In
Ancient India, mathematics was only used in an applied role, and studied for its own stake  primarily to formulate mathematical methods to solve architectural problems or to supplement studies of other branches of science. However, the intensity and depth with which these mathematical studies have been conducted, has amazed and stunned the world of mathematics ever since it has been discovered. Vedic Mathematics, as it is commonly known, was used to understand astronomical phenomena (as in determining timings of festivals and rituals), solve architectural problems (as in the constructions of
Harappa), for astrology (as in the works of
Jain mathematicians), for medicinal compositions (as in
Ayurveda), and even in construction of Vedic altars (as in
Shulbasūtra of
Baudhāyana).
Vedic Mathematics, or simply Vedic Maths, finds its applications not only in arithmatic, but also in algebra, geometry, trigonometry, calculus and applied mathematics. Contrary to common belief, Vedic Mathematics is not an alternate numeral system, or even methodology. It is a set of mathematical rules formulated to supplement the usual methods of calculations. Vedic Mathematical Sūtras provide simple, straightforward alternates to calculate, the otherwise cumbersome and complicated calculations:
Simplify Mathematical Computations and Calculations with these Sūtras, and Explanations »
  Upavidhi Vedic Maths 
We, at Upavidhi, believe that Vedic Mathematics does not find its due respect in the world of mathematics. It is our belief that the principles of Vedic Mathematics should be propagated and excercised by more and more people around the world  for the greater good of mathematics, and for no other vested reason. Our efforts are dedicated to Vedic Mathematics  the oldest, fastest and easiest mathematical methods, and we make our standpoints amply clear in the
Introduction.
And, although we are obviously biased towards the principles of Vedic Mathematics, our
Upavidhi Blog on Mathematics discusses mathematics as a whole subject, and is not constrained by the boundaries of Vedic Mathematics.
The base10 (decimal) numeral system (1,2,3,...,9,10,...) that the entire world uses today is commonly known as the
HinduArabic numeral system  despite the fact that the entire system was developed in
Ancient India, and only spread in the western world through Arabic translations.
The two
Ancient Indian mathematicians who are credited with developing this system are Aryabhata of Kusumapura (for developing the placevalue notation in the 5th century AD) and Brahmagupta of Bhillamala (for introducing the concept and symbol of zero in the 6th century AD). This numeral system, including the concept of zero, spread into various countries due to commercial and military activities with
Ancient India and was adopted, especially by the Arabs  who modified the
SanskritDevanagari texts of numerology and further spread them to the western world. Even today, the Arabs refer their subject of mathematics as
Hindsa, and their numeral system as
Rakam AlHind, or the Hindu numeral system. The Arabic translation of Brahmagupta's
Brahmasputha Siddhanta was titled
Sind Hind, and remained the primary reference for Arab mathematicians  for many centuries. It is, thus, not a coincidence that the numeral system that the world uses today, exhibits great similarities with many of the
SanskritDevanagari notations, which are still used in India.
(Source: Wikipedia 
Numeral system,
Aryabhata,
Brahmagupta)
In
Ancient India, Mathematics was referred as
Ganitam, whereas 'algebraic rules' were referred as
Bijaganitam (
Bija means 'another', or 'second' and
Ganitam means mathematics)  implying a set of rules, or
Sūtras (rules), recognized as a parallel form of computation, in addition to the conventional operations. Reference to
Bijaganitam is found in Aryabhata's treatise on Mathematics in 5th century AD (
Āryabhaṭīya in 449 AD, when he was 23 years old), and again in
SiddhantaShiromani by Bhaskaracharya in 12th century AD  which has a section on
Bijaganitam.
India was subject to invasions from 13th century AD primarily by Arabs, who adapted the system of mathematics practiced in India and termed it
AlJabr (reunion in Arabic) to mean the amalgamation of Indian mathematics into Arabic. During the same time, Christian kingdoms of Europe made several attempts to reclaim the birthplace of Jesus Christ from the then MohammedanArab rulers. Although these 'Crusades' failed their military objectives, there were massive exchanges of ideas. During these exchanges,
AlJabr was renamed as Algebra, adopted by the western world, and developed further during Renaissance in Europe. Incidentally, algebra is known in India as
Bijganit  introduced to Indians by the British Empire!
It is, thus, not a coincidence that Vedic Mathematical Sūtras can be applied on the existing numeral system. On the contrary, they are an obvious set of rules (or formulae) that was developed to supplement calculations based on the commonly used numeral system, which the
Vedic Civilization founded  and not mere tricks for faster arithmetic calculations.
It should be noted that most of the
Vedic Civilization was based on oral traditions, and were only documented in scriptures much later  after the
SanskritDevanagari language was developed. The enormity and depth that the studies have been conducted, especially in astronomy and cosmology, in studying spirituality (nature)  is evident of a thorough understanding of arithmetic, algebra, geometry, trigonometry and many other branches of mathematics  which has been a source, either directly or indirectly, of many findings, inferences and theories. However, there is not much documented scripture or book on mathematics used in the
Vedic Civilization, implying that mathematics was studied for its own stake  primarily to further the studies of other branches of science and nature. Nevertheless, references and traces of mathematics are found in the
Vedas  dating it at early as 10th century BCE.
However, the first known and dated references found are normally credited with a finding. Aryabhata is credited with the placevalue notation, or the positional notation, that uses the same symbol for different orders of magnitude (for example, the 'ones place', 'tens place', 'hundreds place', and so on to define a number as 2512 = 2x1000 + 5x100 + 1x10 + 2x1). In his treatise on mathematics in 5th century AD, he states '
sthānam sthānam daśa guṇam' (from place to place, ten times in value). This was a turning point in the world of mathematics, as it was a far superior numeric system than the Babylonian numeral system or the Roman numeral system  and thus, was readily adopted by the entire world. It is only obvious that based on his positional notation, the Sanskrit nomenclature for power of 10 came into existence 
dasa (10),
sata (100),
sahasra (1,000),
ayuta (10,000),
laksha (100,000),
niyuta (1,000,000),
koti (10,000,000),
vyarbuda (100,000,000),
paraardha (1,000,000,000) and so forth.
 Incidentally, 'The Tale of Two Fishes and a Frog' in Panchatantra dated 3rd century BCE, documents the names of the two fishes as Satabuddhi (HundredWit) and Sahasrabuddhi (ThousandWit)  almost 700 years before Aryabhata's time. Since, the original texts of the Panchatantra has been long lost, one can only wonder if the positional notation was already in use even before Aryabhata documented it. 
 The earliest surviving manuscript that refers to positional notation is the 3rd century Bakhshali manuscript  at least 100 years before Aryabhata's time! 
(Source: Wikipedia 
Aryabhata,
Positional Notation,
Panchatantra,
Bakhshali manuscript)
Even the concept of zero appears in
Brahmasputha Siddhanta written by Brahmagupta in 628 AD to explain 'The Opening of the Universe', where he mentions not only zero, but negative numbers and algebraic rules for elementary arithmatic operations with negative numbers, and zero. This was a historic leap in the world of mathematics, as the inclusion of zero in mathematics opened up a new dimension of negative numerals and provided a standard for measurements.
Nevertheless,
Brahmasputha Siddhanta is a book on astronomy, and not mathematics, where Brahmagupta uses the mathematical concepts to support his astronomical findings  making one wonder if the concept of zero was already in use, much before Brahmagupta documented it:
 The famous Indian grammarian, Pāṇini, provided certain rules in 4th century BCE to add a zero suffix (a suffix with no phonemes in it) to a base to form words, and this can be said somehow to imply the concept of the mathematical zero  almost 1000 years before Brahmagupta's time! 
 In ancient times, this symbol by Pāṇini (zero) was used in computation, and it was indicated by a dot and was termed Pujyam, which was later replaced by the more current term Shunyam meaning a blank. The concept of 'Zero' or Shunyam is derived from the concept of a void, which already existed in philosophy  hence the derivation of a symbol for it. The concept of Shunyata, influenced Southeast Asian culture through the Buddhist concept of Nirvana (attaining salvation by merging into the void of eternity). 
 The 3rd century Bakhshali manuscript uses a dot (Pujyam) to denote zero. Along with many topics covered including fractions, square roots, arithmetic and geometric progressions, solutions of simple equations, simultaneous linear equations, quadratic equations and indeterminate equations of the second degree in Bakhshali manuscript, it also discusses rules for dealing with negative numbers (profit and loss)  almost 300 years before Brahmagupta's time!. 
(Source: Wikipedia 
Brahmagupta,
Pāṇini,
Bakhshali Manuscript)
These observations help in understanding the depth of mathematical influence in
Ancient India  not reflected in recorded history. These observations do not, in any way, discredit the Indian mathematical greatness of Aryabhata and Brahmagupta  especially because their contribution to the world of mathematics is far more than what is generally percieved.
For example, the 'Fibonacci Identity' was discovered by Brahmagupta, who generalized it to 'Brahmagupta Identity' and used it in his study (what is popularly known as Pell's Equation!). His 6th century AD
Brahmasputha Siddhanta containing these works was translated into Arabic by Mohammad alFazari, which was in turn translated into Latin in 1126 AD  and finally, this identity appeared in Fibonacci's 'Book of Squares' in 1225 AD. Thankfully, 'Fibonacci Identity' is now renamed as 'BrahmaguptaFibonacci Identity'  in acceptance to the original contributor.
And, Brahmagupta's
Chakravala Vidhi to solve Pell's Equation (ironically, almost 1000 years before John Pell's time) is commonly attributed to another Indian mathematician, Bhaskara II, for his 12 century contribution to the subject (the first general method for solving the Pell's Equation by extending Brahmagupta's works)  again, almost 400 years before John Pell's time.
(Source: Wikipedia 
Pell's Equation,
Chakravala Method)
One of the most interesting observations in Vedic Mathematical studies is the
Baudhāyana Shulbasūtra, which is part of the
Baudhayana sūtras compiled in 8th century BCE:
 Baudhāyana states 'dīrghachatursrasyākṣaṇayā rajjuḥ pārśvamānī, tiryagmānī, cha yatpṛthagbhūte kurutastadubhayāṅ karoti' (A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together). In the world of mathematics, this is popularly known as Pythagorean theorem, credited to Pythagoras of Samos, a Greek philosopher and mathematician born in 570 BCE  almost 300 years after the Baudhāyana Shulbasūtra was compiled. Pythagoras is said to have 'travelled to Egypt and Greece, and maybe India'! 
 Baudhāyana also provides a nonaxiomatic demonstration using a rope measure for an isosceles right triangle (The cord which is stretched across a square produces an area double the size of the original square.), which produces a sequence  popularly known as the 'Pythagorean triples'! 
 Another Greek mathematician, Euclid, born in 4th century BCE and referred to as the 'Father of Geometry', proceeded to prove the Pythagorean theorem. His book 'Elements', referred as the 'most successful and influential textbook ever written', has influenced scientists like Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Sir Isaac Newton  contains basics of geometry, and even square root of 2.
 Baudhāyana Shulbasūtra explains construction of geometric shapes, and areapreserving transformations of various shapes like square, rectange, isosceles trapezium, isosceles triangle, rhombus and circle. For example, Baudhāyana explains finding a circle whose area is the same as that of a square (Draw half its diagonal about the centre towards the EastWest line; then describe a circle together with a third part of that which lies outside the square.)  almost 400 years before the 'father of geometry' was born! 
 In an explanation to rectangles and squares, Baudhāyana states 'samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet tac caturthenātmacatustriṃśonena saviśeṣaḥ' (The diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.)  which is equivalent of a formula for square root of 2  and calculated to 1.414216, which is correct to five decimals! 

 The mathematical constant, Pi, represented by the Greek letter: π, is an irrational number having a neverending decimal representation  commonly approximated as 3.14159. History records ancient Chinese and Indian mathematicians to have calculated the value of pi to 7 and 5 decimals, respectively  in 5th century AD.
 Two sūtras of Baudhāyana ('If it is desired to transform a square into a circle, [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with onethird [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn', and 'Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired]') provide the values of pi as 3.088 and 3.004, respectively. 
 Another astronomical treatise, Shatapatha Brahmana, dated between 8th century to 4th century BCE, calculated pi as 339/108  a value of 3.139. There are other later Ancient Indian sources dated around 150 BCE that approximates the value of pi to square root of 10  a value of 3.1622. 
 Nevertheless, the historically first exact formula for pi was developed by Indian mathematician Madhava of Sangamagrama only in 14th century AD. Thankfully, the 'Leibniz Series' is now renamed to the commonly known 'Madhava–Leibniz Series'  in acceptance to the original contributor. 

 Other sūtras explained by Baudhāyana include diagonals of rectangle bisect each other, diagonals of rhombus bisect at right angles, area of a square formed by joining the middle points of a square is half of original, the midpoints of a rectangle joined forms a rhombus whose area is half the rectangle, etc.  almost 300 years before Pythagoras, and 400 years before Euclid's time! 
 Evidently, the Brahmi numerals, the ancestors of the current HinduArabic numeral system, appears before the period of Shulbasūtra came to an end  around 3rd century BCE. 
Incidentally, the
Baudhāyana Shulbasūtra (
shulba means rope, and
sūtra means rule) is not a compilation of geometry, or mathematics, or even astronomy. It is only a compilation on vedic rituals, and explains geometry (with a rope, for layman's understanding) for construction of
yajña bhūmikās  vedic altars on which the rituals are to be conducted  including fire offerings (
yajña)!
And, the manner in which the geometric rules are explained serves as a indicator that these rules are already proven, and do not need to be proved  more like statements, or axioms  for reference and guidance. One can only wonder that geometry in
Vedic Civilization was already in use much before 800 BCE! But, not much is known about the
Vedic Civilization, and India has to remain content with the earliest recorded surviving evidence  which is the 800 BCE compilation,
Baudhāyana Shulbasūtra.
(Source: Wikipedia 
Baudhayana Sūtras,
Shulba Sūtras,
Pythagoras,
Pythagorean theorem,
Euclid,
Pi,
Shatapatha Brahmana,
Madhava–Leibniz Series)
Furthermore, geometry in
Ancient India was referred as
RekhaGanitam and applied in the drafting of
Mandalas, meant for architectural purposes. The Arab scholar Mohammed Ibn Jubair al Battani studied the Indian use of ratios and fractions from
RekhaGanitam and spread them to other scholars  notable amongst them is Al Khwarazmi  the chief exponent for an IndoArab amalgam in mathematics. When it was adopted by the western world, the works of Al Khwarazmi was renamed 'Algorismi', which is the root of a modern term 'Algorithm', used in computer science.
To shorten the long story, Vedic Mathematics is a unique contribution of
Ancient India to the world, particularly to the world mathematics  often termed as 'tricks' or 'mental maths' by the western world  which, over thousands of years, remains the oldest and the fastest method for calculations. We, at Upavidhi, believe that one can use the
Sūtras of Vedic Mathematics now, or wait for a hundred odd years for today's mathematicians to 'rediscover' them:
Enjoy Vedic Mathematics for Fastest Mental Calculations » or simply,
Get Introduced to the Tricks »