Indian mathematics

Indian mathematics developed in the Indian subcontinent from around 1200 BCE until the late 1700s. During its classical period, between 400 CE and 1200 CE, Indian scholars made big advances in math. Important figures like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava helped shape what we know today.

One of the biggest contributions from Indian mathematicians was the decimal number system that we use now. They were also early to think of zero as a number, and they studied negative numbers, arithmetic, and algebra. In India, trigonometry grew a lot, and the modern ideas of sine and cosine came from there. These ideas spread to the Middle East, China, and Europe, helping math grow around the world.

Many ancient Indian math texts were written in Sanskrit. They often started with short poems called sutras to make the rules easy to remember. After that came longer explanations in prose. The oldest known math paper from India is the Bakhshali Manuscript, found near Peshawar in modern-day Pakistan. It likely dates to the 7th century CE.

Later, in the 15th century, mathematicians from the Kerala school in India discovered how to expand trigonometric functions into series. They did this work long before calculus was invented in Europe, and their methods were some of the first examples of what we now call a power series.

Prehistory

Cubical weights standardised in the Indus Valley civilisation

Excavations at Harappa, Mohenjo-daro, and other sites of the Indus Valley civilisation show that people used practical math long ago. They made bricks in a special size ratio of 4:2:1 to help buildings stay strong. They also used a standard system of weights with many different sizes, from very small to very large.

These ancient people tried to measure lengths very accurately. They made a ruler, called the Mohenjo-daro ruler, which was divided into ten equal parts. They also found special shell tools that could help measure angles and find where stars were for navigation.

Vedic period

Pingala (300 BCE – 200 BCE)

Pingala was a music theorist who lived around 300–200 BCE. He wrote a book called the Chhandas Shastra, which was about the patterns of sounds in poetry. In his work, Pingala showed early ideas of what we now call Fibonacci numbers. He also described a pattern similar to Pascal's Triangle, which helps us understand combinations in numbers.

Another mathematician named Kātyāyana, who lived around the 3rd century BCE, wrote about geometry. He explained the Pythagorean theorem and calculated the square root of 2 very accurately.

Jain mathematics (400 BCE – 200 CE)

Jain mathematicians helped connect earlier Indian math with later, more advanced math. They were special because they studied very big numbers and even infinity, which means numbers that go on forever. They described five kinds of infinity, like numbers that go on forever in one direction or everywhere.

These mathematicians were also the first to use the word shunya to mean zero. This word is where the English word “zero” comes from. They also created ways to show powers of numbers, like squares and cubes, and solved early algebraic equations. Some important Jain math books include the Sthānāṅga Sūtra, Anuyogadwara Sutra, and Ṣaṭkhaṅḍāgama.

Oral tradition

Ancient Indian mathematicians were usually Sanskrit scholars called pandits. They learned through reciting and memorizing texts. This method helped preserve important texts, including sacred writings and mathematical ideas, for many years.

The design of the domestic fire altar in the Śulba Sūtra

They used special ways to remember texts. One method involved saying words in different orders, like saying "word1word2" then "word2word1". Another method paired the first and last words together. These techniques helped keep texts exactly the same across generations. Even very old texts, like the Ṛgveda, were kept intact this way.

Mathematical ideas were also passed down orally, using short and clever phrases called sūtra. These phrases were very brief, using hints and abbreviations so that students could understand the full meaning when taught by a teacher. This way, important knowledge was shared clearly and accurately.

The written tradition: prose commentary

As mathematics grew more complex, people began writing it down in manuscripts. These manuscripts were copied many times over the years. Today, India has about thirty million manuscripts, the most in the world.

The earliest mathematical book with explanations was about a work called the Āryabhaṭīya, written in 499 CE. It included 33 short rules without proofs. Later, around 600 CE, Bhaskara I began adding explanations and examples to these rules. Students would first memorize the rules and then use chalkboards covered in dust to work through problems and check their answers. This style of learning helped people understand and remember the math better.

Numerals and the decimal number system

The decimal number system we use today was first recorded in India. Indian mathematicians developed a way to write numbers using place values, meaning the position of a digit shows its value. This idea spread from India to the Islamic world and then to Europe.

Early Indian scripts like the Kharoṣṭhī and Brāhmī script had their own numeral symbols, but they did not use place values. The oldest known examples of decimal place value numbers date from around the year 500 CE. Later, Indian scholars used clever methods to represent numbers in poems by linking them to objects in nature or religion.

Bakhshali Manuscript

The Bakhshali Manuscript is the oldest known mathematical manuscript from India. It was discovered in 1881 near Peshawar, in what was then British India and is now Pakistan. Written on birch bark in an ancient script, the manuscript dates back to between 224 and 383 CE.

The manuscript includes rules and examples of arithmetic and algebra, such as solving equations and working with fractions. It also uses a decimal system with a symbol for zero, which was very advanced for its time. One interesting problem involves figuring out the value of different animals by solving equations.

Classical period (400–1300)

The classical period of Indian mathematics, lasting from 400 to 1300, is often called the golden age of this field. During this time, mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama, and Nilakantha Somayaji made significant contributions. Their work laid broader and clearer foundations for many branches of mathematics and spread to Asia, the Middle East, and eventually to Europe.

Mathematics during this period was part of the 'astral science' known as jyotiḥśāstra, which included three main areas: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka), and divination (saṃhitā). These areas are reflected in works like Varahamihira's Pancasiddhantika, which compiled five earlier astronomical texts. The main texts were written in Sanskrit verse and followed by prose commentaries.

This era saw the development of trigonometry, algebra, and calculus concepts, with mathematicians introducing trigonometric functions, solving quadratic and cubic equations, and making accurate astronomical calculations. Notably, the decimal number system in use today was first recorded in Indian mathematics during this period.

Medieval and early modern mathematics (1300–1800)

Main article: Navya-Nyāya

Chain of teachers of Kerala school of astronomy and mathematics

Main article: Kerala school of astronomy and mathematics

In the years between 1300 and 1800, Indian mathematicians made many important discoveries. One group, called Navya-Nyāya, focused on logic and philosophy. They created new ways to think about ideas and solve problems by carefully naming and describing objects.

Another important group was the Kerala school, which started in South India. They worked on astronomy and math and discovered ways to describe curves and angles using special number patterns. Their work included ideas that were later part of calculus, invented in Europe by Isaac Newton and Gottfried Leibniz, but they did not develop the full theory of calculus.

Charges of Eurocentrism

Some people believe that Indian mathematicians did not get enough credit for their work. They think that many important ideas are often credited to Western scholars instead. For example, the infinite series for trigonometric functions, which were later rediscovered in Europe, were actually first described by Indian mathematicians from the Kerala school hundreds of years earlier.

Scholars have noted that both Arab and Indian mathematicians made important discoveries related to what we now call calculus before the 17th century. While they did not combine these ideas in the same way as Newton and Leibniz did, it is possible that some of their work may have influenced later European mathematicians through trade and travel. However, there is no solid evidence to prove this connection.