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2017-09-24 10:05:45 UTC
https://qz.com/1084558/how-ancient-india-pioneered-fundamental-math-concepts-centuries-before-europe/
How ancient India pioneered fundamental math concepts centuries before
Europe
By Christian Yates, senior lecturer in mathematical biology, University
of Bath September 22, 2017 Quartz India
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 (3×10²), six tens
(6×10¹), and five units (5×10⁰), 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 3rd 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 600 AD, 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 7th 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 such 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 seven cows, then effectively the first farmer has negative seven
cows. If the first farmer goes out to buy some animals to repay his
debt, he has to buy seven 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, infinitesimals, 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.
Christian Yates, Senior Lecturer in Mathematical Biology, University of
Bath. This article was originally published on The Conversation. Read
the original article.
How ancient India pioneered fundamental math concepts centuries before
Europe
By Christian Yates, senior lecturer in mathematical biology, University
of Bath September 22, 2017 Quartz India
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 (3×10²), six tens
(6×10¹), and five units (5×10⁰), 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 3rd 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 600 AD, 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 7th 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 such 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 seven cows, then effectively the first farmer has negative seven
cows. If the first farmer goes out to buy some animals to repay his
debt, he has to buy seven 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, infinitesimals, 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.
Christian Yates, Senior Lecturer in Mathematical Biology, University of
Bath. This article was originally published on The Conversation. Read
the original article.