Bhaskara II (12th cent CE) wrote the first work with full and systematic use of the decimal number system. His writings are a compilation of all achievements in Indian maths till his time. He also made many original contributions which anticipated developments in algebra in Europe by few centuries.

One of his books is Lilavati. Latter legend suggests that he wrote it to console his daughter who was destined to remain unmarried. It is unknown how true this legend is, but some problems in Lilavat are addressed to women, using such feminine vocatives as “dear one” or “beautiful one.”

The book has elegant maths riddles. Here is one. “A fifth part of a swarm of bees came to rest on the flower of Kadamba, a third on the flower of Silinda. Three times the difference between these two numbers flew over a flower of Krutaja, and one bee alone remained in the air, attracted by the perfume of a jasmine in bloom. Tell me, beautiful girl, how many bees were in the swarm?”

Source: University of Connecticut, “Lilavati’s Swarm with Solution”; Britannica

The number system includes numerals 1-9 and zero, place value on base 10 and decimals. This package of ideas, indisputably from India, is followed by every person who learns basic maths. Ancient Greeks, Romans & Egyptians did not use place value. Babylonians and Chinese used a base of 60 & Mayans used a base of 20. Rig Veda first expresses numbers in words, using place value with base 10. By the Kushan period (2nd CE), place value system along with use of zero was well known – a common metaphor was that a woman could be a mother, daughter or a wife. Similarly a number may mean differently depending on its place.

A Syrian bishop Severus Sebokth (7th cent CE) cautions the Greeks not to be too proud because Indians may be better at maths. From 8th Century onwards Arabs popularized the number system. Arabs came to India to learn the system and one great name Ibn Sina (Avicenna, 11th cent CE) claims to have learnt it from an Indian vegetable seller, so common was it in India then.

Source: Alok Kumar, “Ancient Hindu Science.”

The Baudhayana Sulbasutra (600BCE at the least) first gave a detailed statement of the Pythagoras theorem about the relationship of side lengths in a right angled triangle. On its history, Dr Manjul Bhargava states that the Egyptians who made the pyramids must have known this. Pythagorean triplets were mentioned by Mesopotamians in 1800BCE. The first formulation is in the Sulbasutras. The Chinese were the first to provide a proof (500BCE). Pythagoras’ contribution was to present all of the above in a rigorous manner around 6th cent BCE.

Sulbasutras were concerned with how to make fire altars for yajnas – including very complex ones shaped like the falcon. They thus explored ideas like how to make a square from a circle or rectangle and vice versa, how to make a square which is the sum or difference of two other squares etc. This they did by devising practical rules based on experimentation – ‘sulba’ means a rope or measuring instrument. In the process they accurately calculated the square root of 2 & 3. The Sulbasutra approach to geometric problems was algebraic and this trend continued. Thus Indian maths contributed more to algebra than geometry, unlike the Greeks who developed geometry greatly via Euclid.

Source: Breakthrough Science Society, “Science in Ancient India: Reality vs Myth”

Aryabhata used the terms ‘ardha-jya’ (half chord) and ‘jya-ardha’ (chord half). He abbreviated the terms to ‘jya’ (chord). From jya the Arabs phonetically derived jiba, omitting vowels as they usually do. It was written as ‘jb’. The word ‘jiba,’ is otherwise meaningless in Arabic.

Later writers who came across the word ‘jb’ substituted it with ‘jaib,’ which contains the same letters, and is a good Arabic word meaning cove or bay. Gherardo of Cremona (1150CE), when he made his translations from the Arabic, replaced the Arabic ‘jaib’ by its Latin equivalent, ‘sinus’ which means cove or bay. From this translation comes the trigonometric term ‘sine’.

Source: Amartya Sen, “Identity & Violence.”

In 1881, 70 leaves of a birch bark manuscript in the Grantha language & Sarada script were found at Peshawar. It was incomplete but contained many illustrations on algebra & arithmetic e.g. summing up fractions by equalizing the denominator, arithmetic & geometric progressions, quadratic surds etc. Interestingly it refers to sunya or zero at many places as a dot.

Earlier this manuscript was dated to 12th cent CE but in 2017, the Oxford Bodleian library where it is stored, announced that the revised dating is 3rd-4th cent CE. This makes it the earliest reference to zero as a number, earlier than the inscription at the Chaturbhuj temple, Gwalior dated to 9th cent CE or the sandstone carving unearthed in Cambodia dated to 605CE.

While the use of zero as a placeholder was seen in many cultures, it is in India that the dot evolved to became the symbol that we use. Secondly, it was only in India that zero developed into a number in its own right, as seen in Brahmagupta’s writings (7th cent CE). Indians have talked about zero for long. Panini (6th cent BCE) called it ‘lopa’ (lupt – unseen), Pingala (3rd cent BCE) mentioned it & Buddhists called it sunya (empty, 1st cent CE).

The Arabs literally translated sunya into ‘sifar’ (8th cent CE). Fibonacci (12th cent CE) called it ‘zephyr’ in Latin & Adelard (11th cent CE) ‘siphrae’. The word zero in English comes from the last two terms.

Source: Bodleian library, https://www.glam.ox.ac.uk/

The Golden ratio is an irrational number approx. 1.618, represented by the Greek symbol ‘phi’. This ratio is said to represent the building block of aesthetic beauty and is found extensively in nature e.g. body proportions of living beings, growth patterns of plants, insects etc. Researchers opine that the beauty of the Brihadeeshwara temple architecture, made in 1018CE, was due to the extensive use of the golden ratio in its design. The golden ratio emerges from the Fibonacci sequence, where each number (with the exception of the first two) is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, 13…).

Fibonacci (1170-1250) was an Italian mathematician who popularized the Indian numeral system in Europe through his book, Liber Abaci. He is said to have stayed at Bugia on the Berbery coast of Africa, where he came into contact with Hindu mathematicians.

The series named after him was in use in India much before him, as the Brihadeeshwarar temple shows. The series was referred to by many Indians writing on musical metres like Virahanka (9th cent CE), Gopala (1135) & Hemachandra (1150).

Source: Velmurgan& Kalaivanan, “Existence of Golden Ratio in Tanjavur Brihadeeshwarar Temple”, Open Journal of Mathematical Science

In defining God, Indians talked about the Nirguna-rupa (formless) leading to the concept of neti-neti (not this, not that) and sunya (empty). Also in contrast, they visualized a saguna-rupa (with form) and started assigning qualities to God. They concluded that such qualities have to be Ananta (infinite), which incidentally became a name for God. This explains the obsession with very large numbers in ancient Indian maths, unlike in other cultures.

The Shanti Mantra of the Isha Upanishad says,

“oṃ pūrṇam adaḥ pūrṇam idam
pūrṇāt pūrṇam udacyate,
pūrṇasya pūrṇam ādāya
pūrṇam evāvaśiṣyate”

“The world there is full, the world here is full.
Fullness from fullness proceeds
After taking fully from the full,
It still remains completely full”

Source: Alok Kumar, “Ancient Hindu Science”.

Prosody (chandas) was one of the basic studies in ancient India and is a Vedanga (limb of Vedas).

Pingala is the best known prosodist with his work Chandashastra. He adopts the conventional method of metrical analysis of padas or verses. In the last chapter of his work, he introduces a new method which leads to binary numbers & combinatorics.

A metre consists of syllables contained in a foot or a pada with four feet to a stanza. A syllable is of two kinds – laghu (light) or guru (heavy). Normally, any verse is divided into units of three syllables and names assigned to metres on the basis of combination of triplets of syllables. This is the conventional method.

Pingala goes a step further and associates to a syllabic foot, a table called Prastara which consists of rows of laghus & gurus, listed horizontally. He assigns the value 0 to guru and 1 to laghu. He then defines rules for constructing prastaras. Thus he gives equivalence between the binary numbers and their decimal equivalents.

This is the classic binary system which was discovered by Leibniz in the 17th cent. Unfortunately, Pingala’s work on binary numbers did not influence the evolution of maths, remaining confined to the world of music.

Source: R Sridharan, “Mathematics in Ancient & Medieval India”, Chennai Mathematical Institute

Panini (~6th cent BCE) is known for systematizing Sanskrit grammar and was the father of linguistics. He also identified an approach which was adopted by other branches of Indian knowledge systems, including Maths. This is similar to Euclid’s approach. He stated axioms (obvious truths) and then deduced theorems (specific rules) from them.

Panini stated that while you go to a potter for a pot, you don’t go to a grammarian for a word. The grammarian only explains the language which is created as part of social usage. He thus devised rules to explain the language – these were algorithmic (how roots combine into words), recursive (describe metres used ) & symbolic (words broken into syllables that stand for sounds). Nothing was supposed to be axiomatic – a theory explains a given problem, not all problems.

The same approach was taken up by Indian mathematicians. They emphasized procedures & rules over proofs. Not that proofs (upapattis) were not presented but they were done in the bhasyas (commentaries) and not in the sutras. That’s why some historians are reluctant to give full credit to ancient Indian maths saying they lack rigour of modern maths.

Source: Prof MD Srinivas, series of lectures on ancient Indian maths at IIT Bombay, available on you tube.

Pi is the ratio of the circumference of a circle to its diameter. It has been known for 4000 years. Babylonians (1900BCE) calculated it as 3 or 3.125. Egyptians (1650BCE) calculated it to 3.1605. Archimedes (250BCE) was the first to present detailed calculation of pi as a fraction. The Chinese (5th cent CE) calculated it to 7 decimal places. In India, the Sulbasutras (600BCE) don’t refer to pi but from its calculations, it can be inferred as 3.08. The Jains were the first to explicitly present the value of pi as square root of 10. Later Aryabhatta in the 5th cent CE calculated it upto 4 places and Madhava (14th cent CE) to 11 places. Jain maths flourished 200BCE onwards. The greatest of them was Umasvati (150BCE) who hailed from Kusumpura (near Patna). Aryabhatta (5th cent CE) also belonged to this school. Jain maths had certain peculiarities and different nomenclature. They studied pi & the circle in connection with their cosmology. They knew very large numbers and hence the place value system. They classified maths under ten heads – arithmetic, fractions, geometry, squaring, cubing, raising numbers to higher powers, use of ‘unknown x’, permutations & combinations.

Source: R Sridharan, “Mathematics in Ancient & Medieval India”, Chennai Mathematical Institute

Madhava of Sangamgrama (born 1340CE) pioneered a school of mathematics which flourished in a small region of Kerala’s Thrissur district for a couple of centuries at least. The other great scholars from this school are Nilakantha Somayaji & Jyesthadeva.

Madhava demonstrated a deep understanding of the properties of infinite series now associated with calculus. These were later discovered in Europe like the series named after Taylor, Newton, Gregory, Leibniz etc, almost 300 years later. For a long time, historians believed that Indian maths stagnated after Bhaskara II (12th cent CE). This was largely because they were looking for Sanskrit texts while after 1200CE a lot of maths was being recorded in regional languages.

Around 1950, the translation of Jyesthadeva’s Yuktibhasa, a Malayalam text composed around 1520 CE, changed this perception and brought Madhava’s work to light. Yuktibhasa is unique as it is the first Indian maths text to state precise propositions along with their proofs. Historians have often contended that Indian mathematicians were empirical in their approach and did not bother with proofs.

Source: R Sridharan, “Mathematics in Ancient & Medieval India”, Chennai Mathematical Institute

Jafar, the grand vizier in the Aladdin tale belonged to the Barmakid family, viziers of Abbasid caliphs. The Barmakids were Sanskrit-literate converts from Buddhism, who had studied Indian maths in Kashmir. They sent missions to India that brought the works of Brahmagupta and Aryabhata to Baghdad in 773. Between the 5th-7th cent CE, Indian mathematicians had worked out a system of arithmetic (pati-ganita) & algebra (bija-ganita). This covers mathematics till Standard 10 of today.

A generation later, Persian genius Khwarizmi summarized all the Sanskrit mathematical texts at the House of Wisdom in Baghdad. His name gave us the word “algorithm.” His book Kitab al-Jabr is the basis of our word “algebra.” The original name of the book points to its inspiration “The Compendious Book on Calculating by Completion and Balancing, According to Hindu Calculation”.

Source: William Dalrymple, writing in The Guardian