Senior Lecturer in Mathematical
Biology, University of Bath

It should come as no surprise that
the first recorded use of the number zero, recently
discovered to be made as early as the 3rd or
4th century, happened in India. Mathematics on the Indian subcontinent has a
rich history going
back over 3,000 years and thrived for centuries before similar
advances were made in Europe, with its influence meanwhile spreading to China
and the Middle East.

As well as giving us the concept of
zero, Indian mathematicians made seminal contributions to the study of trigonometry,
algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we
still employ worldwide today was first seen in India.

**The number system**

As far back as 1200 BC, mathematical
knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations
of powers of ten. For example, 365 might be
expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name
rather than a set of symbols. It is reasonable
to believe that this representation using
powers of ten played a crucial role in the development of the decimal-place
value system in India.

From the third
century BC, we also have written evidence of
the Brahmi
numerals, the precursors to the modern,
Indian or Hindu-Arabic numeral system that most of the world uses today. Once
zero was introduced, almost all of the mathematical mechanics would be in place
to enable ancient Indians to study higher mathematics.

**The concept of zero**

Zero itself has a much longer
history. The recently
dated first recorded zeros, in what is
known as the Bakhshali manuscript, were simple placeholders – a tool to
distinguish 100 from 10. Similar marks had already been seen in the Babylonian
and Mayan cultures in the early centuries AD
and arguably in Sumerian
mathematics as early as 3000-2000 BC.

But only in India did the
placeholder symbol for nothing progress to become a number
in its own right. The advent of the concept of zero
allowed numbers to be written efficiently and reliably. In turn, this allowed
for effective record-keeping that meant important financial calculations could
be checked retroactively, ensuring the honest actions of all involved. Zero was
a significant step on the route to the democratisation
of mathematics.

These accessible mechanical tools
for working with mathematical concepts, in combination with a strong and open
scholastic and scientific culture, meant that, by around 600AD, all the
ingredients were in place for an explosion of mathematical discoveries in
India. In comparison, these sorts of tools were not popularised in the West
until the early 13th century, though Fibonnacci’s book
liber abaci.

**Solutions of quadratic equations**

In the seventh century, the first
written evidence of the rules for working with zero were formalised in the Brahmasputha
Siddhanta. In his seminal text, the
astronomer Brahmagupta introduced rules for solving quadratic equations (so
beloved of secondary school mathematics students) and for computing square
roots.

**Rules for negative numbers**

Brahmagupta also demonstrated rules
for working with negative numbers. He referred to positive numbers as fortunes and
negative numbers as debts. He wrote
down rules that have been interpreted by translators as: “A fortune subtracted
from zero is a debt,” and “a debt subtracted from zero is a fortune”.

This latter statement is the same as
the rule we learn in school, that if you subtract a negative number, it is the
same as adding a positive number. Brahmagupta also knew that “The product of a
debt and a fortune is a debt” – a positive number multiplied by a negative is a
negative.

For the large part, European
mathematicians were reluctant to accept negative numbers as meaningful. Many
took the view that negative
numbers were absurd. They reasoned that numbers were
developed for counting and questioned what you could count with negative
numbers. Indian and Chinese mathematicians recognised early on that one answer
to this question was debts.

For example, in a primitive farming
context, if one farmer owes another farmer 7 cows, then effectively the first
farmer has -7 cows. If the first farmer goes out to buy some animals to repay
his debt, he has to buy 7 cows and give them to the second farmer in order to
bring his cow tally back to 0. From then on, every cow he buys goes to his
positive total.

**Basis for calculus**

This reluctance to adopt negative
numbers, and indeed zero, held European mathematics back for many years.
Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the
negatives in a systematic way in his development
of calculus in the late 17th century. Calculus
is used to measure rates of changes and is important in almost every branch of
science, notably underpinning many key discoveries in modern physics.

But Indian
mathematician Bhāskara had already
discovered many of Leibniz’s ideas over
500 years earlier. Bhāskara, also made major
contributions to algebra, arithmetic, geometry and trigonometry. He provided
many results, for example on the solutions of certain “Doiphantine” equations,
that would
not be rediscovered in Europe for centuries.

The
Kerala school of astronomy and mathematics,
founded by Madhava
of Sangamagrama in the 1300s, was responsible for
many firsts in mathematics, including the use of mathematical induction and
some early calculus-related results. Although no systematic rules for calculus
were developed by the Kerala school, its proponents first conceived of many of
the results that would later
be repeated in Europe including Taylor series expansions,
infinitessimals and differentiation.

The leap, made in India, that
transformed zero from a simple placeholder to a number in its own right
indicates the mathematically enlightened culture that was flourishing on the
subcontinent at a time when Europe was stuck in the dark ages. Although its
reputation suffers
from the Eurocentric bias, the
subcontinent has a strong mathematical heritage, which it continues into the
21st century by providing
key players at the forefront of every branch of mathematics.

(

*Adapted from www.theconversation.com`*)