Showing posts with label INDIAN MATH. Show all posts
Showing posts with label INDIAN MATH. Show all posts

Friday, October 28, 2016

Indian mathematics -Still Amazing

Despite developing quite independently of Chinese (and probably also of Babylonian mathematics), some very advanced mathematical discoveries were made at a very early time in India.
Mantras from the early Vedic period (before 1000 BCE) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century CE Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre).
As early as the 8th Century BCE, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) - 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5 decimal places.
As early as the 3rd or 2nd Century BCE, Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.
Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century CE. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicians) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time.
The Indians were also responsible for another hugely important development in mathematics. The earliest recorded usage of a circle character for the number zero is usually attributed to a 9th Century engraving in a temple in Gwalior in central India. But the brilliant conceptual leap to include zero as a number in its own right (rather than merely as a placeholder, a blank or empty space within a number, as it had been treated until that time) is usually credited to the 7th Century Indian mathematicians Brahmagupta - or possibly another Indian, Bhaskara I - even though it may well have been in practical use for centuries before that. The use of zero as a number which could be used in calculations and mathematical investigations, would revolutionize mathematics.
Brahmagupta established the basic mathematical rules for dealing with zero: 1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0 (the breakthrough which would make sense of the apparently non-sencical operation 1 ÷ 0 would also fall to an Indian, the 12th Century mathematician Bhaskara II). Brahmagupta also established rules for dealing with negative numbers, and pointed out that quadratic equations could in theory have two possible solutions, one of which could be negative. He even attempted to write down these rather abstract concepts, using the initials of the names of colours to represent unknowns in his equations, one of the earliest intimations of what we now know as algebra.
The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries, which has led to some claims of plagiarism by later European mathematicians, at least some of whom were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history.
Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a method of linking geometry and numbers first developed by the Greeks. They used ideas like the sine, cosine and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey the land around them, navigate the seas and even chart the heavens. For instance, Indian astronomers used trigonometry to calculated the relative distances between the Earth and the Moon and the Earth and the Sun. They realized that, when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle, and were able to accurately measure the angle as 1⁄7°. Their sine tables gave a ratio for the sides of such a triangle as 400:1, indicating that the Sun is 400 times further away from the Earth than the Moon.
Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 CE, contains the roots of modern trigonometry, including the first real use of sines, cosines, inverse sines, tangents and secants.
As early as the 6th Century CE, the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine and inverse sine, and specified complete sine and versine tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761.
Bhaskara II, who lived in the 12th Century, was one of the most accomplished of all India’s great mathematicians. He is credited with explaining the previously misunderstood operation of division by zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. So, dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately, therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷ 0 = ∞ (the symbol for infinity).
However, Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his findings predate similar discoveries in Europe by several centuries, and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions.
The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of Sangamagrama, sometimes called the greatest mathematician-astronomer of medieval India. He developed infinite series approximations for a range of trigonometric functions, including π, sine, etc. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that the later European development of calculus was influenced by his work to some extent.
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Saturday, July 25, 2015

Bakhshali Manuscript – Ancient Indian mathematical manuscript on math

The Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on "birch bark" which was found near the village of Bakhshali in 1881 in what was then the North-West Frontier Province of British India (now Khyber Pakhtunkhwa province, in Pakistan).
Bakhshali Manuscript is written in Śāradā script and in Gatha dialect (which is a combination of the ancient Indian languages of Sanskrit and Prakrit). The manuscript is incomplete, with only seventy leaves of birch bark, many of which are mere scraps. Many remain undiscovered. The Bakhshali manuscript, which is currently too fragile to be examined by scholars, is currently housed in the Bodleian Library at the University of Oxford and is too fragile to be examined by scholars.
It does not appear to belong to any specific period. Although that said, G Joseph classes it as a work of the early ‘classical period’, while E Robertson and J O’Connor suggest it may be a work of Jaina mathematics, and while this is chronologically plausible there is no proof it was composed by Jain scholars. L Gurjar discusses its date in detail, and concludes it can be dated no more accurately than ‘between 2nd century BC and 2nd century AD’. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried by R Hoernle). His evidence includes the language in which it was written (‘died out’ around 300 AD), discussion of currency found in several problems, and the absence of techniques known to have been developed by the 5th century.
Historians who have placed the date at pre 450 AD and identified the ‘current’ version as a copy.
Avoiding further debate, L Gurjar states that the Bakshali manuscript is the:
Capstone of the advance of mathematics from the Vedic age up to that period…
Although, as much work was lost between ‘periods’, we cannot fully gauge continuity of progress and it is possible the composer(s) of the Bakhshali manuscript were not fully aware of earlier works and had to start from ‘scratch’. This would make the work an even more remarkable achievement.
The arithmetic contained within the work is of such a high quality that it has been suggested:
…In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic…
Some of the contents of Manuscript are elaborated here.
Examples of the rule of three (and profit and loss and interest).Solution of linear equations with as many as five unknowns.The solution of the quadratic equation (development of remarkable quality). Arithmetic (and geometric) progressions.Compound Series (some evidence that work begun by Jainas continued).Quadratic indeterminate equations (origin of type ax/c = y). Simultaneous equations.Fractions and other advances in notation including use of zero and negative sign.Improved method for calculating square root (and hence approximations for irrational numbers). The improved method (shown below) allowed extremely accurate approximations to be calculated:A = (a2 + r) = a + r/2a – {(r/2a)2 / 2(a + r/2a)}
Example 6.1: Application of square root formula.
Again we can calculate 10, where a = 3 and r = 1.10 = (32 + 1) = 3 + 1/6 – {(1/36)/2(3 + 1/6)}= 3 + 1/6 – {(1/36)/(19/3)}= 3 + 1/6 – 1/228= 3.16228… in decimal form root(10) = 3.16228 when calculated on a calculator and rounded to five decimal places.
Example 6.2: Quadratic equation as found in B. Ms.
If the equation given is dn2 + (2a – d)n -2s = 0Then the solution is found using the equation:n = (- (2a – d) (2a – d)2 +8ds))/2dWhich is the quadratic equation with a = d, b = 2a – d, and c = 2s.
Example 6.3: Linear equation with 5 variables.
The following problem is stated : ”Five merchants together buy a jewel. Its price is equal to half the money possessed by the first together with the money possessed by the others, or one-third the money possessed by the second together with the moneys of the others, or one-fourth the money possessed by the third together with the moneys of the others…etc. Find the price of the jewel and the money possessed by each merchant.
Solution :We have the following systems of equations:
x1/2 + x2 + x3 + x4 + x5
= px1 + x2/3 + x3 + x4 + x5
= px1 + x2 + x3/4 + x4 + x5
= px1 + x2 + x3 + x4/5 +x5
= px1 + x2 + x3 + x4 + x5/6 = p
Then if x1/2 + x2/3 + x3/4 + x4/5 + x5/6 = q ,
the equations become (377/60 )q = p.
A number of possible answers can be obtained. This is the origin of the indeterminate equation of the type ax/c = y, the theory of which was greatly developed, and later perfected by Bhaskara II, four hundred years before it was discovered in Europe. If q = 60 then p = 377 and x1 = 120, x2 = 90, x3 = 80, x4 = 75 and x5 = 72
Ms. Historians of mathematics debate whether true algebra ‘began’ in Greece or Arabia, and little mention is ever made of Indian algebra. In light of my own research I feel that early Arabic algebra (c. 800 AD) in no way surpasses the level of understanding of 6th century Indian scholars.
The Bakhshali manuscript is a unique piece of work and while it not only contains mathematics of a remarkably high standard for the time period, also, in contrast to almost all other Indian works composed before and after, the method of the commentary follows a highly systematic order of:
i. Statement of the rule (sutra)
ii. Statement of the examples (udaharana)
iii. Demonstration of the operation (karana) of the rule.
By the end of the 2nd century AD mathematics in India had attained a considerable stature, and had become divorced from purely practical and religious requirements, (although it is worth noting that over the next 1000 years the majority of mathematical developments occurred within works on astronomy).
The topics of algebra, arithmetic and geometry had developed significantly and it is widely thought that the decimal place value system of notation had been (generally) perfected by 200 AD, the consequence of which was far reaching.
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Sunday, March 29, 2015

Pingala, Inventor of Binary Numbers in 2nd Century BCE

Pingala MathematicianPingala (Devanagari: पिङ्गल) is the author of Chandaḥśāstra (Chandaḥsūtra), the earliest known Sanskrit treatise on prosody.
Very less historical knowledge is available about Pingala, though his works are retained till date.
He is identified either as the younger brother of Pāṇini (4th century BCE), or of Patañjali, the author of the Mahabhashya (2nd century BCE).
His work, Chandaḥśāstra means science of meters, is a treatise on music and can be dated back to 2nd century BCE.
Main commentaries on ‘Chandaḥśāstra‘ are ‘Vrittaratnakara‘ by Kedara in 8th century AD, ‘Tatparyatika‘ by Trivikrama in 12th century AD and ‘Mritasanjivani‘ by Halayudha in 13th century AD. The complete significance of Pingala’s work can be understood by the explanations found in these three commentaries.
Pingala (in Chandaḥśāstra 8.23) has assigned the following combinations of zero and one to represent various numbers, much in the same way as the present day computer programming procedures.
0 0 0 0 numerical value = 1
1 0 0 0 numerical value = 2
0 1 0 0 numerical value = 3
1 1 0 0 numerical value = 4
0 0 1 0 numerical value = 5
1 0 1 0 numerical value = 6
0 1 1 0 numerical value = 7
1 1 1 0 numerical value = 8
0 0 0 1 numerical value = 9
1 0 0 1 numerical value = 10
0 1 0 1 numerical value = 11
1 1 0 1 numerical value = 12
0 0 1 1 numerical value = 13
1 0 1 1 numerical value = 14
0 1 1 1 numerical value = 15
1 1 1 1 numerical value = 16
Other numbers have also been assigned zero and one combinations likewise.
Pingala’s system of binary numbers starts with number one (and not zero). The numerical value is obtained by adding one to the sum of place values. In this system, the place value increases to the right, as against the modern notation in which it increases towards the left.
The procedure of Pingala system is as follows:
  • Divide the number by 2. If divisible write 1, otherwise write 0.
  • If first division yields 1 as remainder, add 1 and divide again by 2. If fully divisible, write 1, otherwise write 0 to the right of first 1.
  • If first division yields 0 as remainder that is, it is fully divisible, add 1 to the remaining number and divide by 2. If divisible, write 1, otherwise write 0 to the right of first 0.
  • This procedure is continued until 0 as final remainder is obtained.
Example to understand Pingala System of Binary Numbers :
Find Binary equivalent of 122 in Pingala System :
        Divide 122 by 2. Divisible, so write 1 and remainder is 61.
          Divide 61 by 2. Not Divisible and remainder is 30. So write 0 right to 1.
            Add 1 to 61 and divide by 2 = 31.
            Divide 31 by 2. Not Divisible and remainder is 16. So write 0 to the right.
              Divide 16 by 2. Divisible and remainder is 8. So write 1 to right.
                Divide 8 by 2. Divisible and remainder is 4. So write 1 to right.
                  Divide 4 by 2. Divisible and remainder is 2. So write 1 to right.
                    Divide 2 by 2. Divisible. So place 1 to right.
              Now we have 122 equivalent to 1001111.
              Verify this by place value system : 1×1 + 0x2 + 0x4 + 1×8 + 1×16 + 1×32 + 1×64 = 64+32+16+8+1 = 121
              By adding 1(which we added while dividing 61) to 121 = 122, which is our desired number.
              In Pingala system, 122 can be written as 1001111.
              Though this system is not exact equivalent of today’s binary system used, it is very much similar with its place value system having 20, 20, 21, 22, 22, 23, 24, 25, 26 etc used to multiple binary numbers sequence and obtain equivalent decimal number.
              Reference : Chandaḥśāstra (8.24-25) describes above method of obtaining binary equivalent of any decimal number in detail.
              These were used 1600 years before westeners invented binary system.
              We now use zero and one (0 and 1) in representing binary numbers, but it is not known if the concept of zero was known to Pingala— as a number without value and as a positional location.
              Pingala’s work also contains the Fibonacci number, called mātrāmeru, and now known as the Gopala–Hemachandra number. Pingala also knew the special case of the binomial theorem for the index 2, i.e. for (a + b) 2, as did his Greek contemporary Euclid.
              Halayudha (10th century AD) who wrote a commentary on Pingala’s work understood and used zero in the modern sense but by then it was commonplace in India and had also begun to make its way to West Asia as well to countries like Indonesia, Cambodia and others in East and Southeast Asia. It took several centuries more before being accepted in Europe. It was Leonardo of Pisa, better known as Fibonacci who seems to have introduced it in Europe in the 13th century. (He learnt it from the Arabs but noted that it came from India. His successors were not so careful, and for centuries they were known as Arabic numerals.)
              Halayudha was himself a mathematician no mean order. His discussion of combinatorics of poetic meters led him to a general version of the binomial theorem centuries before Newton. (This was the integer version only and not the full general version with arbitrary index given by Newton.) This too traveled east and west with the Persian mathematician and poet using the results in the 13th century.
              Halāyudha’s commentary includes a presentation of the Pascal’s triangle for binomial coefficients (called meruprastāra).
              Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables (Short = 0, Long = 1).
              Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used short and long syllables.
              As Pingala’s system ranks binary patterns starting at one (four short syllables—binary “0000”—is the first pattern), the nth pattern corresponds to the binary representation of n-1, written backwards. Positional use of zero dates from later centuries and would have been known to Halāyudha but not to Pingala.

              Wednesday, February 11, 2015

              Ancient Indian inventions

              Ancient Hindu Zero - BindhuZero (0)LINK to other article related to Aryabhatta’s Bhakshali manuscript mentions zero,which was translated by Arabic first then by Europeans.

              Ancient Indians first made ink  by burning tar, pitch, bones. Carbon was the primary pigment.

              The Hindu Vedas written in Ink are among some of the most ancient texts in the world
              The Hindu Vedas written in Ink are among some of the most ancient texts in the world

               Sea Dock -Link to other articleIndia was the first nation to have a dock that dated back to 6000 BCE. Harappa Civilization were the first to build a dock in Lothal. This proves their oceanology and marine engineering. The Lothal Dock proves their precision and vast knowledge about tidal waves and hydrography.
              Historic Lothal Dock

              Diamond MiningDiamonds were first mined in India. Till 18th century, India was the only country where diamonds were found and exported to other countries. Various ancient books have mentioned the use of diamond as a tool and have also mentioned the exquisiteness of this sparkling stone
              Some of the largest diamonds in the world were stolen from Hindu kings and now housed outside of India

              Medical Vaccinations & Treatments (Ayurveda)Link to other articleLeprosy was first noticed by Indians and various ancient remedies are also mentioned in the Atharva Veda. Treatment for  stones was first introduced in India. Small Pox vaccinations were first cured in India and symptoms and ways of immunization against small pox were mentioned in 8 th century by Madhav.Ayurveda and Siddha are the two primitive methods of treatment that originated in India and are still used. Indian medical practitioner  Nobel Laureate Upendra Nath Bramhachari invented methods to treat Visceral Leishmaniasis or Kala Azar.
              The ancient form of medical treatments & foods is called Ayurveda

              Surgery Link to other articleAncient Indian physician Sushruta performed first Cataract surgery and plastic surgery back to >2000 BCE and his work were later translated to Arabic language and gradually passed on to European countries. He used a curved needle and removed the cataract by pushing the lens. People from far off countries came to India to seek treatment.
              Surgery 1000s of years ago in Ancient India
              Wool, Cotton, Plant (Natural Fibers)Natural fibers like wool, cotton and plant fiber originated from India. Evidences show that people of the Indus Valley used cotton and India pioneered the art of cotton spinning and used it in making fabric. Jute, a plant fiber, was cultivated in India since ancient times and was later exported to other countries. Cashmere wool, which is supposed to be the finest wool was first made in Kashmir and was used to make hand- made shawls. These shawls have maintained their richness and exclusivity even today.
              Natural fibers from ancient India
              ButtonsButtons are a major part of our clothing even today. Buttons were invented in India and various historical evidences and excavations prove that buttons were used by the people belonging to the Indus Valley Civilization. Shells were given various shapes and were pierced into a hole. Earlier they were used more as an embellishment but were gradually used to fasten clothes
              Buttons made of stone from Ancient India
              Cotton Gin is a machine used to separate cotton from the seeds. The evidence of this machine was found through the carvings on Ajanta caves where the pictures of these machines were engraved. Dating back to 500 AD, this hand roller machine was locally called Charkha. This machine has undergone changes through the course of time but the most primitive form of cotton gin originated from India.
              Cotton Gin origins from Ancient India
              Crucible SteelHigh-quality steel has been produced in South India since ancient times. They used crucible technique toproduce high quality steel in Ancient India.. Pure wrought iron was first  mixed with glass and charcoal and was heated till the metal melted and absorbed the carbon.
              Steel from ancient India
              Steel from ancient India
              Adopted from