, B_] := A - Abs[B]*Floor[A/Abs[B]] or, using Mathematica Mod, r[A_, B_] := Mod[A, Abs[B]] For each A in Integers, this gives the UNIQUE number r such that A is congruent to r mod B and 0<=r<Abs[B]. So, using the notations above, he following would always return True: myBaseForm[A, B] == BaseForm[A, B, Mod[#1, Abs[#2]] & ] This means that what you want is just myBaseForm[A, -B] Here is a direct implementation of the ideas above: myBaseForm[a_, base_, r_:(Mod[#1, Abs[#2]] & )] := Reverse[Drop[(r[#1, base] & ) /@ FixedPointList[(#1 - r[#1, base])/base & , a], -2]] Some examples for comparison: IntegerDigits[200, 3] {2,1,1,0,2} myBaseForm[200, 3] {2,1,1,0,2} myBaseForm[21, -10] {1,8,1} Carlos César de Araújo Gregos & Troianos Educacional www.gregosetroianos.mat.br Belo Horizonte, MG, Brasil (31) 3283-1122