Continuous invasions by the Mughals had made mathematics to dry in northen and central India. But South India was not much affected by Mughals and hence mathematics kept on growing in south. Most notable contributer from that period in the field of mathematics and astronomy was Madhava. Madhava (1340 – 1425 AD) was born in Sangamagrama, Kerala. He was the founder of Kerala school of Astronomy and Mathematics. Madhava is considered as the greatest mathematician and astronomer from Medieval Period. He had major contributions in fields of mathematics like Trigonometry, Geometry, Algebra and Calclulus. He was first all over the world to use infinite series of approximation for a range of trigonometric functions which led to passage of infinity and which finally led to modern Mathematical Analysis topics like Differentiation, Integration, Limits and Infinite series. Madhava wrote following books: Golavada Madhyamanayanaprakara Mahajyanayanaprakara Lagnaprakara Venavaroha Sphutacandrapati Aganita-grahacara Candravakyani Most of his work (except some of his astronomical books) is lost but his pupils from his school had written books on mathematics and almost all his discoveries are mentioned in those commentaries. Value of pi: In his book Mahajyanayanaprakara he derived a formula for pi by infinite series expansion which is now called as Madhava- Leibnitz series, i.e. by successively adding and subtracting odd number fractions till infinity, he could arrive to exact formula for π. π/4 = 1 - 1/3 + 1/5 -1/7 + ... (-1)n/(2n+1) + ….. and was extended by Nilkanta Somayaji in his book ‘Tantrasangraha’ as tan-1x = x - x3/3 + x5/5 - ... Another his formula for pi, on taking 21 terms, can lead to exact value of pi upto 11 decimals.. π = √12(1 - 1/(3×3) + 1/(5×32) - 1/(7×33) + .. π = 3.14159265359 (Correc upto 11 decimal places). Trignometry and Infinite Series of Trigonometric Functions Following were the 1st algebraic formulae to … [Read more...]

## Hindu Numerals goes West

We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made....Albert Einstein We have already seen how the Indian Numerals (Decimal system) evolved from Harappans to Kharoshti to Brahmi and Bhashali numbers giving different symbols from 1-9. Invention of 0 and Place Value System Many have misconception that 0 was invented by Aryabhata. No, its not true… meaning of 0 as emptiness, nothing, void (used for meditation) was already known to Buddhist and Jains around 600 -700 years before Aryabhata and it was called as “Shunya”. Aryabhata invented "Place Value System". The symbol ‘0’ was seen in Jain book ‘Lokavibhaga’ as early as AD 458 and Aryabhata was born in 476AD. As seen earlier Aryabhata never used any symbol for numbers, he used words. He used ‘kha’ as placeholder to denote nothing(empty). Eg: 306, Aryabhata will denote 0 as kha for separation purpose or to denote nothing (Babylonians used “ or ‘ ‘ for same but were not consistent) and hence differentiating 306 and 36. This gave rise to the place value system which we follow today. Later 'kha' was replaced by 0 which meant same: nothing and later 0 was used by many Indian scholars like Brahmagupta, Bhaskara, etc in their books. Brahmagupta and Bhaskara I were the first to use 0 as a number. So by the time of Brahmagupta and Bhaskara I, we had all numerals from 0-9. Hindu Numerals goes West and more West Brahmagupta is considered as the teacher of Arabs. Brahmagupta’s book ‘Brahmasputa Siddhanta’ is believed to be translated in Arabic/Islamic language. Because of such books, Arab could understand the astronomy and which helped them to prepare their own calendar. Hindu numeral system (0-9) came in notice and fame in 825 AD , when by Persian famous Mathematician Al-Khwarizmi wrote a books On the Calculation with Hindu Numerals and On the Use of Hindu Numerals in the year 830. Since then Arabs started using Hindu numerals in … [Read more...]

## Indian Classical Period and Early Medieval Period

Indian Classical period starts around 400 AD and ends in 1200 AD. This period is often considered as the golden period for Indians in terms of science, technology, mathematics, astronomy, engineering, art, logic, philosophy, and religion. Restricting us to mathematics, people like Aryabhata , Bhramagupta, Bhaskara I and II, Mahavira were born in this period and had made immense contributions in the field of mathematics and astronomy. [wptabs style="wpui-narrow" effect="slide" mode="horizontal"] [wptabtitle] Aryabhata[/wptabtitle] [wptabcontent] It was this period in which Aryabhata was the first to tell many astronomical facts like earth is round, sun is the center point of solar system and all the planets revolve around sun, earth spins around its own axis, cause of day and night, etc. He also made huge contributions in mathematics as well. Although ancient Indians had been following decimal numeral system from the Harappan civilization, still there was no place value system. Decimal place value system was seen in Aryabhata’s Aryabhatiya book and this is considered as the turning point of mathematics and biggest contribution in the field of mathematics which gave rise to ‘0’ . Aryabhata didn’t invent ‘0’ symbol. Infact he did not use any of the symbols (1-9) though Brahmi numerals were widely used around same time but he did invent a place value system where '0' can be fixed (place holder). Also he used words for the numbers and to denote the place value system. He used the word "kha" to denote nothing (0) and it was later used as "ZERO".(see more in Hindu Numerals goes West Section) Earlier Babylonians had Place value system but they were not consistent. They used “ “ and (space) to denote 0. They would right 108 as 1”8 or 1 8 but they didn’t follow it always especially when 0 use to come at the end. So for them 180 was written as 18 and hence there was no differentiation for 180 and 18. Mayan although had separate symbol for 0 (Shell) but they did not follow the … [Read more...]

## Post Vedic Period

Various important events and important contributions were made in the period 500 BC - 400 AD which we call it as Post Vedic period. Religions like Buddhism, Jainism were formed; first Indian scripts- Brahmi scripts (derived from Kharoshti scripts) were recorded and which led the foundation for Devanagari numbers which ultimately led to the numbers which we follow today, epics like Ramayana and Mahabharata were written, complete treatise on Sanskrit grammar which was prepared by Panini and is even followed today, etc. [wptabs style="wpui-narrow" effect="slide" mode="horizontal"][wptabtitle] Jaina & Buddhist Mathematics[/wptabtitle] [wptabcontent] Jaina mathematics is one of the least understood chapters of Indian mathematics, mainly because of the scarcity of the work. Jain religion was formed by Varadhamana (Mahavira) around in the period 570 BC. Jaina mathematics is termed as mathematics which was followed by those who followed Jainism and was seen in the period 400 BC – 200 AD. Jaina mathematics could be seen written in various Sutras like Surya Prajnapati, Bhagabati Sutra, Sthananga Sutra, Jambudvipa Prajnapti , Vaishali Ganit, Uttaradhayyan Sutra, Anuyoga Dwara Sutra, Tiloyapannatti, etc. Many topics on mathematics were discussed in Sthananga Sutra: arithmatic operations like multiplication, division, subtraction, addition; Number Theory; Geometry;Mensuration of solid objects like sphere, cylinders,etc; fractions; Solving simple, quadratic, cubic, biquadratic equations, laws of indices and Permutation & Combinations. Like people from Vedic period, Jaina mathematicians were also interested in cosmology and large numbers. Jain cosmology says universe, as an uncreated entity, existing since infinity,the shape of the universe as similar to a man standing with legs apart and arm resting on his waist. This Universe, according to Jainism, is narrow at the top, broad at the middle and once again becomes broad at the bottom. Infinty: This brings the … [Read more...]

## Vedic Period

Vedic period is a period in which oldest scripts of world, in Sanskrit, called as Vedas were made which were based on Hinduism. Initially for many years these Vedic scripts, texts, hymes were transferred orally. The Vedic period starts around in 1700 BC and ends in the year 500 BC. People initially were present in north-west part of ancient India but slowly moved towards eastern India and also towards south (Deccan). People from the Vedic period cannot be termed as mathematicians by professions; they were priests/ pandits/ rishis who made use of mathematics for carrying out their religious rituals. Thus emphasis was not given on the proofs but used as properties/sutras. Vast knowledge of mathematics was also used in fields of astronomical and astrological topics like calculation of year duration, eclipses, zodiac signs etc. Vedic period can be divided in 2 periods (approx dates): [wptabs style="wpui-narrow" effect="slide" mode="horizontal"] [wptabtitle] Early Vedic Period[/wptabtitle] [wptabcontent] Early Vedic Period (1700 - 1000 BC): No mathematical texts are found from this period as in those days each information was transferred orally and was recited number of times so that they are well remembered. But still they used mathematics on large basis and used mathematical knowledge for carrying out their rituals. As stated in Rigveda, like Babylonians, they were able to independently predict the date of solar eclipse. Vedic people did name very large numbers (upto till 1062) and surprisingly they used all these big numbers just for their rituals.For one of the ritual, one of the Rishi counts till 1012. He after preparing bricks for a Vedic ritual, prays to the Lord of fire (Agni). Imā me Agna istakā dhenava Santvekā ća desa ća satam ća Sahasram ćāyutam ća niyutam ća Prayutam ćārbudam ća nyarbudam ća Samudrasća madhyam ćāntasća Parārdhasćaita me agna ishtakā Dhenavasantvamutrāmushmimlloke. Meaning: Oh Agni! Let these bricks be … [Read more...]