1299
IEEF TRANSACTIONS ON COMPUThRS, DECFMBER 1974
[2] C. V. Page, "Equivalence between probabilistic and deterministic sequential machines," Inform. Contr., vol. 9, 1966.
[3] G. H. Ott, "Reconsidering the state minimization problem for
stochastic finite state systems," in IEEE 8th Symp. Switching
and Automata Theory, Oct. 1966, pp. 267-273.
[4] M. Depeyrot, "Operand investigation of stochastic systems,"
Stanford Univ., Stanford, Calif., Ph. D. dissertation, May 1968.
[5] C. V. Page, "Strong stability problems for probabilistic sequential machines," Inform. Contr., vol. 15, pp. 487-509, 1969.
[61 A. Paz, Introduction to Probabilistic Automata. New York:
Academic, 1971.
Francois Bancllhon was born in France in
1948. He graduated from the Ecole des Mines,
Paris, France in 1970 and received the
Docteur-Ingenieur degree from the University of Paris, Paris. He is currently a
Ph.D. student at the University of Michigan,
Ann Arbor, on leave from IRILA, Domaine de
Voluceau, Rocquencourt, France.
In 1971 and 1972 he was with IRIA, Domaine de Voluceau, Rooquencourt, France.
Correspondence
Serial Binary Division by Ten
RICHARD L. SITES
Abstract-A technique is presented for dividing a positive binary
integer by ten, in which the bits of the input are presented serially,
low-order bit first. A complete division by ten is performed in two
word times (comparable to the time needed for two serial additions).
The technique can be useful in serial conversions from binary to
decimal, or in scaling binary numbers by powers of 10.
Index Terms-Binary division, binary-to-decimal conversion,
decimal floating point, serial arithmetic, serial division.
INTRODUCTION
In many applications, decimal numbers are stored in a machine
in binary form, making it necessary to convert between binary and
decimal representations, and perhaps to scale binary numbers during
internal arithmetic. A common operation in such applications is
multiplication or division of a binary number by ten. Multiplication
by ten can easily be done by shifting and adding (2 X B + 8 X B).
This paper deals with the inverse operation: division of a binary
number by ten, specifically in a bit-serial machine where the bits
of a number are presented in low-order bit first, the same order used
for serial addition. For simplicity, only positive integers are treated.
The problem, then, is to design a black box that accepts a serial
number B and produces two serial numbers Q = LB/10j and
R = B mod 10.
Manuscript received January 29, 1973; revised March 4, 1974. This
work was supported by the Fannie and John Hertz Foundation.
The author is with the Computer Science Department, Stanford
University, Stanford. Calif. 94305.
' [A] is the floor of A-the largest integer A.
REDUCTION TO DIVISION BY 5
The remainder R = B mod 10 can be expressed in terms of
division by 5:
R = B mod 10 = 2 X (LB/2j mod 5) + B mod 2.
This says that the remainder can be found by appending the loworder bit of B (= B mod 2), on the right of the remainder of dividing
the rest of B (= LB/2J) by 5.
In a similar way, the quotient Q = LB/10J can be expressed in
terms of dividing by 2 and then by 5: Q = LB/2 X 5J = LLB/2j/5j.
In a binary machine, LB/2J can be produced by just throwing away
the low-order bit of B.
The problem of division by ten is now reduced to the process of
saving the low-order bit of B for the remainder, then using the
quotient and remainder from dividing LB/2J by 5.
A LOOK AT MULTIPLICATION BY 5
Division by 5 is a process that "undergoes" multiplication by 5.
To gain some insight of how division by 5 works, it is useful to look
at multiplication by 5. The long multiplication of a number, A =
a4 a3 a2 al aO, by 5 looks like this:
x
a4 a3 a2 al aO
1 0 1
a4 a3 a2 al aO
a4 a3 a2 al aO
P = p7 p6 p5 p4 p3 p2 pl p0.
If we were given a known multiple of five, P = A X 5, and asked to
find A, we could immediately read off the two low-order bits of Athey are just the two low-order bits of P. Using al and aO, the next
two bits of A can be found by subtraction:
IEEE TRANSACTIONS ON COMPUTERS, DECEMBER 1974
1300
SERIAL BINARY DIVISION BY TEN
Fig. 1.
Divide by five circuit. The delay (Di and D2) outputs are
initially zero.
TABLE I
NEW VALUE OF R IF A 1-BIT IS INPUT; 0-BITS DO NOT CHANGE R
Bit Time
0 4 8
1 5 9
2 6 1O
3 7 11
p7 p6 p5 p4 p3 p2
-
al aO
Current
000 001
...
..
...
...
001
010
100
0o1
010
011
000
100
011
100
001
000
t7 t6 t5 t4a3 a2.
In turn, a3 a2 can be subtracted to find the last bit of A:
t7 t6 t5 t4
-
a3 a2
-
since B
L (B -B mod 5) /5J
- B mod 5
=
(B -B mod 5) /5,
is an exact multiple of five. For example,
L37/5i
2/5j = 35/5 = 7.
So if B mod 5 is known, the quotient Q = LB/5J can be found with
=
L37
-
two subtractors and two bit delays2 (Fig. 1).
DETERMINING B MODULO FIVE
The problem of finding B mod 5 can be solved with a 5-state
sequential machine. This is done by viewing B as the sum of the
powers of two that the individual bits represent, and using the
identity (X + Y) mod 5 = (X mod 5 + Y mod 5) mod 5. For example:
2 Edward S. Davidson has suggested that one of the serial subtractors
can be eliminated by using B mod 5 to set the initial states of the two
delay flip-flops and the "borrow" flip-flop in the remaining subtractor:
B mod 5
Borrow
D2
Di
0
1
2
3
4
0
1
0
1
1
0
o
0
0
1
1
0
1
1
The bits of a binary number have the following place values modulo 5:
b5
b4
b3
b2
bl
bO
64 mod 5 32 mod 5 16 mod 5 8 mod 5 4 mod 5 2 mod 5 1 mod 5
...
4
2
1
3
4
2
1 .
This is the key to the division by five process: given a multiple of 5,
read off the two low-order bits as the two low-order bits of the
quotient, then subtract these two bits from the rest of the input to
produce two more quotient bits, and repeat this process until the
input is exhausted. This can be implemented with a serial subtractor
and two bit delays.
The above process only works if the input is an exact multiple of
five. For an arbitrary number, B,
=
(32 +2 + 1) mod5
(32mod5 +2mod5 + 1 mod5) mod5
- (2 + 2 + 1) mod 5
= 5 mod 5
=
=
*
0a4 0a4.
LB/5i
000
001
011
010
- 0.
b6
a4a3 a2
+ a4
-a3 a2
100
000
010
001
35mod5
a4 a3 a2 al aO
+ a4a3 a2
al aO
-
R in binary
010 011 100
0
The pattern * * .3 4 2 1 ... repeats indefinitely. If R designates
B mod 5, and initially R = 0, then every one-bit in positions bO,b4,
b8,... adds 1 to R, every bit in positions bl,b5,b9, .. adds 2 to R,
every bit in positions b2,b6,blO,.. adds 4 to R, and every bit in
positions 3, b7,bl 1, * adds 3 to R (the additions are done modulo 5).
R has the transition diagram given in Table I. There is an analogy
between the familiar casting out nines process in decimal arithmetic
and this process of casting out 15's.
- -
THE COMPLETE DIVIDER
The divide-by-ten black box is now complete except for some
clocking, the logic for the low-order remainder bit, and the conversion
of the static remainder flip-flop outputs to a serial bit train. The full
circuit appears in Fig. 2.
APPLICATIONS
This technique can be used to convert integers from binary to
decimal by repeated divisions. Unlike methods based on the ideas of
Couleur [1}]-3], this method uses a fixed amount of hardware,
independent of the word size; longer words just take more time for
the successive divisions. This technique can also be used for the fast
scaling of binary numbers by powers of 10, such as in decimal fraction or floating decimal point calculations. The technique allows
division by ten to be performed in essentially the same amount of
time as multiplication by ten: one or two add times instead of the
much longer time needed for a general division.
Since half the logic is unused during each word time, it is possible
to pipeline the process: the remainder part of a second division can
be calculated while the first quotient is being developed. Thus the
effective speed can be as high as one division per word time. Of
course, for converting a number to decimal by successive divisions,
it is necessary to wait for the first quotient before using it as input
to the second division. But two different numbers could be converted
in almost the same time needed to convert one.
1301
CORRESPONDENCE
SERIAL BINARY DIVISION BY TEN
lst word time: develop R = B mod 10
2nd word time: output
R3 R2 R1
Ro:
Dl, D2 :
R
=
B mod 10, 0 = LB/1OJ
Flip-flops holding the remainder, 0000 to 1001
1-bit delay flip-flops
-
TIME
J-L.... BIT0
BIT1
BIT2
BIT3
-~~~
Fig.
T2
X
r-L...T4
2.Copetdvdebtnit
TS
Fig. 2. Complete divide by ten circuit.
REFERENCES
BRUCE W. SUTER
any modulus and any radix. This leads to the consideration of using
modular arithmetic on arbitrarily long sequences with a finite state
machine. A finite state machine can perform modular addition, subtraction, multiplication, and, if defined, division of a pair of arbitrarily
large numbers, using any modulus and any radix.
Two algorithmic procedures are presented for the evaluation
of the residue of an arbitrarily long sequence. The first technique
provided is a general procedure employing any modulus and
any radix; a second technique provides an improvement of the
previous technique for moduli of the form (rn- rQ), where r > 2
and 0 < q <n-1.
Applications are presented for the almost simultaneous calculation of both the sum and residue sum and both the difference and
residue difference of any pair of arbitrarily large r-ary numbers,
using any modulus.
Abstract-Some fundamental results in the area of computability
theory are presented. These include the fact that a finite state
machine can find the residue of an arbitrarily long sequence using
Index Terms-Computability theory, computer arithmetic, modular addition, modular arithmetic, modular division, modular multiplication, modular subtraction, modulus, residue.
[11 J. F. Couleur, "BIDEC-a binary-to-decimal or decimal-to-binary
converter," IRE Trans. Electron. Comput., vol. EC-7, pp. 313-316,
Dec. 1958.
con[21 V. T. Rhyne, "Serial binary-to-decimal and decimal-to-binary 1970.
version," IEEE Trans. Comput., vol. C-19, pp. 808-812, Sept.
[3] M. S. Schmookler, "High speed binary to decimal conversion,"
IEEE Trans. Comput. (Short Notes), vol. C-17, pp. 506-508, May
1968.
The Modular A thmetic of Arbitrarily Long Sequences
of Digits
Manuscript received July 16, 1973; revised January 23, 1974.
The author is with Honeywell, Inc., St. Petersburg, Fla.
The preliminary work with regard to the determining of remainders
of binary sequences of finite length was conducted by Rothstein [1]