Do you know that India is the birthplace of many great intellectual inventions including the number system, algebra, trigonometry, calculus?

The Invention of the Fundamentals of Maths

Invention of Decimal System

Did you know that Hindus gave us the method of expressing numbers by means of a decimal system?

The so called Arabic numerals are actually Hindu numerals and even many Arab mathematicians admit that. During the 700's, the Arabs learned Hindu arithmetic from scientific writings of the Hindus and the Greeks. Then, in the 800's, a Persian mathematician wrote a book that was translated into Latin about 300 years later. This translation brought the Hindu-Arabic numerals into Europe.

Hindu mathematicians used a system based on 10. The Hindus had symbols for each number from one to nine. They had a name for each power of 10, and used these names when writing numerals. For example, Hindus wrote "1 sata, 3 dasan, 5" to represent the number we write as 135. They wrote "1 sata, 5" for the number we write as 105. Hindus found a way of eliminating place names. They invented the symbol shunya (meaning empty), which we call zero. With this symbol, they could write "105" instead of "1 sata, 5."

The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 1053 (i.e 10 to the power of 53) with specific names (Tallakshana) as early as 5000 B.C. during the Vedic period. Even today, the largest used number is Tera: 1012 (10 to the power of 12).

"It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity."
— French mathematician Pierre Simon Laplace (1749 - 1827)

The Place Value System

The place value system is built into the Sanskrit language and so whereas in English we only use thousand, million, billion etc, in Sanskrit there are specific nomenclature for the powers of 10, most used in modern times are dasa (10), sata (100), sahasra (1,000=1K), ayuta (10K), laksha (100K), niyuta (106=1M), koti (10M), vyarbuda (100M), paraardha (1012) etc. Results of such a practice were two-folds. Firstly, the removal of special importance of numbers. Instead of naming numbers in grops of three, four or eight orders of units one could use the necessary name for the power of 10. Secondly, the notion of the term "of the order of". To express the order of a particular number, one simply needs to use the nearest two powers of 10 to express its enormity.

The Word-Numeral System

The word-numeral system was the logical outcome of proceeding by multiples of ten. Thus, in an early system, 60,799 is denoted by the Sanskrit word sastim (60), shsara (thousand), sapta (seven) satani (hundred), navatim (nine ten times) and nava (nine). Such a system presupposes a scientifically based vocabulary of number names in which the principles of addition, subtraction and multiplication are used. It requires:
  1. the naming of the first nine digits (eka, dvi, tri, catur, pancha, sat, sapta, asta, nava);
  2. a second group of nine numbers obtained by multiplying each of the nine digits in 1 by ten (dasa, vimsat, trimsat, catvarimsat, panchasat, sasti, saptati, astiti, navati): and
  3. a group of numbers which are increasing integral powers of 10, starting with 102 (satam sagasara, ayut, niyuta, prayuta, arbuda, nyarbuda, samudra, Madhya, anta, parardha…).
To understand why word numerals persisted in India, even after the Indian numerals became widespread, it is necessary to recognize the importance of the oral mode of preserving and disseminating knowledge. An important characteristic of written texts in India from times immemorial was the sutra style of writing, which presented information in a cryptic form, leaving out details and rationale to be filled in by teachers and commentators. In short pithy sentences, often expressed in verse, the sutras enabled the reader to memorize the content easily.

The concept of 'zero'

Did you know that the ancient Hindus originated the concept 'zero'?

The concept of zero is referred to as shunya in the early Sanskrit texts and it is also explained in the Pingala’s Chandah Sutra (200 AD). In the Brahma Phuta Siddhanta of Brahmagupta (400-500 AD), the zero is lucidly explained. The Hindu genius Bhaskaracharya proved that x divided by 0 = 4 (infinity) and that infinity however divided remains infinity. This concept was recognized in Hindu theology millennia earlier. The earliest recorded date for an inscription of zero (inscribed on a copper plate) was found in Gujarat (585 – 586 AD). Later, zero appeared in Arabic books in 770 AD and from there it was carried to Europe in 800 AD.

The Indian place-value numeration with zero sign ranks among humanity's fundamental discoveries.

Origin of Geometry, Algebra, Trigonometry and Calculus

Did you know that Geometry, Trigonometry, Calculus and Algebra are studies which originated in India?

The word Geometry seems to have emerged from the Sanskrit word gyaa-miti which means "measuring the Earth". And the word Trigonometry is similar to tri-kona-miti meaning "measuring triangular forms". Euclid is credited with the invention of Geometry in 300 BCE while the concept of Geometry in India emerged in 1000 BCE, from the practice of making fire altars in square and rectangular shapes. The treatise of Surya Siddhanta describes amazing details of Trigonometry, which were introduced to Europe 1200 years later in the 16th century by Briggs. All Hindu as well as Buddhist mandalas and yantras are complex forms of Geometrical shapes.

Bhaskaracharya otherwise known as Bhaskara is probably the most well known mathematician of ancient Indian today. Bhaskara wrote his famous Siddhanta Siroman in the year 1150 A.D. It is divided into four parts; Lilavati (arithmetic), Bijaganita (a treatise on algebra), Goladhyaya (celestial globe), and Grahaganita (mathematics of the planets). An Arabic Scholar Al Zabar translated a Bhaskara's work Bijaganita from Sanskrit. It was later known as Algebra in European languages.

From India the sine function was introduced to the Arab world in the 8th century, where the term jya was transliterated into jiba or jyb. Early Latin translations of Arabic mathematical treatises mistook jiba for the Arabic word jaib, which can mean the opening of a woman's garment at the neck. Accordingly, jaib was translated into the Latin sinus, which can mean "fold" (in a garment), "bosom," "bay," or even "curve." Hence our word "sine."

The word “Algorithm” was actually supposed to be pronounced “Al-Khwarizmi”, which was the name of an eminent 9th century Arab scholar, who played important roles in importing knowledge on arithematic and algebra from India to the Arabs. In his work, De numero indorum (Concerning the Hindu Art of Reckoning), it was based presumably on an Arabic translation of Brahmagupta where he gave a full account of the Hindu numerals which was the first to expound the system with its digits 0,1,2,3,…,9 and decimal place value which was a fairly recent arrival from India. The new notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm.

The Value of Pi

Did you know that the ratio of the circumference and the diameter of a circle known as Pi (a value of 3.141592657932…) was first calculated by Hindus?

The Sanskrit text, by the famous Hindu mathematician, Baudhayana in his Baudhayana Sulbha Sutra of the 6th century BC mentions this ratio as approximately equal to 3. The Hindu mathematician, Aryabhatta, in 499 AD worked out the value of Pi to the fourth decimal place as [3x (177/1250) = 3.1416]. In 825 AD one Arab mathematician Mohammad Ibna Musa said: This value has been given by the Hindus [Indians] (62832/20,000 = 3.1416).

Binary System of Number Representation

A Mathematician named Pingala developed a system of binary enumeration convertible to decimal numerals. He described the system in his book called Chandahshaastra. The system he described is quite similar to that of Leibnitz, who was born in the 17th century.

A binary number system was used by Pingala (450 BC, if we accept the tradition that he was Panini's brother) to represent meters of songs. The structure of this number system may have helped in the invention of the sign for zero that, took place around 50 BC - 50 AD. Without the binary system, the development of computers would be much harder; and without a sign for zero, mathematics would have languished. It is of course true that the binary number system was independently invented by Leibnitz in 1678, but the fact that the rediscovery had to wait almost 2,000 years only emphasizes the originality of Pingala's idea.

Can you imagine today’s computers without the invention of zero?

Baudhayana’s Theorem

Did you know that the so-called Pythagoras Theorem that the square of the hypotenuse of a right-angled triangle equals to the sum of the square of the other two sides was documented by the famed Hindu mathematician Baudhayana in his 6th century BC treatise called Baudhayana Sulba Sutra?

Baudhayana states:
"The area produced by the diagonal of a rectangle is equal to the sum of area produced by it on two sides."
Bhaskaracharya’s Law of Gravity

Did you know that the famous Hindu astronomer, Bhaskaracharya in his Surya Siddhanta wrote:
"Objects fall on the earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon and sun are held in orbit due to this attraction."
It was not until 1687, 1200 years later did Issac Newton "rediscover" the Law of Gravity.

In Surya Siddhanta, dated 400-500 AD, the ancient Hindu astronomer Bhaskaracharya states,
"Objects fall on the earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit due to this force."
Approximately 1200 years later (1687 AD), Sir Isaac Newton rediscovered this phenomenon and called it the Law of Gravity.
Astronomy and Cosmology

Do you know Indian astronomers had mapped the sky 4000 years ago?

Earth is Round and Revolves Around the Sun

One thousand years before Copernicus (1543) published his theory of the revolution of the earth, the famous Hindu mathematician, Aryabhatta in the 5th century (400-500 AD) clearly stated this fact:
"Just as persons traveling on a boat feel that the trees on a bank are moving, people on earth feel that the sun is moving."
In Aryabhatta’s treatise (Aryabhateean) on this subject matter he clearly states that the earth is round; it rotates on its axis, orbits the sun and is suspended in space. Aryabhatta, in his treatise also explained that lunar and solar eclipses occur by the interplay of the shadows of the sun, the moon and the earth. India's first man made satellite was named Aryabhatta.

Copernicus published his theory of the revolution of the earth in 1543.

Time Taken for Earth to Orbit Sun

The famous Hindu mathematician, Bhaskaracharya, in his treatise Surya Siddhanta, calculated the time taken for the earth to orbit the sun to nine decimal places (365.258756484 days).

Bhaskaracharya rightly calculated the time taken by the earth to orbit the sun hundreds of years before the astronomer Smart. His calculations was - Time taken by earth to orbit the sun: ( 5th century ) 365.258756484 days.

Today’s accepted measurement is 365.2564 days. Therefore, assuming that today’s figures are correct, it means that Bhaskaracharya was off by only 0.0002%.

Smallest and largest measuring units of Time

The ancient Hindus had given the world the idea of the smallest and largest measuring units of Time. Astonishingly, the ancient Hindus used the following units of time:

Krati34,000th of a second
1 Truti300th of a second
2 Truti1 Luv
2 Luv1 Kshana
30 Kshana1 Vipal
60 Vipal1 Pal
60 Pal1 Ghadi24 minutes
2.5 Gadhi1 Hora1 Hour
24 Hora1 Divas1 Day
7 Divas1 Saptaah1 Week
4 Saptaah1 Maas1 Month
2 Maas1 Rutu (season)
6 Rutu1 Varsh1 Year
100 Varsh1 Shataabda1 Century
10 Shataabda1 Sahasraabda10 Centuries or 1000 Years
432 Sahasraabda1 Yuga4320 Centuries or 432000 Years
10 Yuga1 Mahayuga43200 Centuries or 4320000 Years
1000 Mahayuga1 Kalpa43200000 Centuries or 4.32 Billion Years
India gave the largest measurement of time as 8.64 billion years.

The Gregorian calendar on your desk simply adds on one day for every 4 years and is not in coherence with the movement of sun. But, Hindu calendar is in coherence as the short fall is corrected in the month itself by adding Adhikamasa as postulated by Maharshi Vishwamitra. Rig Veda 1.164.1, 2, 14 and 15 describe sun's motion, ritus and colours of spectrum. Kalyana varma, Varahamihira, Jaimini, Vidyanatha Deekshita, Kalidasa, Mantreshwara, Satyacharya, Venkatadri, Parashara, Ramadayalu and Garga have immensely contributed for the development of Hindu astrology.

An Extremely Old Universe

The idea that the universe is very old is quite startling, when one notes that humanity's collective memory doesn't go further than a few thousand years. The Puranas speak of the universe going through cycles of creation and destruction of 8.64 billion years, although there are longer cycles as well. The figure of 8.64 billion years is about right based on current astrophysical estimates. The revolutionary nature of this idea becomes clear when one notes that only a couple of hundred years ago the dogma in most Eurasia was that the world was created in 4004 BC.

Life-Cycles of the Universe

The Hindus view that the Universe has no beginning or end, but follows a cosmic creation and dissolution. Hindus are the only one who propounds the idea of life-cycles of the universe. It suggests that the universe undergoes an infinite number of deaths and rebirths. Hindus views the universe as without a beginning (anadi = beginning-less) or an end (ananta = end-less). Rather the universe is projected in cycles. Hindu scriptures refer to time scales that vary from ordinary earth day and night to the day and night of the Brahma that are a few billion earth years long.

According to Carl Sagan,
"A millennium before Europeans were wiling to divest themselves of the Biblical idea that the world was a few thousand years old, the Mayans were thinking of millions and the Hindus billions".
Continues Carl Sagan,
"… is the only religion in which the time scales correspond… to those of modern scientific cosmology."
Its cycles run from our ordinary day and night to a day and night of the Brahma, 8.64 billion years long, longer than the age of the Earth or the Sun and about half the time since the Big Bang". One day of Brahma is worth a thousand of the ages (yuga) known to humankind; as is each night." Thus each kalpa is worth one day in the life of Brahma, the God of creation. In other words, the four ages of the mahayuga must be repeated a thousand times to make a "day ot Brahma", a unit of time that is the equivalent of 4.32 billion human years, doubling which one gets 8.64 billion years for a Brahma day and night. This was later theorized (possibly independently) by Aryabhata in the 6th century. The cyclic nature of this analysis suggests a universe that is expanding to be followed by contraction… a cosmos without end. This, according to modern physicists is not an impossibility.

An Atomic World and the Subject/Object Dichotomy

According to the atomic doctrine of Kanada, there are nine classes of substances: ether, space, and time that are continuous; four elementary substances (or particles) called earth, air, water, and fire that are atomic; and two kinds of mind, one omnipresent and another which is the individual. This system also postulates a subject/object dichotomy, which is a part of the systems of Sankhya and Vedanta as well. In these systems, the conscious subject is separate from the material reality but he is, nevertheless, able to direct its evolution. The atomic doctrine of Kanada is much more interesting than that of Democritus.

It is the recognition of the subject/object dichotomy that led to the creation of modern physics.

Relativity of Time and Space:

That space and time need not flow at the same rate for different observers is a pretty revolutionary notion. We encounter it in Puranic stories and in the Yoga Vasishtha. Obviously, we are not speaking here of the mathematical theory of relativity related to an upper limit to the speed of light, yet the consideration of time acting different to different observers is quite remarkable. To see the significance of this idea a couple of thousand years ago, note that modern relativity theory was forced upon scientists a hundred years ago by certain equations related to the transmission of electromagnetic waves.

Here's a passage on anomalous flow of time from the Bhagavata Purana:
Taking his own daughter, Revati, Kakudmi went to Lord Brahma in Brahmaloka, and inquired about a husband for her. When Kakudmi arrived there, Lord Brahma was engaged in hearing musical performances by the Gandharvas and had not a moment to talk with him. Therefore Kakudmi waited, and at the end of the performance he saluted Lord Brahma and made his desire known. After hearing his words, Lord Brahma laughed loudly and said to Kakudmi, "O King, all those whom you may have decided within the core of your heart to accept as your son-in-law have passed away in the course of time. Twenty-seven chaturyugas have already passed. Those upon whom you may have decided are now gone, and so are their sons, grandsons and other descendants. You cannot even hear about their names."
There are other stories, less dramatic, where an observer returns from a journey to another loka, and finds that people he loves have aged many more decades than he has.

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