Indian mathematics

Indian mathematics grew in the Indian subcontinent from around 1200 BCE until the late 1700s. During a special time from 400 CE to 1200 CE, Indian thinkers made big steps in math. Great people like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava helped shape math as we know it today.

One major gift from Indian mathematicians is the decimal number system we use now. They were among the first to think of zero as a number. They also studied negative numbers, arithmetic, and algebra. In India, trigonometry grew a lot, and the ideas of sine and cosine began there. These ideas traveled to the Middle East, China, and Europe, helping math grow everywhere.

Many old Indian math books were written in Sanskrit. They often began with short poems called sutras to help people remember the rules. After the poems came longer explanations. The oldest known math book 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 learned how to expand trigonometric functions into series. They did this long before calculus was created in Europe. Their work was one of the earliest 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 a very long time 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 thinker who lived around 300–200 BCE. He wrote a book called the Chhandas Shastra, about patterns in poetry sounds. In his book, Pingala showed early ideas of what we now call Fibonacci numbers. He also described a pattern like Pascal's Triangle, which helps us understand number combinations.

Another mathematician named Kātyāyana, who lived around the 3rd century BCE, wrote about geometry. He talked about the Pythagorean theorem and found 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 studied very big numbers and even infinity, which means numbers that go on forever. They described five kinds of infinity.

These mathematicians were 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 Sanskrit scholars called pandits. They learned by reciting and memorizing texts. This helped keep important writings and math ideas safe for many years.

They had special ways to remember texts. One way was to say words in different orders, like saying "word1word2" and then "word2word1". Another way was to pair the first and last words together. These tricks helped keep texts the same for many years. Even very old texts, like the Ṛgveda, were kept this way.

Math ideas were also shared orally, using short, clever phrases called sūtra. These phrases were brief, using hints and abbreviations. Students learned the full meaning from their teachers. This made sharing knowledge clear and accurate.

The written tradition: prose commentary

As math became more complicated, people started writing it down in books. These books were copied many times over the years. Today, India has about thirty million books, the most in the world.

The earliest math book with explanations was about a work called the Āryabhaṭīya, written in 499 CE. It had 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 to work through problems and check their answers. This way of learning helped people understand and remember the math better.

Numerals and the decimal number system

The decimal number system we use today began in India. Indian mathematicians created a way to write numbers using place values. This means 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 symbols for numbers, but they did not use place values. The oldest known examples of decimal place value numbers are from around the year 500 CE. Later, Indian scholars found clever ways to show numbers in poems by linking them to objects in nature or religion.

Bakhshali Manuscript

The Bakhshali Manuscript is the oldest known math book from India. It was found in 1881 near Peshawar, in a place that was part of British India but is now Pakistan. It was written on birch bark in an old script and dates from between 224 and 383 CE.

The book has rules and examples of arithmetic and algebra, like solving equations and using fractions. It also uses a decimal system with a symbol for zero, which was very advanced for its time. One fun problem is about finding the value of different animals by solving equations.

Classical period (400–1300)

The classical period of Indian mathematics, from 400 to 1300, is often called its golden age. During this time, mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama, and Nilakantha Somayaji made important contributions. Their work helped build stronger foundations for many areas of mathematics and spread to Asia, the Middle East, and Europe.

Mathematics during this time was part of the 'astral science' called jyotiḥśāstra. This included three main areas: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka), and divination (saṃhitā). These areas are shown in works like Varahamihira's Pancasiddhantika, which gathered five older astronomical texts. The main books were written in Sanskrit verse and later explained in prose commentaries.

This era saw the growth of trigonometry, algebra, and ideas that led to calculus. Mathematicians introduced trigonometric functions, solved quadratic and cubic equations, and made precise astronomical calculations. Importantly, the decimal number system we use today was first written down 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

Between 1300 and 1800, Indian mathematicians made many important discoveries. One group, called Navya-Nyāya, studied logic and philosophy. They found new ways to think about ideas and solve problems by naming and describing things clearly.

Another important group was the Kerala school from South India. They studied astronomy and math and found ways to describe curves and angles with special number patterns. Their ideas were similar to parts of calculus, which was later developed in Europe by Isaac Newton and Gottfried Leibniz. However, they did not create the full theory of calculus.

Charges of Eurocentrism

Some people think that Indian mathematicians did not get enough credit for their work. They believe that many important ideas are often given to Western scholars instead. For example, important math ideas about curves and angles were first found by Indian mathematicians from the Kerala school long before they were found again in Europe.

Scholars know that both Arab and Indian mathematicians discovered important ideas about calculus before the 17th century. They did not put these ideas together the same way Newton and Leibniz did, but their work might have helped later European mathematicians. However, there is no proof of this connection.