Kim Plofker
Dept. of History of Mathematics
Brown University
> THE ZERO STORY:
> -----------------------------
> Source: Unknown, from the Siemens Network Magazine.
[...]
Note also that a circular symbol for zero is apparently used
on an Indus Valley civilization decimal measuring-stick or ruler
dated to the 3rd millennium BCE; but there is, not surprisingly,
no chain of physical or textual evidence available to link this
definitely with the use of a circle for a sunya-symbol in Sanskrit
texts 3000 years later. Datta and Singh consider the use of a dot
for zero in the classical Indian tradition to be earlier than the
open-circle symbol.
> Indians became adept mathematicians around 3000BC, when
> the Mohenjadaro and Harappa civilizations flourished. Its
> usage became well known around the 6th century when
> Brahmagupta of Multan formulated the rules of operation of
> Aero in his treatise, Brahmasphutasiddantha, in which he
> treated Zero as just another number: A+0=A, A-0=A; Ax0=0
> A/0=0. He went wrong on the last count. Any quantity divided
Brahmagupta wrote in the seventh century CE (but as I noted before,
the use of zero must have been "well known" many centuries before
that). The _Brahmasphutasiddhanta_ mentions a ruler of a dynasty
whose capital was at Bhillamala in modern Rajasthan (and a ninth-century
commentator calls Brahmagupta "the teacher from Bhillamala"); I don't
understand the author's reference to Multan. Brahmagupta does not state
that A/0=0; he says that "a positive or negative [quantity] divided by
zero is [a fraction] having that as a divisor ["taccheda"]." This was
a common algebraic interpretation of division by zero throughout many
subsequent centuries, as it permitted "cancellation" of zeroes in
numerator and denominator.
> by zero is infinity. The mistake was corrected some centuries
> later when Bhaskara (AD 1114) of Bijapur wrote Leelavati.
Again, we shouldn't assume that Bhaskara was responsible for this
development just because his famous work (one of thousands of
mathematical works in medieval India) is the first in which our
limited knowledge has encountered it. Also, in the _Lilavati_
Bhaskara uses essentially the same definition of division by zero
as Brahmagupta had: "In addition, zero [produces a result] equal to the
added [quantity], in squaring and so forth [it produces] zero. A
quantity divided by zero has zero as a denominator; [a quantity] multiplied
by zero is zero, and [that] latter [result] is [considered] "[that] times
zero" in subsequent operations. A [finite] quantity is is understood to
be unchanged when zero is [its] multiplier if zero is subsequently [its]
divisor, and similarly [if it is] diminished or increased by zero"
[_Lilavati_ 45--46]. (Bhaskara, by the way, lived in Vijjadavida in
the Sahyadri range; I don't know what the connection with modern Bijapur
is supposed to be.)
> He claimed that the division of any quantity by zero is an
> infinity, or immutable god, a god who does not change when
> worlds are created or destroyed. Only the tangible changes;
> zero the intangible, immutable.
This is a rather garbled version of what Bhaskara says not in his
arithmetic text _Lilavati_, but in the _Bijaganita_ on algebra. (The
immutability of a quantity divided by zero is there considered to be
analogous to that of the deity; Bhaskara does not imply that infinity,
or zero, is itself divine.) Combined with the quote above, this makes
it evident that different concepts of division by zero might co-exist
for different mathematical purposes (which makes the reference to
Brahmagupta's "mistake" a little dubious).
> For 400 years from the 6th
> century, India was foremost in maths and zero began its journey
> around the world.
Other responders have already pointed out the existence of concepts
and symbols for zero in other parts of the world before the Indian
"sunya" was transformed (accompanied by the rest of the decimal numerals
and their place-value system) into the Arabic "sifr".