On-line multiplication in real and complex base

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1 On-line multiplication in real complex base Christiane Frougny LIAFA, CNRS UMR place Jussieu, Paris Cedex 05, France Université Paris 8 Christiane.Frougny@liafa.jussieu.fr Athasit Surarerks Computer Engineering Department Faculty of Engineering Chulalongkorn University Phayathai road, Bangkok 10330, Thail athasit@cp.eng.chula.ac.th Abstract Multiplication of two numbers represented in base is shown to be computable by an on-line algorithm when is a negative integer, a positive non-integer real number, or a complex number of the form Ô Ö, where Ö is a positive integer. 1 Introduction On-line arithmetic, introduced in [24], is a mode of computation where opers results flow through arithmetic units in a digit serial manner, starting with the most significant digit. To generate the first digit of the result, the Æ first digits of the opers are required. The integer Æ is called the delay of the algorithm. This technique allows the pipelining of different operations such as addition, multiplication division. It is also appropriate for the processing of real numbers having infinite expansions : it is well known that when multiplying two real numbers, only the left part of the result is significant. On-line arithmetic is used for special circuits such as in signal processing, for very long precision arithmetic. One of the interests of on-line computable functions is that they are continuous for the usual topology on the set of infinite sequences on a finite digit set. To be able to perform on-line computation, it is necessary to use a redundant number system, where a number may have more than one representation (see [24, 5]). An example of such a system is the so-called signed-digit number system. It is composed of an integer base a signed-digit set of the form, with ½. In this sytem addition can be performed in constant time in parallel [1, 4]. On-line multiplication is also feasible [24, 5]. Parallel addition is used internally in the multiplication algorithm. On-line algorithms for addition, subtraction, multiplication, division square-root in integer base are well studied, see [2, 6, 19]. In this paper we study the multiplication in base when is a negative integer, or a non-integer real number, or a complex number of the form Ô Ö. It is known that any real number can be represented in negative integer base without a sign [12, 13, 15], that, with a signed-digit set, addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7]. We show that the on-line multiplication used in the signed-digit number system can be applied in negative base. Negative base is related to some complex number systems, see below. When the base is a real number ½, by the greedy algorithm of Rényi [23], one can compute a representation in base of any real number belonging to the interval ¼ ½, called its -expansion, where the digits are elements of the canonical digit set ¼ if is not an integer, or of ¼ ½ if is an integer. In such a representation, when is not an integer, not all the patterns of digits are allowed (see [21] for instance). For instance in base ½ Ôµ Ôthe golden ratio, the - expansion of the number Ü is ½¼¼½¼¼¼. The pattern ½½ is forbidden. Different -representations of Ü are ¼½½½¼¼¼, or ½¼¼ ¼½µ ¼½µ ¼½µ for instance. This system is thus naturally redundant. Representation of numbers in non-integer base is encountered for instance in coding theory, see [10], in the modelization of quasicrystals, see [3]. In [8] we have studied the problem of the conversion in a real base from a digit set ¼ with to the canonical digit set, without changing the numerical value. Addition multiplication by a fixed positive integer are particular cases of digit set conversion. We have proved that the digit set conversion is on-line computable. 1

2 Moreover, if the base is a Pisot number 1 this conversion is realizable by an on-line finite automaton. This means that in the Pisot case, digit set conversion needs only a finite storage memory, independent of the size of the data. However, addition is not computable in constant time in parallel. In this work, we prove that multiplication of real numbers represented in real base is on-line computable with a certain delay explicitly computed. Our algorithm can be applied to the particular case that is an integer. With digit set ¼, this is the Carry- Save representation. The delay of our on-line multiplication algorithm in the Carry-Save representation is greater than the delay of on-line multiplication in the signed-digit representation, but the Carry-Save representation takes less memory. We then consider the Knuth complex number system, which is composed of the base Ô Ö, with Ö an integer, canonical digit set ¼ Ö ½. Every complex number has a representation in this system [12]. This allows a unified treatment of the real imaginary parts of a complex number. We show that in the Knuth number system with a signed-digit set, with Ö Ö ½, multiplication is on-line computable. It is known that addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7]. The case together with the signed-digit set, has been considered in [17, 16] with practical applications. Other complex numeration systems have been considered. For instance the Penney complex number system consists of the complex base ½ the canonical digit set ¼ ½. Every complex number is representable in this system, the representation is unique finite for the Gaussian integers [22]. It is shown in [25] that in base ½ with signed-digit set ½ ¼ ½ multiplication is online computable. The techniques are different from the ones used in the Knuth number system. 2 Preliminaries 2.1 On-line computability Let be two finite digit sets, denote by Æ the set of infinite sequences of elements of. Let ³ Æ Æ µ ½ µ ½ The function ³ is said to be on-line computable with delay Æ if there exists a natural number Æ such that, for each ½ there exists a function Æ such that 1 A Pisot number is an algebraic integer such that its algebraic conjugates are strictly less than ½ in modulus. The golden ratio the natural integers are Pisot numbers. ½ Æ µ, where Æ denotes the set of sequences of length Æ of elements of. This definition extends readily to functions of several variables. It is well known that some functions are not on-line computable, like addition in the binary system with canonical digit set ¼ ½. Addition is considered as a conversion from ¼ ½ to ¼ ½. Denote by Ú the infinite concatenation ÚÚÚ, by Ú Ò the word Ú concatenated Ò times. Since ¼½ Ò ¼ µ ½¼ ¼½ Ò ¼ µ ¼½ Ò ¼ for any Ò ½, one sees that the most significant digit of the result depends on the least significant digits of the input. Recall that a distance can be defined on Æ as follows: let Ú Ú µ ½ Û Û µ ½ be in Æ, Ú Ûµ Ö where Ö ÑÒ Ú Û if Ú Û, Ú Ûµ ¼ otherwise. The set Æ is then a compact metric space. This topology is equivalent to the product topology. Then any function from Æ to Æ which is on-line computable with delay Æ is Æ -Lipschitz, is thus uniformly continuous [8]. Let be a digit set. We say that multiplication is on-line computable with delay Æ in base on the digit set if there exists a function such that Æ Æ Æ Ü µ ½ Ý µ ½ µ Ô µ ½ Ô ½ ½ Ü ½ Ý which is on-line computable with delay Æ. Note that apriori, because of redundancy, the result of such a process is not unique, but the algorithms we shall consider later on are deterministic, thus compute a function. In the following, we will make the assumption that the opers begin with a run of Æ zeroes. This allows to ignore the delay inside the computation. 2.2 Number representation A survey on numeration systems can be found in [14, Chapter 7]. Let be a finite digit set of real or complex digits let be a real or complex number such that ½. A -representation of a real or complex number Ü with digits in is a finite or a right infinite sequence Ü µ Ò with Ü such that Ü È ½ Ò Ü. It is denoted by Ü Ò Ü ¼ Ü ½Ü µ In this paper we consider only representations of the form Ü µ ½ Æ representing a number Ü equal to Ƚ Ü. When a representation ends with infinitely many zeroes, it is said to be finite, the zeroes are usually omitted. 2

3 2.3 Signed-digit number system The base is a positive integer ½. With a signed-digit set of the form Ë, ½ the representation is redundant addition can be performed in constant time in parallel, is computable by an on-line finite automaton, [1, 4, 19]. 2.4 Negative base numeration systems Let the base be a negative integer ½. It is well known (see [12, 13, 15]) that any real number can be represented without a sign in base with digits from the canonical digit set ¼ ½. With a signed-digit set of the form Ì, with ½ the representation is redundant addition can be performed in constant time in parallel, is computable by an on-line finite automaton [7]. 2.5 Representations in real base Let be a real number ½, generally not an integer. Any real number Ü ¼ ½ can be represented in base by the following greedy algorithm, [23] : denote by by the integral part the fractional part of a number. Let Ö ¼ Ü È for ½ let Ü Ö ½ Ö Ö ½. Thus Ü Ü ½, where the digits Ü are elements of the canonical digit set ¼ if Æ, ¼ ½ otherwise. The sequence Ü µ ½ of Æ is called the -expansion of Ü. When is not an integer, a number Ü may have several different -representations on : this system is naturally redundant. The -expansion obtained by the greedy algorithm is the greatest one in the lexicographic order. It is shown in [8] that addition in real base is on-line computable. When the base is a Pisot number, addition is computable by an on-line finite automaton. 2.6 Knuth number system Here the base is a complex number of the form Ô Ö, where Ö is an integer. Any complex number is representable in base with digits in the canonical digit set ¼ Ö ½ (see [12, 11, 9]). If Ö is a square then every Gaussian integer has a unique finite representation of the form ¼ ½,. Since Ö, we have Þ Thus, ½ ½ Öµ Ô Ö ¼ Þµ Ü ½ Öµ ½ Öµ ½ Þµ Ý Ô Ö ¼ ½ Öµ ½ So the -representation of Þ can be obtained by intertwinning the Öµ-representation of Ü the Öµrepresentation of Ý Ô Ö. Most studied cases are ¼, strongly related to base, Ô ¼ ½ ([12, 13, 20, 17, 16]). With a signed-digit set of the form Ê, Ö Ö ½, addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7]. 3 On-line multiplication algorithm in integer base 3.1 Classical on-line multiplication algorithm First we recall the classical algorithm for on-line multiplication in the signed-digit number system, see [24, 5]. We give our own presentation. THEOREM 1 Multiplication of two numbers represented in integer base ½ with digits in Ë, ½, is computable by an on-line algorithm with delay Æ, where Æ is the smallest positive integer such that Æ ½µ ½ (1) Proof. Denote by the partial sum È ½ Ü ( respectively È ), denote by round(þ) the closest integer to Þ Classical on-line multiplication algorithm Å Ë. Input: two sequences Ü Ü µ ½ Ý Ý µ ½ of Ë Æ such that Ü ½ Ü Æ ¼ Ý ½ Ý Æ ¼. Output: a sequence Ô Ô µ ½ of Ë Æ such that Ƚ Ô È ½ Ü È ½ Ý begin 1. Ô ½ ¼,..., Ô Æ ¼ 2. Ï Æ ¼ 3. Æ ½ 4. while Æ ½ do 5. Ï Ï ½ Ô ½ µ Ý Ü ½ 6. Ô round Ï µ 7. ½ end First let us prove by induction that for any Æ Ï È ½ 3

4 By Line 5 of the algorithm, Ï ½ Ï ½ Ô ½ µ Ý Ü ½ µ By induction hypothesis, Ï ½ ½ È ½ Ô ½ Ý Ü ½ µ, the result follows from the fact that ½ Ü, the similar relations for È. Thus at step Ò Æ, Ò Ò Ò Ï Ò È Ò ½ Ò Ï Ò Ô Ò µ È Ò. Since Ï Ò Ô Ò ½, Ò Ò È Ò Ò the algorithm is convergent. The sequence Ô ½ Ô Ò is a -representation of the most significant half of the product Ò Ò. Now it remains to prove that the digits Ô s computed in Line 6 of the algorithm are in the digit set Ë. It is enough to show that Ï ½. From Line 5 the fact that ½ are less than follows that Æ ½µ Ï Æ ½µ ½ by (1). Note that additions multiplications by a digit in Line 5 are performed in constant time in parallel. COROLLARY 1 The delay Æ for Algorithm Å Ë takes the following values. If ½, Æ. If, Æ. If then Æ. If if ½, Æ ½. 3.2 On-line multiplication in negative base Now we consider the case where the base is a negative integer ½ the digit set is Ì, ½. PROPOSITION 1 Multiplication of two numbers represented in negative base ½ digit set Ì, ½, is computable by the classical on-line algorithm Å Ë with delay Æ, where Æ is the smallest positive integer such that Æ ½µ ½ (2) Proof. At step Ò Æ of the algorithm we get that Ò Ò È Ò Ò Ï Ò Ô Ò µ. Since Ï Ò Ô Ò ½, Ò Ò È Ò Ò To show that Ô is in Ì, we have to show that Ï ½. From Line 5 the fact that ½ are less than Æ ½µ by (2). follows that Ï Æ ½µ ½ 4 On-line multiplication in real base Let ¼ be a digit set containing, that is,. THEOREM 2 Multiplication of two numbers represented in base with digits in is computable by an on-line algorithm with delay Æ, where Æ is the smallest positive integer such that Æ ½ (3) ½µ Proof. Clearly a number Æ satisfying (3) exists, because. Real base on-line multiplication algorithm ÅÊ. Input: two sequences Ü Ü µ ½ Ý Ý µ ½ of Æ such that Ü ½ Ü Æ ¼ Ý ½ Ý Æ ¼ 2. Output: a sequence Ô Ô µ ½ of Æ such that Ƚ Ô È ½ Ü È ½ Ý begin 1. Ô ½ ¼,..., Ô Æ ¼ 2. Ï Æ ¼ 3. Æ ½ 4. while Æ ½ do 5. Ï Ï ½ Ô ½ µ Ý Ü ½ 6. Ô Ï 7. ½ end As above, at step Ò Æ, Ò Ò È Ò Ò Ï Ò Ô Ò µ. Since ¼ Ï Ò Ô Ò ½ we get ¼ Ò Ò È Ò Ò the algorithm is convergent. It remains to prove that the Ô s are in, i.e. ¼ Ô. It is enough to show that Ï ½. From Line 5 the fact that ½ are less than by (3). Ï Æ ½µ Æ ½µ ½ follows that 2 This implies that È ½ Ü È ½ Ý are in ¼ ½. 4

5 Ô EXAMPLE 1 Let ³ ½ µ be the golden ratio. Then the canonical digit set is ³ ¼ ½. Multiplication on ³ is on-line computable with delay Æ by (3). This delay does not seem to be optimal, we conjecture that is optimal. We give in Table 1 below the detail of a computation with Ü Ý.¼ ½¼½¼½. The numerical value of Ü is equal to ³ ³ ½ ³ ³ µ. The result is Ô.¼ ½¼ ½¼½¼¼¼½¼¼¼¼½. Computations are represented in base ³, in a symbolic way. Ï µ ³ Ô 6.¼¼¼¼¼½ ¼ 7.¼¼¼¼½ ¼ 8.¼¼½¼¼¼½¼¼½ ¼ 9.¼½¼¼¼½¼¼½ ¼ 10.½¼½¼¼¼½¼¼¼¼½ ¼ 11 ½.¼½¼¼¼½¼¼¼¼½ ½ 12.½¼¼¼½¼¼¼¼½ ¼ 13 ½.¼¼¼½¼¼¼¼½ ½ 14.¼¼½¼¼¼¼½ ¼ 15.¼½¼¼¼¼½ ¼ 16.½¼¼¼¼½ ¼ 17 ½.¼¼¼¼½ ½ 18.¼¼¼½ ¼ 19.¼¼½ ¼ 20.¼½ ¼ 21.½ ¼ 22 ½.¼ ½ 23.¼ ¼ Table 1. On-line multiplication in base ³ with delay 5 Note that the result is not the greedy -expansion in general. For instance with Ü.¼ ½¼½¼¼½¼½¼½¼½¼½ Ý.¼ ¼½¼½¼¼½¼½¼½¼½¼½, Algorithm ÅÊ gives the result Ô.¼ ½½ ½¼½¼¼¼¼½½¼¼¼¼½¼½¼¼¼½¼¼½, which contains the forbidden pattern ½½. Ô EXAMPLE 2 Let µ be the square of the golden ratio. Then the canonical digit set is ¼ ½. Multiplication on is on-line computable with delay Æ. This delay is optimal for our algorithm : suppose that the delay is achievable, take Ü Ý.¼¼. The result given by Algorithm ÅÊ would then be Ô.¼¼¼½¼ ¼½¼½½¼½½ which is not on the alphabet. 5 Carry-Save versus Signed-Digit In this section is an integer ½. Take ¼, then the representation on is redundant. We call it the Carry-Save representation, because it is used in computer arithmetic under that name in the case that for internal additions in multipliers, see [18]. By the real base algorithm ÅÊ, multiplication in base 2 on ¼ ½ is on-line computable with delay Æ. This delay is optimal, as shown by the following example. Suppose that the delay is 2, take Ü ¼¼ Ý ¼¼½. The result computed by the algorithm would be equal to Ô ¼¼¼½ ¼½. If, multiplication on ¼ is on-line computable with the optimal delay Æ. Relation (3) is never satisfied for integer ½, which is not surprising because it is known that multiplication in integer base on the canonical digit set is not on-line computable. Internal additions multiplications by a digit in Algorithm ÅÊ can be performed in parallel when is an integer. This is well known when. We give below the algorithm for addition in the general case. PROPOSITION 2 Addition in the Carry-Save representation can be performed in constant time in parallel. Proof. Input: Ü Ò ½ Ü ¼ Ý Ò ½ Ý ¼ with Ü Ý in ¼ for ¼ Ò ½. Output: Ò ¼ with in such that Ü Ý ¼Ò ¼Ò ½ ¼Ò ½ begin 1. In parallel for ¼ Ò ½ do 2. Þ Ü Ý 3. if Þ then ½ ; Ö ¼ 4. if Þ ½ then if Þ ½ then ½ ; Ö ½ else ½ ½; Ö ½ 5. if Þ ( ½) then ½ ½; Ö 6. if Þ then ½ ½; Ö ¼ 7. if Þ ½ then if Þ ½ then ½ ½; Ö ½ else ½ ¼; Ö ½ 8. if ¼ Þ then ½ ¼; Ö Þ 9. Ö 10. Ò Ò end Clearly, Ü Ý ¼Ò ¼Ò ½ ¼Ò ½ 5

6 One has to prove that the digits s are elements of. Since ½ Ö ½ ¼ then ½ ½. The case ½ can happen only if Ö ½ ¼, which is impossible. The case ½ can happen only if Ö ½, which is impossible as well. The delay for multiplication in the signed-digit representation is better than in the Carry-Save representation, but note that the digit-set in the signed-digit representation has cardinality ½, to be compared to card µ ½, the digits in being nonnegative take less memory to be stored. 6 On-line multiplication in the Knuth complex number system THEOREM 3 Multiplication of two complex numbers represented in base Ô Ö, with Ö an integer, digit set Ê, Ö Ö ½, is computable by an on-line algorithm with delay Æ, where Æ is the smallest odd integer such that Ö Ö Æ ½ ½ Ö (4) ½µ Proof. On-line multiplication algorithm Å. Input: two sequences Ü Ü µ ½ Ý Ý µ ½ of Ê Æ such that Ü ½ Ü Æ ¼ Ý ½ Ý Æ ¼. Output: a sequence Ô Ô µ ½ of Ê Æ such that Ƚ Ô È ½ Ü È ½ Ý begin 1. Ô ½ ¼,..., Ô Æ ¼ 2. Ï Æ ¼ 3. Æ ½ 4. while Æ ½ do 5. Ï Ï ½ Ô ½ µ Ý Ü ½ 6. Ô Ò Ï µµ Ï µ ½ 7. ½ end The digit Ô will belong to Ê if Ï µ ½. By Line 6, for all, Ï Ô µ ½ Ï Ô µ Ï µ. Thus, by Line 5, Ï µ Ô Ö Ï ½ µ µ ½ µ Ô Ö Ï µ µ ½ µ First suppose that Æ is odd. Then µ Ö Æ ½ Ö ½µ µ Ô Ö Ö Æ ½ Ö ½µ the same holds true for ½. Thus Ï µ Ö Ö Æ ½ Ö ½µ ½ by (4). Suppose now that a better even delay Æ ¼ could be achieved. Then thus This delay will work if µ µ Ô Ö Ö Æ¼ Ö ½µ Ö Æ¼ Ö ½µ Ï µ Ö Ö ½µ Ö Æ¼ Ö ½µ Ö Ö ½µ ½ Ö Æ¼ Ö (5) ½µ Suppose that the delay in (4) is of the form Æ ½ the delay in (5) is of the form Æ ¼ ¼, set Ö ½µ ½ Öµ Then is the smallest positive integer such that ÐÓ µ ÐÓ Öµ ¼ is the smallest positive integer such that obviously ¼. Since for Ò Æ Ï Ò ¼ ÐÓ Ö ½µµ ÐÓ Öµ Ò Ò È Ò Ò Ï Ò Ô Ò µ Ï Ò Ô Ò µ ½ Ô Ö Ô Ò µ Ï Ò µ ÔÖ Ö Æ ½ Ö ½µ the algorithm is convergent, Ô ½ Ô Ò is a - representation of the most significant half of Ò Ò. 6

7 COROLLARY 2 The delay Æ for Algorithm Å takes the following values. If Ö ½, Æ. If Ö or Ö Ö ½ then Æ. If Ö ½¼ then Æ. In the other cases, for Ö ½¼ the delay is Æ. EXAMPLE 3 Let Ê ½ ¼ ½. By Corollary 2 the delay Æ is equal to 5. Let Ü.¼ ½¼½¼½ Ý.¼ ½½¼¼½½. The result is Ô.¼ ½¼ ½½½½ ½½ ½ ½ ½. Ï µ Ô 6.¼¼¼¼¼½ ¼ 7.¼¼¼½½½ ¼ 8.¼¼½½½ ¼ 9.¼½½½ ½½ ¼ 10.½½½½¼¼¼¼ ½ ¼ 11 ½.½½½¼½¼ ½ 12 ½.½½ ½½ ½ ½ ½ ½ ½½ ½ 13 ½.½ ½½ ½ ½ ½ ½ ½½ ½ 14 ½. ½½ ½ ½ ½ ½ ½½ ½ 15 ½.½ ½ ½ ½ ½ ½½ ½ 16 ½.½½½½½½ ½ 17 ½. ½ ½ ½ ½½ ½ 18. ½ ½ ½ ½½ 19 ½.½½½½ ½ 20 ½. ½ ½½ ½ Table 2. On-line multiplication in base with delay 5 Acknowledgements We are pleased to thank the anonymous referee who indicated us references [17] [16]. References [1] A. Avizienis, Signed-digit number representations for fast parallel arithmetic. IRE Transactions on electronic computers 10 (1961), [2] J.-C. Bajard, J. Duprat, S. Kla J.-M. Muller, Some operators for on-line radix 2 computations. J. of Parallel Distributed Computing 22 (1994), [3] Č. Burdík, Ch. Frougny, J.-P. Gazeau R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. of Physics A: Math. Gen. 31 (1998), [4] C.Y. Chow J.E. Robertson, Logical design of a redundant binary adder. Proc. 4th Symposium on Computer Arithmetic, I.E.E.E. Computer Society Press (1978), [5] M.D. Ercegovac, On-line arithmetic: An overview. Real time Signal Processing VII SPIE 495 (1984), [6] M.D. Ercegovac, An on-line square-rooting algorithm. Proc. 4th Symposium on Computer Arithmetic, I.E.E.E. Computer Society Press (1978). [7] Ch. Frougny, On-line finite automata for addition in some numeration systems. Theoretical Informatics Applications 33 (1999), [8] Ch. Frougny, On-line digit set conversion in real base. Theoret. Comp. Sci. 292 (2003), [9] W. Gilbert, Radix representations of quadratic fields. J. Math. Anal. Appl. 83 (1981), [10] W.H. Kautz, Fibonacci codes for synchronization control. I.E.E.E. Trans. Infor. Th. 11 (1965), [11] I. Kátai J. Szabó, Canonical number systems. Acta Sci. Math. 37 (1975), [12] D.E. Knuth, An imaginary number system. C.A.C.M. 3 (1960), [13] D.E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 2nd ed., Addison- Wesley, [14] M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, [15] D.W. Matula, Basic Digit Sets for Radix Representation. J.A.C.M. 29 (1982), [16] R. McIlhenny, Complex number on-line arithmetic for reconfigurable hardware : algorithms, implementations, applications. PhD Dissertation, Computer Science Department, UCLA, [17] R. McIlhenny M.D. Ercegovac, On-line algorithms for complex number arithmetic. Proc. 32nd Asilomar Conference on Signals, Systems, Computers (1998). [18] J.-M. Muller, Arithmétique des ordinateurs, Masson, [19] J.-M. Muller, Some characterizations of functions computable in on-line arithmetic. I.E.E.E. Trans. on Computers 43 (1994),

8 [20] A.M. Nielsen J.-M. Muller, Borrow-Save Adders for Real Complex Number Systems. In Proceedings of the Conference Real Numbers Computers, Marseilles, 1996, [21] W. Parry, On the -expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), [22] W. Penney, A binary system for complex numbers. J.A.C.M. 12 (1965), [23] A. Rényi, Representations for real numbers their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), [24] K.S. Trivedi M.D. Ercegovac, On-line algorithms for division multiplication. I.E.E.E. Trans. on Computers C 26 (1977), [25] A. Surarerks, On-line arithmetics in real complex base. Ph.D. Dissertation, University Paris 6, LIAFA report 2001/06, june