198 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSChina. Chinese counting rods were red or black according as the numbers repre-
sented were positive or negative, yet no zero occurs, since obviously it would be
absurd to have a rod representing no rods at all. A Chinese work on astronomy and
the calendar written in the late second century CE (Li and Du, 1987, p. 49) gives
rules for adding and subtracting "strong" (positive) and "weak" (negative): When
adding, like signs add and opposite signs subtract; when subtracting, like signs
subtract and opposite signs add. The same kinds of rules are given in Chapter 8
("Rectangular Tabulation") of the Jiu Zhang Suanshu. Yet it was a full thousand
years after that time when the rules for multiplying and dividing signed numbers
first appeared in the Suanshu Chimeng of 1303. When one is using a counting
board or an abacus, no symbol for zero is needed, since it is visually apparent that
a given square has no numbers written in it or that the beads on a string are in
their "empty" position. The first known occurrence of the symbol 0 for zero occurs
in a work dating to the year 1247.India. Around the year 500, Aryabhata I, used a place-value decimal system with-
out zero. A century later Brahmagupta introduced zero in connection with the
kuttaka method described above. Although he used the word sunya (empty) for
this concept and it really does denote an empty place in that method, the idea
that the algorithm χ >—> ax + b can be executed as χ ·—> ax when no b is present
suggests the use of a neutral element for addition, and that is what the zero is.
Brahmagupta gave complete rules for addition, subtraction, multiplication, and
division of both positive and negative quantities and zero. As we know, division
by zero must be considered separately and either rejected or given some special
meaning. Brahmagupta (Colebrooke, 1817, pp. 339-340) showed some puzzlement
about this, and he wrote:Cipher, divided by cipher, is nought. Positive, divided by neg-
ative, is negative. Negative, divided by affirmative, is negative.
Positive or negative, divided by cipher, is a fraction with [cipher]
for denominator, or cipher divided by negative or affirmative [is a
fraction with the latter for denominator].The word cipher here translates the Sanskrit sunya or kha, both meaning empty
space. The last rule given here is not a happy effort at a definition; it is rather
like saying that a jar contains its contents. Not much new information is conveyed
by the sentence. But the obscurity is natural due to the complete absence of any
human experience with situations corresponding to division by zero. Five hundred
years later Bhaskara was still having trouble with this concept (Colebrooke, 1817,
p. 19):
A definite quantity divided by cipher is the submultiple of nought
[that is, a fraction with zero for its denominator, just as Brah-
magupta had said]. The product of cipher is nought: but it must
be retained as a multiple of cipher, if any further operation impend.
Cipher having become a multiplier, should nought afterwards be-
come a divisor, the definite quantity must be understood to be
unchanged.Although these principles might be more clearly stated, it seems that Bhaskara
may have in mind here some operations similar to those that occur in limiting