Maarg Manthan · Topic 10.2

Mathematics (Zero, Decimal System, Aryabhata, Brahmagupta)

Art & Culture › Science & Technology in Ancient India · Topic 10.2

Ancient Indian mathematics gave the world the decimal place-value system and zero as a number. The Sulba Sutras of Baudhayana, Apastamba and Katyayana set out geometry for altar building, including the Pythagorean relation. Aryabhata wrote the Aryabhatiya in 499 CE and gave pi as about 3.1416. Brahmagupta, in the Brahmasphutasiddhanta of 628 CE, stated the first clear rules for arithmetic with zero and negative numbers. Bhaskara II wrote the Lilavati in the 12th century, and the Kerala school of Madhava of Sangamagrama developed infinite series for trigonometric functions about two centuries before Europe.

6 min read · UPSC Prelims · Art & Culture · Science & Technology in Ancient India

At a Glance

Big gifts Zero and decimal place-value
Aryabhata Aryabhatiya, 499 CE
Brahmagupta Brahmasphutasiddhanta, 628 CE
Bhaskara II Lilavati, 12th century
Kerala school Madhava, infinite series
Exam link UPSC Prelims, Art and Culture: Science & Technology in Ancient India

Where Does It Fit?

This topic follows metallurgy in the macro on science in ancient India. Its mathematicians were also astronomers, so the same names return in the next topic on astronomy. Previous topic: Metallurgy (Iron Pillar of Delhi, Harappan Bronze Work). Next topic: Astronomy (Aryabhatiya, Varahamihira, Observatories).

Who Were the Great Mathematicians of Ancient India?

Mathematician	Period and region	Work and contribution
Baudhayana, Apastamba, Katyayana	Vedic period	Sulba Sutras with geometry of altars, rules like the Pythagorean theorem and square roots
Pingala	c. 200 BCE	Chandahshastra on metre, with a binary-like system and a pattern linked to Pascal triangle
Aryabhata	Born 476 CE, Kusumapura (Pataliputra)	Aryabhatiya (499 CE), pi near 3.1416, sine table, Earth rotation, place-value
Varahamihira	6th century CE, Ujjain	Pancha Siddhantika, trigonometric tables and combinatorics
Brahmagupta	598 to c. 668 CE, Bhillamala, Rajasthan	Brahmasphutasiddhanta (628 CE), rules for zero, negative numbers, quadratic equations
Mahavira	9th century, Rashtrakuta court	Ganita Sara Sangraha, with early work on fractions and permutations
Bhaskara II	Born 1114 CE, Vijayapura region	Siddhanta Shiromani including Lilavati and Bijaganita, early ideas of calculus
Madhava of Sangamagrama	14th century, Kerala	Infinite series for sine, cosine and arctangent, founder of the Kerala school
Nilakantha Somayaji	15th to 16th century, Kerala	Tantrasamgraha and a revised planetary model

What Did India Contribute on Zero and the Decimal System?

Place-value: the idea that a digit takes value from its position, with a symbol for an empty place, was fully used in India by the Gupta age
Brahmi numerals: the ancestors of modern numerals appear in the Ashokan edicts, though without place-value
Bakhshali manuscript: a birch-bark manuscript found in 1881 near Peshawar, now in the Bodleian Library, Oxford, which uses a dot for zero, with carbon dating by Oxford placing parts of it in the 3rd to 4th century CE
Brahmagupta on zero: he treated zero as a number, gave rules for adding, subtracting and multiplying with it, and wrote the first rules for negative numbers
Gwalior inscription: the Chaturbhuj temple at Gwalior, Madhya Pradesh, holds an inscription of 876 CE with zero in a place-value number, and older examples are known from Cambodia
Name of zero: the word shunya meant void, and was rendered as sifr in Arabic and later gave the words cipher and zero
Spread abroad: Arab scholars such as Al-Khwarizmi learned the system, and Europe took it as Hindu-Arabic numerals through Fibonacci in the 13th century
Large numbers: ancient Indian texts used names for very large powers of ten, up to tens of crores and beyond
Piece of evidence: place-value Sanskrit number words appear in astronomical works from the 5th century

What Else Should You Remember?

Sulba Sutras: ritual geometry for fire altars, the earliest geometry text tradition of India, giving a value for the square root of two with great accuracy
Aryabhata and pi: he called the value approximate, and the Aryabhatiya has 121 verses in four sections
Aryabhata satellite: India named its first satellite in 1975 after him
Brahmagupta rules: he gave solutions of quadratic equations and a formula for the area of a cyclic quadrilateral
Lilavati: a work of arithmetic written as problems posed to a young girl, and taught for centuries
Bhaskara II and calculus: he used ideas close to the derivative while studying planetary motion
Kerala school: the Kerala school of astronomy and mathematics, with Madhava, Nilakantha Somayaji and Jyeshthadeva (author of the Malayalam work Yuktibhasha), worked out series expansions
Jain mathematics: Jain scholars studied large numbers, sets and permutations, and Mahavira belonged to this tradition
Modern link: Srinivasa Ramanujan (1887 to 1920) of Tamil Nadu is the best-known modern Indian mathematician, and 22 December is National Mathematics Day
Common mistake: Aryabhata I of 499 CE should not be confused with Aryabhata II, a later author
Link to astronomy: these authors are also covered in Astronomy (Aryabhatiya, Varahamihira, Observatories)

Why Does It Matter for the Exam?

Author and work: matching Aryabhata, Brahmagupta, Bhaskara II and Mahavira with their texts is the most common question
Zero: questions ask who first gave rules for zero as a number, and what the Bakhshali manuscript shows
Kerala school: Madhava is asked for infinite series
Dates: Aryabhatiya 499 CE and Brahmasphutasiddhanta 628 CE
Common error: crediting Arab mathematicians with the invention of the place-value system

As an analytical point, Indian mathematics grew from practical and ritual needs, such as altar building, calendars and trade, and it was written in verse for memory. Its global spread through Arab scholars shows a long chain of scientific exchange.

Why Is This Topic Important?

Authors Match the pairs
Zero A leading topic
Dates Fixed anchor years

What Came Next?

The next topic covers astronomy, where the same authors applied their mathematics to the sky. Next topic: Astronomy (Aryabhatiya, Varahamihira, Observatories).

Key People and Terms

Shunya Zero or void
Place-value Digit value depends on position
Sulba Cord for measuring altars
Lilavati Bhaskara II arithmetic work
Yuktibhasha Malayalam proof-based text of Kerala school
Siddhanta A treatise on astronomy

Study this topic together with these linked topics from other subjects.

Science and Technology in the Gupta Age (Ancient History)

Exam Corner

Points to Remember

Aryabhatiya: 499 CE, Aryabhata.
Brahmasphutasiddhanta: 628 CE, Brahmagupta.
Lilavati: Bhaskara II.
Infinite series: Madhava, Kerala school.
Bakhshali manuscript: dot for zero.

Do Not Confuse With

Aryabhata and Varahamihira Mathematician-astronomer of Kusumapura versus the Ujjain scholar of Pancha Siddhantika
Brahmagupta and Bhaskara II Seventh century versus twelfth century
Sulba Sutras and Siddhantas Vedic altar geometry versus later astronomical treatises

Memory Hook

Aryabhata 499, Brahmagupta 628, Bhaskara twelve, Madhava made the series.

Mains Angle

Questions ask about the global impact of Indian science and why it matters. These points give a frame.

Global reach Numerals reached Europe through Arab scholars
Continuity From altars to calculus
Today Teaching heritage science with evidence

A Question You May Face

An original practice question, not a past paper question.

“Discuss the contribution of ancient Indian mathematicians to world knowledge, with reference to zero, the decimal system and the Kerala school.”

How to Answer

Base Sulba Sutras, place-value.
People Aryabhata, Brahmagupta, Bhaskara.
Spread Arabs, then Europe.
Way Teach with evidence.

GS Relevance

Prelims: mathematicians and their works, zero, decimal system, Bakhshali manuscript, Kerala school. GS Paper 1: Indian heritage and culture, contribution of India to world knowledge.

Frequently Asked Questions

Who invented zero in India?

No single inventor is known. Place-value with a symbol for an empty place developed in India by the Gupta age, and Brahmagupta in 628 CE gave the first clear rules for treating zero as a number. The Bakhshali manuscript, carbon dated to the 3rd or 4th century CE, shows an early zero dot.

What did Aryabhata write and when?

Aryabhata, born in 476 CE and linked with Kusumapura near Pataliputra, wrote the Aryabhatiya in 499 CE. It has 121 verses and covers arithmetic, algebra, trigonometry and astronomy, including an approximate value of pi near 3.1416 and the idea that the Earth rotates on its axis.

What is the Brahmasphutasiddhanta?

The Brahmasphutasiddhanta is a Sanskrit treatise on astronomy and mathematics written by Brahmagupta in 628 CE. It gives rules for zero and negative numbers, methods for quadratic equations, and a formula for the area of a cyclic quadrilateral, and it was later translated into Arabic.

What was the Kerala school of mathematics?

The Kerala school was a line of scholars from the 14th to 16th century, founded by Madhava of Sangamagrama, with Nilakantha Somayaji and Jyeshthadeva. They found infinite series for sine, cosine and arctangent, about two centuries before similar results were obtained in Europe.

What are the Sulba Sutras?

The Sulba Sutras are Vedic texts of Baudhayana, Apastamba and Katyayana that give rules for building fire altars with cords. They contain geometry, including the Pythagorean relation, constructions of squares and rectangles, and a close value of the square root of two.

PYQ Practice — Statement Analysis

1 Brahmagupta gave the first rules for treating zero as a number.

True

In the Brahmasphutasiddhanta, 628 CE.

2 The Lilavati was written by Aryabhata.

False

It was written by Bhaskara II.

3 The Kerala school was founded by Madhava of Sangamagrama.