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Tag: Vedic Maths

  • Vedic Sutras 29 On Mathematics Explained.With Translator

    Vedic Sutras 29 On Mathematics Explained.With Translator

    This is the third part of my series on History of Mathematics. In my earlier articles, I had explored the concept of mathematics and how and where it emerged from. Though mathematics had made its way in India very early ,in Vedic texts,among mathematicians who lived before Christ,in Jainism/ Buddhism and in the language I am familiar with (in which I can read ancient texts)Tamil. I shall write about them. Now I shall attempt at the references to Mathematics in Vedic texts. One may note that the oldest literature known to man, Rig Veda is dated about 5000 years. My view is that it is older at least by few thousand years. I had written articles on this topic in this blog.

    Vedas are called the Sabda Pramana. It means that Vedas are one of the tools of knowledge and it represents testimony. Vedas are Testimonies of Seekers of Truth and they had recorded their experiences in the Form of Sound. Vedas are one of the tools of knowledge recognised by Hinduism,along with Perception, Inference,Comparison ,Logic,Intuition,and Experience.

    Sanatana Dharma recognised two levels of Knowledge.One is Para Vidya which is Absolute and another is Apara .It is lower knowledge in that it deals knowledge that is applicable and useful for day to day activities. It is lower knowledge as any thing that binds one to the world in the form of Birth ,death and the cycle involved , is treated as Inferior Knowledge.The knowledge that helps one realise Self is Real Knowledge and it is Absolute.

    So references to Sciences as we know of today , as defined by Western thought, are not pursued as distinctly as the western system of Education. The information relating to these concepts are made in the passing so long they are relevant in Realising Brahman .So the scientist concepts as we know of know today are found in as much as they are found useful in the conduct of Vedic Rituals ,when King’ Prowess or skills are explained in Purana or Ithihasa. And in many an instance, the principles are inferred either from the results or understanding the Aphorisms. And this needs special skill set. As Nature is personified Attributes are ascribed and they reveal the level of scientific knowledge of the Vedic people. For instance, Sun is called Surya and Attributes of Surya include the composition of Light, movement of Sun in the Universe.One can find information about the planets, Galaxies,Stars , Blackholes, Wormhole, Concept of Time as Cyclic, which is now validated by Quantum Theory.And we have the Vaiseshika system deals extensively about Atomic Theory, Particles,sub particles. Nyaya system deals with advanced Logical systems. There are concepts which are applicable to Computer Languages. Binary principles, Binomial theorem, Pythogoras Theorem Trigonometry,Calculus…all are found in ancient Indian texts.

    You may have noticed that I have referred to some systems which are not a part of Vedas. Reason is that Vedas alone do not contribute to Hinduism. Vedas are a Source of Knowledge. Other systems like Nyaya,Vaiseshika Sankhya,Yoga,Purvamimasa and Uttaramimasa are other systems of Indian philosophy. Of these Uttara Mimamsa is Upanishads which is a part of Vedas.Having this point in our approach, let us see what Vedas mention ,in the passing, about scientific, especially Mathematical Concepts.

    Chamaka expresses Mitochondrial Pairs of DNA on the form of Mathematics.

    Whe have, Vedic Mathematics. This helps one to handle fundamental mathematics easily.

    Vedic Mathematics Basic Sixteen Sutras, Aphorisms.
    1. Ekadhikina Purvena
      (Corollary: Anurupyena)
      Meaning: By one more than the previous one
    2. Nikhilam Navatashcaramam Dashatah
      (Corollary: Sisyate Sesasamjnah)
      Meaning: All from 9 and the last from 10
    3. Urdhva-Tiryagbyham
      (Corollary: Adyamadyenantyamantyena)
      Meaning: Vertically and crosswise
    4. Paraavartya Yojayet
      (Corollary: Kevalaih Saptakam Gunyat)
      Meaning: Transpose and adjust
    5. Shunyam Saamyasamuccaye
      (Corollary: Vestanam)
      Meaning: When the sum is the same that sum is zero
    6. (Anurupye) Shunyamanyat
      (Corollary: Yavadunam Tavadunam)
      Meaning: If one is in ratio, the other is zero
    7. Sankalana-vyavakalanabhyam
      (Corollary: Yavadunam Tavadunikritya Varga Yojayet)
      Meaning: By addition and by subtraction.Puranapuranabyham
    8. Puranapuranabyham(Corollary: Antyayordashake’pi Meaning: By the completion or non-completion.
    9. Chalana-Kalanabyha(Corollary: Antyayoreva)Meaning: Differences and Similarities.
    10. Yaavadunam(Corollary: Samuccayagunitah)Meaning: Whatever the extent of its deficiency.
    11. Vyashtisamanstih(Corollary: Lopanasthapanabhyam)Meaning: Part and Whole.
    12. Shesanyankena Charamena(Corollary: Vilokanam)Meaning: The remainders by the last digit.
    13. Sopaantyadvayamantyam(Corollary: Gunitasamuccayah Samuccayagunitah)Meaning: The ultimate and twice the penultimate.
    14. Ekanyunena Purvena(Corollary: Dhvajanka)Meaning: By one less than the previous one.
    15. Gunitasamuchyah(Corollary: Dwandwa Yoga)Meaning: The product of the sum is equal to the sum of the product.
    16. Gunakasamuchyah(Corollary: Adyam Antyam Madhyam Meaning: The factors of the sum is equal to the sum of the factors. Source.Das, Subhamoy. “The 16 Sutras of Vedic Math.” Learn Religions, Aug. 26, 2020, learnreligions.com/vedic-math-formulas-177068 0. https://www.learnreligions.com/vedic-math-formulas-1770680

    Check these sites.

    Vedic Maths in Hindi
    Vedic Mathematics English

    The secularists dismiss this as Tricks!

    These statements have been since rejected in their entirety. Krishna Tirtha failed to produce the sources, and scholars unanimously note it to be a mere compendium of tricks for increasing the speed of elementary mathematical calculations with no overlap with historical mathematical developments during the Vedic period. However, there has been a proliferation of publications in this area and multiple attempts to integrate the subject into mainstream education by right-wingHindu nationalist governments. https://en.m.wikipedia.org/wiki/Vedic_Mathematics

    Does Truth Matter? I Offer A ₹5,00,000 INR ‘Vedic’ Maths Reward  … For Serious Primary Evidence And Proof The 16 Mathematics Sutras Of Bharati Krishna Tirtha Irrefutably Exist In Any Extant Written Vedas.…

    Jagadguru Shankaracharya Swami Bharati Krishna Tirtha’s view was that Vedic maths is NOT to be approached from a factual standpoint. If it is accepted that the Vedas are the source of all knowledge, then the 16 Sutras of Vedic Maths should have been contained in them.

    Dr. V.S. Agrawala, friend of Bharati Krishna Tirtha and Editor of the book titled ‘Vedic Maths’.
    Are The 16 Sutras Of Vedic Maths From India’s Ancient Vedas? What Do Indian Scholars Say?
    VEDIC SCHOLAR PROF. DR. SUBHASH KAK

    “First question: What is Vedic mathematics? It is not mathematics from the Vedic period. Rather it consists of many clever mathematical sutras and algorithms that were devised by Swami Bharati Krishna Tirtha (1884–1960) who for a long time was the Shankaracharya of Govardhan Matha in Puri.

    The teaching of Vedic Mathematics cannot be justified on the grounds that it tells the students something about India’s ancient mathematical heritage. This heritage without even counting the invention of zero was brilliant and unique but Swami Bharati Krishna Tirtha’s Vedic mathematics has nothing to do with it.” Source…

    MATHEMATICIAN PROF. DR. C. K. RAJU

    “Advocating ‘Vedic mathematics’ as a replacement for traditional Indian arithmetic is hardly an act of nationalism; it only shows ignorance of the history of mathematics.

    But where in the Vedas is “Vedic mathematics” to be found? Nowhere. Vedic mathematics has no relation whatsoever to the Vedas. It actually originates from a book misleadingly titled Vedic Mathematics by Bharati Krishna Tirtha. The book admits on its first page that its title is misleading and that the (elementary arithmetic) algorithms expounded in the book have nothing to do with the Vedas. This is repeated on p. xxxv: “Obviously these formulas are not to be found in the present recensions of Atharvaveda.”. https://kreately.in/is-vedic-maths-vedic/

    And more from ‘open’ site.

    As stories go, this is not a bad one, but the evidence does nothing to support it. The 16 sutras expounded by Tirathji do not appear in any known edition of the Atharva Veda. Tirathji’s defenders have claimed that Tirathji was so immersed in Vedic thought that he was able to glean what the Vedic seers had in mind even if it was not explicitly so stated anywhere in the Vedic corpus. If one were to actually concede this meeting of minds between Tirathji and the ancient Vedic seers, it would have the unfortunate consequence of implying that not just Tirathji but even these seers were limited in their mathematical understanding.

    All the sutras largely do is make the burden of addition and multiplication faster (though never nearly as fast as the cheapest pocket calculator), and even that, they do at a cost. Students studying the traditional method of multiplication should ideally understand (and bad teachers themselves fail to grasp this) what multiplication is, how it works, and how it is in essence an act of repeated addition. Tirathji’s methods are just rules that make mathematics seem like a bunch of tricks which are easy to implement but difficult to understand.

    Take, for example, the multiplication of 9 and 7. Line them along with their difference from 10. That is:

    9–10 = –1 and 7–10 = –3

    9–1

    7–3

    ——

    6   3

    You obtain the answer in the following fashion: the unit’s digit is the two differences multiplied together, –1 x –3 = 3 and the other digit 6 is just either of the diagonals added together, that is, 9–3 = 7–1 = 6. This method can be extended to much larger numbers. It is a neat trick, but it does not make multiplication easier to fathom, quite the contrary….open the magazine

    None of these scholars have disproved the fact that Vedic Mathematics sutras work.

    How ridiculous can one be? The author of the book produced Non sense? The formulas work and as I mentioned at the beginning of this article that to understand Indian science, one has to understand Hinduism ‘s Concept of Knowledge first, before jumping into right wing bashing.

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  • Kerala Calculus Precedes Western Calculus

    Kerala Calculus Precedes Western Calculus

    Vedic mathematics and science are ancient Indian systems of knowledge that have been passed down through generations. These systems originated in the Vedas, which are a collection of ancient Indian scriptures. Vedic mathematics is a system of mathematical calculations based on 16 sutras, or word-formulas, and 13 sub-sutras, or corollaries. This system is known for its simplicity and efficiency, as it allows for faster calculations and mental math techniques.

    Vedic science, on the other hand, is a system of knowledge that encompasses a wide range of subjects, including astronomy, astrology, medicine, and philosophy. It is based on the principles of the Vedas and the Upanishads, which are ancient Indian texts that contain philosophical and spiritual teachings.

    One of the key principles of Vedic science is the idea of interconnectedness between all things, known as the principle of unity. This principle suggests that everything in the universe is connected and that there is a fundamental unity underlying all of existence.

    Another important principle of Vedic science is the idea of balance, or the principle of equilibrium. This principle suggests that in order for an individual or society to achieve health and well-being, there must be a balance between different aspects of life, such as work and rest, or physical and spiritual pursuits.

    Overall, Vedic mathematics and science offer unique insights into ancient Indian knowledge systems and can provide useful tools and perspectives for modern-day learning and living.

    Before I write on what Kerala Calculus is, I feel it is necessary to remind us Indians of the fact that our continued ignorance, indifference reluctance to learn our highly scientific texts have resulted in the Myth of Aryan Invasion Theory by the Eurocentric colonialists gained currency and it has taken a lot of efforts to deny the theory which attempted to deny India its glorious past.

    There are other insidious attempts too.

    That the scientificancient texts are vague, they do not really talk about science and people use it, after the modern science discovers something,Indians say that the facts were mentioned in the Hindu Texts.

    And another ingenious attempt to deny India its rich scientific heritage is by stating that the data referred to in the tests were not used by the ancient Indians !

    The only way one can counter these arguments is for us to study the original texts and bring into light the scientific thoughts found in our Texts.

    I am trying to instill a sense of pride in our heritage by posting articles on various astounding facts in out texts, our stupendous Temples Astronomy.

    It is for the specialists to delve deep into the Texts and reveal what they contain.

    Unfortunately those who know Sanskrit are not generally aware of advanced modern Scientific Theories and those familiar with modern concepts do not know Sanskrit.

    My request is that both the groups must get familiar with what they do not know or get together to highlight the highly advanced scientific nature of Hindu Thought.

    I shall continue to post articles giving general directions.

    Now to Calculus.

    ‘

    Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves),and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.

    Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called “the calculus of infinitesimals”, or “infinitesimal calculus”. The word “calculus” comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.”

    The Kerala Calculus.

    The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1500 – c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.

    Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala ..

    The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:

    \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots for <img class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/1/1/2/112f97d0a5732861bb9297b7d2c3e631.png&quot; alt="|x|

    This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (965-1039).[8]

    The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result:

    1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1} for large n. This result was also known to Alhazen.

    They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for \sin x, \cos x, and \arctan x.TheTantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:

    r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^5} - \cdots , where y/x \leq 1.
    r\sin \frac{x}{r} = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot
    r(1 - \cos \frac{x}{r}) = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots , where, for r = 1 , the series reduce to the standard power series for these trigonometric functions, for example:

    \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots and
    \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots (The Kerala school did not use the “factorial” symbolism.)

    The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e.computation of area under the arc of the circle), was not yet developed.) They also made use of the series expansion of \arctan x to obtain an infinite series expression (later known as Gregory series) for \pi:

    \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots

    Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, f_i(n+1), (for n odd, and i = 1, 2, 3) for the series:

    \frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)

    where f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.
    They manipulated the terms, using the partial fraction expansion of :\frac{1}{n^3-n} to obtain a more rapidly converging series for \pi:[1]

    \frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots

    They used the improved series to derive a rational expression, 104348/33215 for \pi correct up to nine decimal places, i.e. 3.141592653 . They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.

    The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists another work, namely Kala Sankalita by J. Warren from 1825 which briefly mentions the discovery of infinite series by Kerala astronomers. According to Whish, the Kerala mathematicians had “laid the foundation for a complete system of fluxions” and these works abounded “with fluxional forms and series to be found in no work of foreign countries.” However, Whish’s results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers, a commentary on the Yuktibhasa’s proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary)..

    The following notes by westerners would justify what I had written at the beginning of the Post.

    “

    1. (Bressoud 2002, p. 12) Quote: “There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use.”
    2. Jump up^ Plofker 2001, p. 293 Quote: “It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula (… in the 10th century)” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” (Joseph 1991, 293), or that Bhaskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” (Bag 1979, 294). … The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). … It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here”
    3. Jump up^ Pingree 1992, p. 562 Quote: “One example I can give you relates to the Indian Mādhava’s demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians’ discovery of the calculus. This claim and Mādhava’s achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish’s article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava’s mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.”

    Reference and citation.

     

    https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics