# Vedic Mathematics: the mathematics that emerged in the Akhand Bharath subcontinent

Vedic Mathematics or Hindu Mathematics refers to the mathematics that emerged in the Indian subcontinent, from ancient Vedic times until the period of Mathematical genius Ramanujam and HH Jagadguru Swami Bharati Krishna Tirth in the 20th century.

Eurocentrism
Many scholars feel that Indian contributions to science, technology, and mathematics have not been given due acknowledgment in modern history and that many discoveries and inventions by Indian mathematicians were known to their Western counterparts, copied by them, and presented as their own original work; and further, that this mass plagiarism has gone unrecognized due to Eurocentrism.
Hindus contribution in Mathematics include

01. Arithmetic:
Decimal system, Negative numbers (Brahmagupta), Zero (Hindu-Arabic numeral system), Binary numeral system, the modern positional notation numeral system, Floating point numbers (Kerala School), Number theory, Infinity (Yajur Veda), Transfinite numbers, Irrational numbers (Shulba Sutras)

02. Geometry:
Square roots (Bakhshali approximation), Cube roots (Mahavira), Pythagorean triples (Baudhayana and Apastamba in Shulba Sutras), Transformation (Panini), Pascal's triangle (Pingala)

03. Algebra:
Quadratic equations (Sulba Sutras, Aryabhata, and Brahmagupta), Cubic equations and Quartic equations (biquadratic equations) (Mahavira and Bhaskara II)

04. Mathematical logic:
Formal grammars, formal language theory, the Panini-Backus form (Panini), Recursion (Panini)

05. General mathematics:
Fibonacci numbers (Pingala), Earliest forms of Morse code (Pingala), Logarithms, indices (Jain mathematics), Algorithms, Algorism (Aryabhata and Brahmagupta)

06. Trigonometry:
Trigonometric functions (Surya Siddhanta and Aryabhata), Trigonometric series (Madhava and Kerala School of Mathematics)

Zero is derived from Sanskrit word Shunya
The Hindu word for zero was shunya, meaning empty, or void; this word, translated and transliterated by the Arabs as a cipher, is the root of the English words cipher and Zero.

Infinity was known to Vedic Rishis
The Isha Upanishad of the Yajurveda states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". (Purnamidah Purnamidam …..). In some Buddhist imagery, a mala is twisted in the middle to form a figure of 8. This represents the endless (infinite) cycle of existence, of birth, death and rebirth, i.e. the [infinity of] samsara.

Transfinite Numbers
Transfinite numbers are numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor (1845- 1918). The Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
• Enumerable: lowest, intermediate, and highest
• Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
• Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished -asamkhyāta ("countless, innumerable") and Ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

Modern Positional Notation Numeral System originated in Bharat
Georges Ifrah, the French author and historian of Mathematics, concludes in his Universal History of Numbers:

“The Brahmi notation of the first nine whole numbers are incontestably the graphical origin of our present-day numerals and there can be no doubt that our decimal place-value system was born in India and was the product of Indian civilization alone.”

Aryabhata stated "sthānam sthānam daśa gunam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras.

Binary Number System of the Vedic Period
The Vedic scholar Pingala (5th-2nd century BC or earlier) developed advanced mathematical concepts for describing prosody, and in so doing presented the first known description of a binary numeral system. He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.

Vedic Rishi Manava discovered Irrational Numbers
The concept of irrational numbers was implicitly accepted by Indian mathematicians since the 7th century BC when Manava (c. 750–690 BC) believed that the square roots of certain numbers such as 2 and 61 could not be exactly determined. Hippasus an ancient Greek Mathematician of 5th century BC was drowned at sea for working with irrational numbers.

Negative Numbers or numbers less than zero
For a long time until the 17th century, negative solutions to problems were considered "false or absurd” in the West. The use of negative numbers was known in early India, and their role in situations like mathematical problems of debt was understood. Consistent and correct rules for working with these numbers were formulated. The diffusion of this concept led the Arab intermediaries to pass it to Europe. The ancient Indian Bakhshali Manuscript, which Pearce Ian claimed was written sometime between 200 B.C. and A.D. 300, carried out calculations with negative numbers, using "+" as a negative sign.

During the 8th century A.D., the Islamic world learned about negative numbers from Arabic translations of Brahmagupta's works, and by A.D. 1000 Arab mathematicians were using negative numbers for debts.

Vedic Rishis solved Square and Square Roots
In Ancient India, the knowledge of theoretical and applied aspects of the square and square root was at least as old as the Sulba Sutras, dated around 800-500 B.C. (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits. According to the historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546.

Classical Indian Number Theory
Hindu Mathematicians were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations.
• Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. He also found the general solution to the indeterminate linear equation using this method.
• Brahmagupta in 628 used the chakravala method to solve more difficult quadratic Diophantine equations, including forms of Pell's equation, such as 61x2 + 1 = y2.
• His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126.
• In Europe, the equation 61x2 + 1 = y2 was solved in 1727 by Leonhard Euler, while the general solution to Pell's equation was found much later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method. Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later.
• Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic, and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations.
Vedic Origin of Pythagorean Theorem and Pythagorean Triplets Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
In India, the Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BC and the 5th century BC, contains a list of Pythagorean triples discovered algebraically. The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean Theorem, using an area computation. According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it.

Panini Ashtadhyayi’s contribution to Computer Science Backus–Naur Form (BNF) Examples
Since 1963 in computer science, Backus–Naur Form (BNF) is widely used as a notation for the grammars of computer programming languages, instruction sets, and communication protocols, as well as a notation for representing parts of natural language grammars. The Backus–Naur Form or BNF grammars have significant similarities to Panini's grammar rules (500 BC), and the notation is sometimes also referred to as Panini–Backus Form. Many textbooks for programming language theory and/or semantics document the programming language in Panini-Backus Form. Sulba Sutras Example
In the Sulba Sutras in ancient India, 8th century BC quadratic equations of the form ax2 = c and ax2 + bx = c were explored using geometric methods. Babylonian mathematicians from circa 400 BC, Chinese mathematicians from circa 200 BC and Euclid, the Greek mathematician around 300 BC solved quadratic equations with positive roots but did not have a general formula. In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit solution of the quadratic equation

ax2 + bx = c

(Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)”

The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y).

The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), before it was known in Europe. Developments have been attributed to Pingala (200 BCE), Virahanka (6th century CE), Gopāla (c.1135 CE), and Hemachandra (c.1150 CE). In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202). Fibonacci numbers are the numbers in the following sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …..

Indian inventions and Foreigners` claims – Few examples of Eurocentrism
Many scholars feel that Indian contributions to science, technology, and mathematics have not been given due acknowledgment in modern history and that many discoveries and inventions by Indian mathematicians were known to their Western counterparts, copied by them, and presented as their own original work; and further, that this mass plagiarism has gone unrecognized due to Eurocentrism.

Pascal Triangle

Indian invention

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Foreigners` claim

Trilostak (Pascal’s Triangle) is explained in Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala between the 5th and 2nd century BCE. Commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru".

Pell’s Equation

Indian invention

Brahmagupta (628 AD ) N x2 + 1 = y2

Foreigners' claim

Pell's equation. John Pell (1610-1685)

Pell's equations were studied as early as 1000 BC in India.

They were mainly interested in the equation

X2 – 2Y2 = 1

because of its connection to the square root of two. Indeed, if x and y are integers satisfying this equation, then x / y is an approximation of √2. For example, Vedic Rishi Baudhayana discovered that

x = 17, y = 12 & x = 577, y = 408

are two solutions to the Pell’s equation, and give very close approximations to the square root of two.

Fibonacci Series

Indian invention

Foreigners` claim

Fibonacci series (1170-1250)

The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), long before it was known in Europe.

Developments have been attributed to Vedic Scholar Pingala (400 BC), Virahanka (6th century AD), Gopāla (c.1135 AD), and Hemachandra (c.1150 AD).

The motivation came from Sanskrit prosody, where long syllables have length 2 and short syllables have length 1. Any pattern of length n can be formed by adding a short syllable to a pattern of length n − 1, or a long syllable to a pattern of length n − 2; thus the prosodists showed that the number of patterns of length n is the sum of the two previous numbers in the sequence. Donald Knuth reviews this work in The Art of Computer Programming.

Indian inventions Foreigners’ Claims

5. Mahavira formula (850 AD) Herigone`s formula (1634 AD) for combinations n Cr = (n)! / ( r!) (n-r)! ( ! stands for factorial)

6. Bhaskaracharya (1114-1193) Rolle`s theorem (1652-1719) Formula for relative difference (retrograde motion)

7. Madhav`s theorem (1340-1425) Gregory Series(1638-1675) x = tan x / 1 – tan 3 x / 3 + tan 5 x / 5 - .......

II (pie) = 1-1/3 + 1/5 - 1/7 +............ Leibnitz `s expansion (1646-1716)

9. Narayan Pandit (1356 AD) Fermat`s result (1601-65) -
factorization method for divisors of a number

10. Bhaskaracharya (1114-1193) Euler’s division algorithm -
method of finding the greatest common divisor

12. Nilkanth Somyaji (1444-1545) Euler`s results (1707-1783)
Summations ∑n, ∑n2, and ∑n3

13 Nilkanth Somyaji (1444-1545) Euler`s results
r sine rule: a / sin A =b / sin B = c / sin C

volumes of a frustum of cone and of a pyramid

15 Jyeshtha Deo (1500 AD) Euler
formulae for sin(x+y) and cos(x+y) in the text `Yuktibhasha`

16 Jyeshtha Deo (1500 AD), Leibnitz (1646-1716)
Linear equations,

17 Jyeshtha Deo (1500 AD) Liebnitz, by the method of integration
volume and surface area of a sphere

18. Shankar Variar (1500-60) Gauss(1777-1855)
Values of II/4, II/16 in series

There is an urgent need for young Hindus to do research in these areas and restore the glory to these forgotten Hindus.