- NUMBER SYSTEMS 199
operations, for example, considering the appropriate value of a fraction such as
(5x^2 + Ax)/(3x^2 — 2x) when χ becomes zero. One can formally cancel the χ without
thinking about whether or not it is zero. After cancellation, setting χ = 0 gives the
fraction the value —2. Bhaskara is explicit in saying that zero added to any number
leaves that number unchanged. Hence for him it is more than a mere placeholder;
arithmetic operations can be performed with it. The use of an empty circle or a
circle with its center marked as a symbol for zero seems to be culturally invariant,
since it appears in inscriptions in India from the ninth century, in Greek documents
from the second century, and in Chinese documents from the thirteenth century.
Islamic number systems. The transmission of Hindu treatises to Baghdad led ulti-
mately to the triumph of the numerals used today. According to al-Daffa (1973,
p. 51) the Sanskrit words for an empty place were translated as the Arabic word
sifr, which became the English words cipher and zero and their cognates in other
European languages. Al-Daffa also points out. that the earliest record of the symbol
for zero in India comes from an inscription at Gwalior dating to 876, and that there
is a document in Arabic dating from 873 in which this symbol occurs.
2.2. Irrational and imaginary numbers. In a peculiar way, the absence of a
place-value system of writing numbers may have stimulated the creation of math-
ematics in ancient Greece in the case of irrational numbers. Place-value notation
provides approximate square roots in practical form, even when the expansion does
not terminate.^10 A cuneiform tablet from Iraq (Yale Babylonian Collection 7289)
shows a square with its diagonals drawn and the sexagesimal number 1;24,51,10,
which gives the length of the diagonal of a square of side 1 to great precision. But
in all the Chinese, Mesopotamian, Egyptian,^11 and Hindu texts there is nothing
that can be considered a theoretical discussion of "numbers" whose expansions do
not terminate.
The word numbers is placed in inverted commas here because the meaning of
the square root of 2 is not easy to define. It is very easy to go around in circles
making the definition. The difficulty came in a clash of geometry and arithmetic,
the two fundamental modes of mathematical thinking. From the arithmetical point
of view the problem is minimal. If numbers must be what we now call positive
rational numbers, then some of them are squares and some are not, just as some
integers are triangular, square, pentagonal, and so forth, while others are not. No
one would be disturbed by this fact; and since the Greeks had no place-value system
to suggest an infinite process leading to an exact square root, they might not have
speculated deeply on the implications of this arithmetical distinction in geometry.
But in fact, they did speculate on both the numerical and geometric aspects of the
problem, as we shall now see. We begin with the arithmetical problem.
The arithmetical origin of irrationals: nonsquare rational numbers. In Plato's dia-
logue Theatetus, the title character reports that a certain Theodorus proved that
the integers 2, 3, 5, and so on, up to 17 have no (rational) square roots, except of
course the obvious integers 1, 4, and 9; and he says that for some reason, Theodorus
stopped at that point. On that basis the students decided to classify numbers as
(^10) In the case of Chinese mathematics the end of a nonterminating square root was given as a
common fraction, and Simon Stevin likewise terminated infinite decimals with common fractions.
(^11) Square roots, called corners, are rarely encountered in the Egyptian papyri, and Gillings (1972,
p. 214) suggests that they were found from tables of squares.