X:The Elegant Hindu Arabic Numerals

X. The Elegant Hindu Arabic Numerals
        [Contd.A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri]

So far, we have travelled a lot at different parts of the world in search of numerals and counting systems of the civilizations of the ancient times. Now let us explore the most elegant and   perfect numerals and counting system,  prevalent at preseent in whole of the world, namely, the “Hindu Arabic numerals” or in a true sense the “Hindu numerals”.

The Hindu Arabic numeral system is a unique decimal place-value numeral system which uses a specific symbol or glyph for zero. Its glyphs are descended from Brahmi* numerals. The full system emerged by the 8^th to 9^th centuries and was first described in Al-Khwarizmi’s  book "The calculation with Hindu Numerals” (825 AD) and Al-Kindi’s  four volume work “on the use of the Hindu Numerals” (850 AD). Today the name “Hindu Arabic numerals” is usually used. But these numerals were invented by the Indian Mathematicians. So the Persians and Arabic Counter parts called “Hindu numerals” where “Hindu” meant Indian. Later they came to be called “Arabic numerals” in Europe, because they were introduced to the West by the Arab merchants.

[* “Brahmi” is the modern name given to one of the oldest writing system used in South and Central Asia during the final centuries BC and the early centuries CE. Like its contemporary Kharosthi was used  which is now Afganistan and Pakistan. The best-known Brahmi inscriptions are the rock-cut edicts of Ashoka in north-central India, dated 250 – 232 BC.

Historians trace modern numerals in most languages to Brahmi numerals, which were      used around the middle of the 3^rd century BC. The place system, however, developed later. The Brahmi numerals have been found in inscriptions in caves and coins in regions near Pune, Mumbai and Uttar Pradesh in India. These numerals were in use with slight variations, over quite a long time span up to the 4^th century]

It will be worthy to know, at least briefly, who were these two scholars and renowned mathematicians of the Arab World.

Al-Khwarizmi or Latinized as Algoritmi was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad. In the 12^th century, Latin translations of his book on Indian numerals introduced the decimal positional number system to the Western World.

And Al-Kindi known as “the philosopher of the Arab’s” was a Muslim Arab philosopher, polymath, mathematician, physician and musician. He was born in Basra and educated in Baghdad, two well-known Iraqi cities. Al-Kindi became a prominent figure in the House of Wisdom and he was entrusted with the job of translations of Greek scientific and philosophical texts into Arabic language. It enriched him greatly with Greek philosophy. In the field of mathematics Al-Kindi played an important role in introducing Indian numerals to the Islamic and Christian World.

During Vedic period (1500 – 500 BC), motivated by geometric construction of the fire altars and astronomy, the use of a numerical system and basic mathematical operations developed in northern India. Hindu Cosmology required the mastery of very large numbers, such as ‘Kalpa’, the life time of the universe and said to be 4,320,000,000 years. The expanse or ‘orbit of the universe’, as they calculated to be 18,712,069,200,000,000 yojanas, according to modern measure a yojana is a distance about 8 km, controversially may be a distance between 8 and 13 km. So, what an unimaginably large number they could think of!

Numbers were expressed using names for the place-value in powers of 10 as dasa, satha, sahasra, ayuta, nijuta, prajuta, arbuda, nyarbuda, samudra, madhya, anta, parardha etc. These places, as per our modern notation in powers of 10 were 10¹, 10², ------ upto 10¹², the last of these being the name for trillion. For example, the number 46352 was expressed as 4 ayuta six sahasra 3 satha 5 dasa 2. The extreme right hand numeral 2, as per modern usage, is in unit’s place having a place value of 10⁰ or 1 and is a multiple of 1.

It is a long history how the present form of glyphs or symbols had come into prevalence. Various symbol sets were used to represent numbers in Hindu-Arabic numeral system, all of which were developed from Brahmi numerals. The symbols used to represent the system had split into various typographical variants since the Middle Ages may be classified into three main groups as shown in Table No.-9a.

Table No.9a                                                      

                                                     Brahmi numerals.

Firstly, the wide spread Western “Arabic numerals” used with Latin, Cyrillic and Greek alphabets, sometimes called “European” descended from the “West Arabic numerals”.

Secondly, the “Arabic-India” or “Eastern Arabic numerals” used with Arabic scripts was developed primarily in what is now Iraq.

Thirdly, the “Hindu numerals” in use with scripts of the Brahmi family was in India and South East Asia. Each of the roughly dozen major scripts of India has its own numeral glyphs.

Some of those major glyphs are as the followings: [Table No. : 9(b) and : 9(c)]

Table No. : 9(b) : Numeral glyphs descend from Brahmi family

Table No. : 9(c) : Hindu Arabic numerals

It is without doubt that mathematics today owes a huge debt to the outstanding contributions on beautiful number system invented by the Indian mathematicians over many hundred years, on which much of mathematical development rested. What is quite surprising is that many famous historians of mathematics are reluctant to recognize this, though it was so clear in front of them.

The history of earliest development of Indian mathematics may be traced back to Indus valley civilization flourished about 2500 BC and survived until 1700 BC or later. This earliest known urban Indian culture was first identified in 1921 at Harappa and then one year later at Mohenjo-Daro, near the Indus River in Sindh. Both of these are now in Pakistan but this is still covered by the term “Indian mathematics”.

The next mathematics of importance on the Indian subcontinent was associated with the religious texts of Vedas which were composed in Vedic Sanskrit between 1500 BC and 800 BC. At first these texts, consisting of hymn, spells and ritual observations were transmitted orally. Later the texts became written works for the use of those practicing the Vedic religious. It consisted of the Sulbasutras which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed not for its own sake, but purely for religious purposes. The mathematics contained in these texts is studied in some detail in the separate article on Sulbasutras.

Ancient and medieval mathematical works, all composed in Sanskrit, usually consisted of a section of Sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solutions. All mathematical works were transmitted orally until approximately 300 BC; there after they were transmitted both-orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is “Bakshali Manuscript”*

[*The Bakshali Manuscript is a mathematical script found by a peasant in the village of Bakshali which is near Peshawar, now in Pakistan, in 1881. It is notable for being the oldest extant, likely from the 7^th century CE, manuscript of Indian mathematics. The manuscript is incomplete consisting of seventy leaves of birch bark. The intended order of those leaves is indeterminate. It is currently housed in the Bodleian Library of the University of Oxford.

The manuscript is a collection of rules and illustrative examples. The sample problems are in verse and the commentary is in prose associated with calculations and solutions. The problems involved arithmetic, algebra, geometry including mensuration. The topics covered include fractions, square roots, arithmetic and geometric progressions, solution of simple equations, simultaneous linear equations, quadratic equations and indeterminate equations of second degree.]

The development of positional decimal system takes its origin in Hindu Mathematics during Gupta period*. Around 500 AD, the astronomer Aryabhata used the word “Kha”, meaning emptiness, to mark zero in tabular arrangements of digits. The 7^th century Brahamasphuta Siddhanta contained a comparatively advanced understanding of the mathematical role of zero. The first dated and undisputed inscription showing the use of a symbol for zero appears on a stone inscription found on the Chaturbhuja Temple at Gwalior in India, dated 876.

[* Gupta Empire : Gupta Empire was an ancient Indian empire founded by Sri Gupta which existed at its zenith from approximately 320 to 550 CE and covered much of the Indian subcontinent. The peace and prosperity created under the leadership of the Guptas enabled the persuit of scientific and artistic endeavours. This period is called the Golden Age of India and was marked by extensive inventions and discoveries in science, technology, engineering, art, dialectic, literature, logic, mathematics, astronomy, religion and philosophy that crystallized the elements of what is generally known as Hindu Culture.

Chandragupta – I, Samudragupta and Chandragupta – II were the most notable rulers of the Gupta dynasty. The Guptas conquered about 21 kingdoms, both in and outside India. The 4^th century CE legendary Sanskrit poet Kalidasa was one of the nine scholars, known as “Navaratnas” of the court of the legendary ruler Chandragupta – II, famously known as Vikramaditya.]

Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, algebra. In addition, trigonometry was further advanced in India.

In a more developed form of the Hindu numeral positional decimal system, there was also use of decimal marker, at first a mark over one’s digit, but now more usually a decimal point. Moreover a symbol for what we now call “recurring decimal” or to specify “these digits recur ad infinitum” was used. In modern usage, this later symbol is usually a vinculum, a horizontal line placed over the recurring digits. In this more developed form, the numeral system can symbolize any rational number, namely, a number which can be expressed as a ratio of two numbers, using only 13 symbols. Those are ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), decimal marker, vinculum and a prepended dash to indicate a negative number.

Indian mathematics emerged in Indian subcontinent from 1200 BC until the end of the 18^th century. In classical period of Indian mathematics ranging from 400 CE to 1600 CE important contributions were made by scholars* like Aryabhata, Varahamihira, Brahamagupta, Bhaskara-I, Bhaskara-II. Madhaba of Sangagrama, Nilkantha Somayaji and many others.

[*Aryabhata : The first most important name of mathematician and astronomer of early classical period of Indian mathematics is Aryabhata (476 – 550 CE).He is also known as Aryabhata – I or Aryabhata the Elder to distinguish him from a 10^th century Indian mathematician of the same name. The theories that he came up at that time present a wonder to the scientific world today. His works were used by the Greeks and the Arabs to develop further.

Aryabhata born in 476 CE but provides no information about his place of birth. According to scholars he was born in Tarenaga, a town in Bihar, India. It is however definite that he travelled to Kusumapara which is modern day Patna for studies and flourished there. It is mentioned in a few places that Aryabhata was the head of the educational institute in Kusumapara. The famous University of Nalanda had an observatory in its premises and it is hypothesized that Aryabhata was the principal of the university as well. On the other hand some other commentaries mention that he belonged to Kerala. But this hypothesis has no strong evidence.

Aryabhata wrote many mathematical and astronomical treatises. His chief work was the ‘Aryabhatiya’ which was a compilation of mathematics and Astronomy. It covers several branches of mathematics such as algebra, arithmetic, plane and spherical trigonometry. Also included in it are theories on continued fractions, sum of power series, sine tables and quadratic equations.

Aryabhata worked on place value system using letters to signify numbers and stating qualities. He also came up with an approximation of pi (π) which is the ratio of the circumference to its diameter of a circle. He introduced the concept of sine in his work called ‘Ardha-jya’ which is translated as ‘half-chord’.

Aryabhata also did a considerable amount of work in astronomy. The Aryh-Siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhat’s contemporary, Varahamihira and later mathematicians and commentators, including Brahamagupta and Bhaskara – I. He knew that the earth is rotating on its axis around the Sun and the moon rotated around it. This heliocentric idea was not known until the 16^th century, more than a millennium later, when a geometric mathematical model of a heliocentric system was presented by Renaissance mathematicians and astronomers and a Polish Catholic Cleric Nicholus Copernicus, leading to the Copernican Revolution.

In some texts, he seems to ascribe the apparent motions of the heavenly bodies to the Earth’s rotation. He also discovered the position of the nine planets and stated that these also revolve around the Sun. He may have believed that the planets’ orbits as elliptical rather than circular which was discovered much later, in 1609, by a famous German mathematician and astronomer Johannes Kepler.

He scientifically explained the eclipses, both solar and lunar. He stated that the moon and planets shine by reflected sunlight. Instead of prevailing Cosmology based on common belief that eclipses are caused by Rahu and Ketu, two demons, he explained the eclipses in terms of shadows cast by and falling on Earth.

Considered in modern English units of time, Aryabhata calculated the diurnal rotation or time of full rotation of Earth about its axis with respect to the Sun (Sidereal rotation) as 23 hours, 56 minutes, 4.1 seconds, while the modern value is 23:56:4.091.

Similarly, his value for the time Earth takes to complete its orbit around the Sun with respect to fixed stars (Sidereal year) was 365 days, 6 hours, 12 minutes and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds (365.25636 days); astonishing! Unbelievable! What a genius Aryabhata was! India’s first satellite launched in 1975 was named after this great astronomer Aryabhata.

He was the first person to mention that the Earth was not flat but in a spherical shape. He also gave the circumference and diameter of the Earth and radius of orbits of planets.

Brahama Gupta :

Brahamagupta (598 – 670 CE) was one of the most significant mathematicians of ancient India. He introduced extremely influential concepts to basic mathematics, including the use of zero in mathematical calculations and the use of mathematics and algebra in describing and predicting astronomical events.

Brahamagupta was born in 598 AD in Bhinmal, a state of Rajasthan in India. He spent most of his life in Bhinmal. He was the head of the astronomical observatory at Ujjain which was the centre of mathematics in India witnessing the work of many extra-ordinary mathematicians. The astronomically significant ancient Indian city of Ujjain was a place near the tropic of cancer that occupies a place in Indian history somewhat comparable to that of Greenwich in England. It was a central reckoning point for ideas of time and space, and it became a major astronomical and mathematical centre.

Little else is known of the life of this mathematician and astronomer who flourished 1400 years ago. It is known that, though he was a devout Hindu, yet he however, rejected the ancient Hindu ideas that the Earth was flat or bowl-shaped; like ancient Greek thinkers, including Aristotle, he realized that it was a sphere and rotate about an axis.

Brahamagupta is known mostly through his writings, which cover mathematical and astronomical topics and significantly combine the two. The first of his two surviving treaties was “Brahama Sphuta – Siddhanta”, often translated as “The opening of the universe” was written in 1628 when he was about 30 years old. His second, the “Khandakhadyaka” which means something like “Edible Bite”, is less well known. It expands on the work of the earlier astronomer, Aryabhata, whose chief contribution was the idea of beginning of each day at midnight. It was written in 665 AD, near the end of his life.

Brahamagupta’s first manuscript the Brahamasphutasiddhanta, was a revision of an older astronomy book “Brahamasiddhanta”. It opened with three chapters on the position and motions of the planets and stars, and on the cycle of day and night. Two chapters dealt with lunar and solar eclipses respectively and one with heliacal rising and setting of stars, planets and moon – the seasonal reappearances and disappearances of these celestial bodies as they pass the horizon line before being hidden by the sun. He also discussed phases of the moon, close approaches or conjunctions of planets in the sky and conjunctions between planets and stars. Brahamagupta also calculated the length of the solar year as 365 days, 6 hours, 5 minutes, and 19 seconds, among the most accurate of early reckonings and remarkably close to actual value of 365 days, 5 hours, 48 minutes and about 45 seconds. It should be remembered that these were remarkable estimates in an era that had no telescope or scientific instruments in the modern sense.

After discussion of astronomy, Brahamagupta turned to mathematics, discussing what would now be called arithmetic and algebra which he termed as “pati-ganita”, or mathematics of procedures, and “bija-ganita”, or mathematics of equations. These ideas laid the foundation for much of the later development of mathematics in India. Some of Brahamagupta’s discussions will sound familiar to the modern students of mathematics. His direction for multiplication of large numbers, in a sense, is close to what students are taught today. His preferred multiplication method, according to the mathematics history website maintained by St. Andrews University in Scotland, is given the name “gomutrika” by Brahamagupta, meaning “like the trajectory of a Cow’s Urine”.

 Brahamagupta also introduced new methods for solving quadratic equations that will be recognizable to modern students of mathematics. He also devised formulas for calculating the area and length of the diagonals of a cyclic quadrilateral. His method is still known as Brahamagupta’s theorem. Brahamagupta investigated various higher functions of algebra and geometry, in each case building on and refixing the mathematical heritage of the ancient world. A curious feature of Brahamagupta’s treatise is that it is largely written in Sanskrit verse.

Although it is difficult to pinpoint single inventor of the concept of zero, Brahamagupta is a reasonable contender for the title. Perhaps Brahamagupta’s most important innovations, however, pertained to his treatment of the number zero. Several different discoveries converged to from the concept of zero. The circular symbol for the number and the idea of representing orders of magnitude in a number through the use of places arose at different times and places prior to Brahamagupta’s work. Brahamagupta, however, was the first to propose rules for the behavior of zero in common arithmetical equations, relating to zero to positive and negative number which he termed ‘fortunes’ and ‘debts’. He correctly stated that multiplying any number by zero yields a result of zeros but erred, as did many other ancient mathematicians, in attempting to division by zero. Nevertheless, Brahamagupta is sometimes called the “Father of zero”.

The impact of Brahamagupta’s discoveries was felt in Islamic world. Brahamagupta’s writings were translated into Arabic in 771 and they had a major impact on subsequent writers in the Arab world, including al-Khwarizmi, the ‘father of algebra’. The mathematical thought of medieval and early modern Europe was influenced by Arabic models that had been in existence for centuries. Distant from the modern mathematics in time and place, Brahamagupta nevertheless exerted a definite influence on mathematics as the discipline is known today.

Bhaskaracharya : (1114 – 1185 CE)

Bhaskaracharya is also known as Bhaskara – II to avoid confusion with the same name of mathematician of the 7^th century AD. He was a great Indian mathematician and astronomer. Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12^th century. He has been called the greatest mathematician of medieval India.

He was born in 1114 CE near Vijjadavida, believed to be Bijjaragi of Bijapur in modern Karnataka. Bhaskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical centre of medieval India. It goes to the credit of Varahamihira and Brahamagupta, the leading mathematician who worked there and built up this school of mathematical astronomy.

Bhaskaracharya’s main work Siddhanta Shiromani consists of four parts called Lilavati, Bijganita, Grahagamita and Goladhyaya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of planets and spheres respectively. He also wrote another treatise named Karana Kautuhala.

It is surprising that Bhaskara-II had conceived the basic ideas of differential calculus, a brilliant and effective mathematical tool for future development of applied sciences in general and mathematics in particular, predates Newton in England and Leibnitz in Germany half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibnitz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus.

He wrote Siddhanta Shiromani at the age of 36. This colossal work consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book. Lilavati has 298, Bijganita has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated the simple ways of calculation from Arithmetic to Astronomy in this book. He wrote Lilavati in an excellent lucid and poetic language. It has been translated in various languages throughout the world.

                                                                                                                      [To continue]

  Reference: Internet.

 Image credit:Table No 9a: Bramhi numerals: Creative Cosmos (https://commons.wikimedia.org)/wiki/File:Brahmi_numeral_signs.svg)
                                      9b:Glyphs of numerals descended from Bramhi:                                           Wikipedia (https://en.wikipedia.org/wiki/History_of_science_and_technology_in_the_Indian_subcontinent )
                                       9c:Hindu Arabic numerals: ( http://archimedes-lab.org/ )