Hindus contribution in Mathematics



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 20th century.

Eurocentrism
Many scholars feel that Indian contributions to science, technology and mathematics have not been given due acknowledgement 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 sunya, meaning empty, or void; this word, translated and transliterated by the Arabs as sifr, 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, 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 Sytem 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 till 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 some time 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 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 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

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

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.


Vedic Origin of Quadratic Equations

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 acknowledgement 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
Varahamihir (488-587AD) Tri-Lostaka


1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1



Foreigners` claim

Pascal Triangle. B.Pascal (1623-1662 AD)


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


Virahank`s ( 600AD) series 0,1,1,2,3,5,8,13,21.....


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 - .......

8. Madhav`s series (1340-1425)

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 divisiors of a number

10. Bhaskaracharya (1114-1193) Euler’s division algorithm

method of finding greatest common divisor

11. Permeshwara`s (1360 AD) Huiler`s formula (1782AD)

Formula for finding circum-radius of a cyclic quadrilateral


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


14. Brahmagupta (628 AD Kepler

volumes of frustum of cone and of 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), Liebnitz (1646-1716)

Linear equations,

17 Jyeshtha Deo (1500 AD) Liebnitz, by 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.

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व्यक्ति जिसने जीत ली दुनिया प्रेम से - प्रभुपाद (इस्कॉन - International Society for Krishna Consciousness - ISKCON)



इस्कॉन या अंतर्राष्ट्रीय कृष्णभावनामृत संघ (International Society for Krishna Consciousness - ISKCON), को "हरे कृष्ण आंदोलन" के नाम से भी जाना जाता है। इसे १९६६ में न्यूयॉर्क नगर में भक्तिवेदांत स्वामी प्रभुपाद ने प्रारंभ किया था। देश-विदेश में इसके अनेक मंदिर और विद्यालय है।

ISKCON Abhay Caranaravinda Bhaktivedanta Svami Prabhupada


स्थापना एवं प्रसार
कृष्ण भक्ति में लीन इस अंतरराष्ट्रीय सोसायटी की स्थापना श्रीकृष्णकृपा श्रीमूर्ति श्री अभयचरणारविन्द भक्तिवेदांत स्वामी प्रभुपादजी ने सन् १९६६ में न्यू यॉर्क सिटी में की थी। गुरू भक्ति सिद्धांत सरस्वती गोस्वामी ने प्रभुपाद महाराज से कहा तुम युवा हो, तेजस्वी हो, कृष्ण भक्ति का विदेश में प्रचार-प्रसार करों। आदेश का पालन करने के लिए उन्होंने ५९ वर्ष की आयु में संन्यास ले लिया और गुरु आज्ञा पूर्ण करने का प्रयास करने लगे। अथक प्रयासों के बाद सत्तर वर्ष की आयु में न्यूयॉर्क में कृष्णभवनामृत संघ की स्थापना की। न्यूयॉर्क से प्रारंभ हुई कृष्ण भक्ति की निर्मल धारा शीघ्र ही विश्व के कोने-कोने में बहने लगी। कई देश हरे रामा-हरे कृष्णा के पावन भजन से गुंजायमान होने लगे।

अपने साधारण नियम और सभी जाति-धर्म के प्रति समभाव के चलते इस मंदिर के अनुयायियों की संख्या लगातार बढ़ती जा रही है। हर वह व्यक्ति जो कृष्ण में लीन होना चाहता है, उनका यह मंदिर स्वागत करता है। स्वामी प्रभुपादजी के अथक प्रयासों के कारण दस वर्ष के अल्प समय में ही समूचे विश्व में १०८ मंदिरों का निर्माण हो चुका था। इस समय इस्कॉन समूह के लगभग ४०० से अधिक मंदिरों की स्थापना हो चुकी है।

नियम एवं सिद्धान्त

पूरी दुनिया में इतने अधिक अनुयायी होने का कारण यहाँ मिलने वाली असीम शांति है। इसी शांति की तलाश में पूरब की गीता पश्चिम के लोगों के सिर चढ़कर बोलने लगी। यहाँ के मतावलंबियों को चार सरल नियमों का पालन करना होता है-
उन्हें तामसिक भोजन त्यागना होगा ( तामसिक भोजन के तहत उन्हें प्याज, लहसुन , मांस, मदिरा आदि से दूर रहना होगा)
अनैतिक आचरण से दूर रहना (इसके तहत जुआ, पब, वेश्यालय जैसे स्थानों पर जाना वर्जित है)
एक घंटा शास्त्राध्ययन (इसमें गीता और भारतीय धर्म-इतिहास से जुड़े शास्त्रों का अध्ययन करना होता है)
हरे कृष्णा-हरे कृष्णा नाम की १६ बार माला करना होती है।

ISKCON Abhay Caranaravinda Bhaktivedanta Svami Prabhupada
योगदान
भारत से बाहर विदेशों में हजारों महिलाओं को साड़ी पहने चंदन की बिंदी लगाए व पुरुषों को धोती कुर्ता और गले में तुलसी की माला पहने देखा जा सकता है। लाखों ने मांसाहार तो क्या चाय, कॉफी, प्याज, लहसुन जैसे तामसी पदार्थों का सेवन छोड़कर शाकाहार शुरू कर दिया है। वे लगातार ‘हरे राम-हरे कृष्ण’ का कीर्तन भी करते रहते हैं। इस्कॉन ने पश्चिमी देशों में अनेक भव्य मन्दिरेवं विद्यालय बनवाये हैं। इस्कॉन के अनुयायी विश्व में गीता एवं हिन्दू धर्म एवं संस्कृति का प्रचार-प्रसार करते हैं।

International Society for Krishna Consciousness
ISKCON Abhay Caranaravinda Bhaktivedanta Svami Prabhupada
The International Society for Krishna Consciousness (ISKCON), known colloquially as the Hare Krishna movement, is aGaudiya Vaishnava religious organization. It was founded in 1966 in New York City by "His Divine Grace" Abhay Caranaravinda Bhaktivedanta Svami "Prabhupada". Its core beliefs are based on traditional Hindu scriptures such as the Śrīmad Bhāgavatamand the Bhagavad-gītā, both of which, according to the traditional Hindu view, date back more than 5,000 years. The distinctive appearance of the movement and its culture come from the Gaudiya Vaisnava tradition, which has had adherents in India since the late 15th century and Western converts since the early 1930s. ISKCON was formed to spread the practice of bhakti yoga, in which aspirant devotees (bhaktas) dedicate their thoughts and actions towards pleasing the Supreme Lord, Krishna. ISKCON today is a worldwide confederation of more than 400 centers, including 60 farm communities, some aiming for self-sufficiency, 50 schools and 90 restaurants. In recent decades the movement's most rapid expansions in terms of numbers of membership have been within Eastern Europe (especially since the collapse of the Soviet Union) and India.

Beliefs and history
ISKCON devotees follow a disciplic line of Gaudiya Bhagavata Vaishnavas and are the largest branch of Gaudiya Vaishnavism.Vaishnavism means 'worship of Vishnu', and Gauḍa refers to the area where this particular branch of Vaishnavism originated, in the Gauda region of West Bengal. Gaudiya Vaishnavism has had a following in India, especially West Bengal and Orissa, for the past five hundred years. Bhaktivedanta Swami disseminated Gaudiya Vaishnava Theology in the Western world through extensive writings and translations, including the Bhagavad Gita, Srimad Bhagavatam (Bhagavata Purana), Chaitanya Charitamrita, and other scriptures. These works are now available in more than seventy languages and serve as the canon of ISKCON. Many are available online from a number of websites.
Early Western conversions to monotheistic Krishna Vaisnavism or the Bhagavata Vaisnava line which forms the basis of the ISKCON philosophy were recorded by the Greeks and are reflected in the archaeological record.
Krishna is described as the source of all the avatars. Thus ISKCON devotees worship Krishna as the highest form of God,svayam bhagavan, and often refer to Him as "the Supreme Personality of Godhead" in writing, which was a phrase coined by Prabhupada in his books on the subject. To devotees, Radha represents Krishna's divine female counterpart, the original spiritual potency, and the embodiment of divine love. The individual soul is an eternal personal identity which does not ultimately merge into any formless light or void as suggested by the monistic (Advaita) schools of Hinduism. Prabhupada most frequently offers Sanatana-dharma and Varnashrama dharma as more accurate names for the religious system which accepts Vedicauthority. It is a monotheistic tradition which has its roots in the theistic Vedanta traditions.

Hare Krishna mantra
The popular nickname of "Hare Krishnas" for devotees of this movement comes from the mantra that devotees sing aloud (kirtan) or chant quietly (japa) on tulsi mala. This mantra, known also as the Maha Mantra, contains the names of God Krishna andRama.
The Maha Mantra:
Hare Krishna Hare Krishna
Krishna Krishna Hare Hare
Hare Rama Hare Rama
Rama Rama Hare Hare

Seven purposes of ISKCON
When Srila Prabhupada first incorporated ISKCON in 1966, he gave it seven purposes:
1. To systematically propagate spiritual knowledge to society at large and to educate all people in the techniques of spiritual life in order to check the imbalance of values in life and to achieve real unity and peace in the world.
2. To propagate a consciousness of Krishna, as it is revealed in the Bhagavad-gita and the Srimad-Bhagavatam.
3. To bring the members of the Society together with each other and nearer to Krishna, the prime entity, thus to develop the idea within the members, and humanity at large, that each soul is part and parcel of the quality of Godhead (Krishna).
4. To teach and encourage the sankirtana movement, congregational chanting of the holy names of God as revealed in the teachings of Lord Sri Caitanya Mahaprabhu.
5. To erect for the members, and for society at large, a holy place of transcendental pastimes, dedicated to the personality of Krishna.
6. To bring the members closer together for the purpose of teaching a simpler and more natural way of life.
7. With a view towards achieving the aforementioned purposes, to publish and distribute periodicals, magazines, books and other writings.


Four regulative principles
Bhaktivedanta Swami prescribed four regulative principles, in relation to the four legs of dharma, as the basis of the spiritual life:
No eating of meat (including fish) or eggs (lacto-vegetarianism)
No illicit sex
No gambling
No intoxication (including alcohol, caffeine, tobacco and other recreational drugs).
The four legs of Dharma are:
Daya: Mercy
Tapas: Self-Control or Austerity
Satyam: Truthfulness
Śaucam: Cleanliness of body and mind

Preaching activities
ISKCON is known for their energetic active preaching. Members try to spread Krishna consciousness, primarily by singing the Hare Krishna mantra in public places and by selling books written by Bhaktivedanta Swami. Both of these activities are known within the movement as Sankirtan. A study conducted by E. Burke Rochford Jr. at the University of California found that there are four types of contact between those in ISKCON and prospective members. Those include: individually motivated contact, contact made with members in public arenas, contact made through personal connections, and contact with sympathizers of the movement who strongly sway people to join. According to the doctrine of Chaitanya Mahaprabhu, one does not need to be born in a Hindufamily to take up the practice of Vaishnavism. There are ISKCON communities around the world with schools, restaurants and farms. In general, funds collected by ISKCON are treated as communal property and used to support the community as a whole and to promote the preaching mission. Many temples also have programs (like Food for Life) to provide meals for the needy. Also, ISKCON has recently brought the academic study of Krishna into western academia as Krishnology.

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17 Days That Shook The World : Pokhran 1998



The 17 Days in May : Chronology of Indian nuclear weapons tests

3 February: Bharatiya Janata Party releases its manifesto for the February-March Indian elections. In the manifesto, the party pledges to “exercise the nuclear option” and “declare India a nuclear weapon state” after coming to power.

4 March: BJP-led coalition wins the largest block of parliamentary seats with a total of 270 out of 545 seats.
18 March: BJP wins vote of confidence in parliament with 274 votes cast in its favour and 261 cast against it. The same day, the party adopts “National Agenda for Governance”, in which it promises to establish a National Security Council to undertake India’s first strategic defense review, re-evaluate India’s nuclear policy, and exercise the option to induct nuclear weapons.

19 March: Prime minister Atal Behari Vajpaee declares India will induct nuclear weapons only if necessary.

21 March: Defence minister George Fernandes announces that the decision to induct nuclear weapons will depend upon a thorough strategic review to be undertaken by India’s National Security Council.

26 March: Foreign minister Gohar Ayub Khan appeals to the world community that it should impose sanctions against India in order to contain its nuclear ambitions. His statement comes a week after The New York Times quotes Western intelligence sources that “India has stockpiled about 100 nuclear warheads, and can rapidly assemble them.”

2 April: Prime minister Nawaz Sharif addresses letters to world leaders, including president Clinton, drawing their intention to India’s pronouncements which “connote a giant leap towards fully operationalizing Indian nuclear capability.” He also warns them that “Pakistan will be obliged to take cognizance of these alarming developments, and it cannot but exercise its sovereign right to adopt appropriate measures to safeguard its security.”

5 April: India unveils a new supercomputer, Param 10,000, that is able to simulate nuclear test-explosions.

21 April: Indian army chief General Ved Malik openly demands a nuclear and missile deterrent for India.

23 April: Defence minister Fernandes revives the Defence Minister’s Committee to directly involve the armed services chiefs of staff in the national security decision-making process. His scientific advisor Abdul Kalam makes a presentation before the committee, which discusses India’s threat perceptions and possible future course of action.

24 April: N N Jha, convener of the Foreign affairs Committee of the BJP, declares that India’s National Security Council may examine the option to induct nuclear weapons into the armed forces on the basis of laboratory testing.

25 April: Pakistan warns the international community that India is softening world opinion before openly deploying nuclear weapons. 
4 May: Defence minister Fernandes declares that China is India’s “potential threat number one”. He says that if the defense review leads the government to believe that India should exercise its nuclear option, then India will do so.

5 May: India expands its Atomic Energy Commission (AEC). The new members are: Dr Raja Ramanna, former AEC chairman, Brajesh Mishra, principal secretary, and Professor S R Jashim, member of the Planning Commission.

11 May: India conducts three nuclear explosions at its Pokhran nuclear test-site. These include a fission-device, a low-yield device, and a thermonuclear device. Prime minister Vajpaee declares that the yields from the explosions are “in line with expected values.” “The people of India now have a credible nuclear deterrent,” it is officially stated.



12 May: BJP’s new president Kushabhau Thakre asserts that India will not give in to “blackmail” by any country and take whatever steps necessary for its security.

13 May: India conducts tests of two sub-kiloton nuclear devices at Pokhran, which, the government states, complete its “planned series of nuclear explosions.”

14 May: BJP president Thakre says, “There is no room for any concern because of the blasts, all that India wants is to have its territories vacated.” Prime minister Vajpaee says India is ready to face sanctions. “If such steps are taken, then we Indians will face it. We are ready for any difficulty.” In a public opinion poll, majority Indians favour the country’s nuclear build-up. 

The BJP calls for nation-wide celebrations on May 16 to mark India’s entry into the Nuclear Club. Congress-I president Sonia Gandhi tacitly supports the nuclear explosions by saying “the nuclear question is a national matter” and, on this, “every Indian is united.” But, anti-nuclear activists in India say they are disappointed by the public euphoria over the nuclear tests and the lack of public awareness about the dangers of nuclear arms race. The UN Security Council also “strongly deplores” Indian nuclear tests.

15 May: India Today quotes prime minister Vajpaee as saying India has a “big bomb.” This is widely interpreted to mean that India has formally declared itself to be a nuclear weapon state. Vajpaee also says that India will use nuclear weapons in case of any external “aggression.”

16 May: BJP celebrates “National Day of Pride.” Celebrations at 139 Mandals across Delhi are marked by the presence of party leaders, distribution of sweets among the jubilant crowd, and display of fireworks. Meanwhile, anti-nuclear campaigners hold protest rallies in the Capital, carrying placards: “We want ‘roti’ and ‘pani’ not ‘bombs’, the nation’s priorities are misplaced.” Prominent Indian scholars and writers—including Kuldip Nayyar, Rajni Kothari, Medha Patkar, Praful Bidwai, Achin Vinayak, Bittu Sehgal, Ravi Agarwal, Nityanand Jayaraman—condemn nuclear tests, saying the “need today was not to enter the club of five nations but to get out of the club of ten least socially developed countries. In a joint statement, they say: “We strongly condemn India’s recent nuclear testing, the aggressive jingoism unleashed by it and the insecurity caused by it in the region. Nuclear weapons cause trans-generational and large-scale destruction…It is shameful that we should celebrate an achievement that announces to the world that India is now capable of killing millions of people with nuclear weapons.”

17 May: India’s top scientists confirm India has become a nuclear weapon state, and that the five nuclear tests at Pokhran were a “culmination of a “weaponisation programme” jointly undertaken by the defense and atomic energy establishment. “The nuclear tests have boosted the confidence of the nation,” says prime minister Vajpaee.

18 May: L K Advani, home minister and former BJP president, warns Pakistan “to roll back its anti-India policy, especially with regards to Kashmir.” He announces the government’s new “pro-active” approach to “deal firmly and strongly with Pakistan’s hostile designs and activities in Kashmir.” “India’s decisive step to become a nuclear weapon state has brought about a qualitatively new stage in Indo-Pak relations, particularly in finding a lasting solution to the Kashmir problem,” says Advani, adding: “Islamabad should realise the change in the geo-strategic situation.” Vishwa Hindu Parishad announces its plan to build a shrine, ‘Shakti Peeth’ (abode of cosmic energy) near the site of the five nuclear explosions in Rajhastan.

19 May: BJP spokesman Krishan Lal Sharma warns Pakistan that it will pay a heavy price for “fuelling the conflict” in Kashmir. “Time has come when the government should take a tough stand and foil Pakistan’s attempts to stoke militancy,” he says. The United States warns India not to disturb the status quo on the Line of Control, saying “we urge both countries to respect it and refrain from provocative actions.”

20 May: Home minister Advani says India is determined to quell proxy war by Pakistan against “our nation. “Until now, India’s policy to make such provocation has been essentially reactive. But it has been decided that further misadventure on Indian territory shall be dealt with enough provocative basis.”
22 May: BJP spokesman Sharma again warns Pakistan that “if it continues with its anti-India policy, Pakistan should be prepared for India’s wrath.”

23 May: Prime minister Nawaz Sharif warns India against any “misadventure”, which, he adds, “will meet a resolute response….Any act that Pakistan takes will be in the supreme national interest.” The same day, Advani is given the charge of Kashmir affairs.

24 May: Farooq Abdullah, chief minister of the disputed Indian state of Jammu and Kashmir, says, “a major offensive against foreign mercenaries in the State will start soon.” BJP president Thakre says the nuclear tests were staged out of “necessity.”

25 May: “India is a nuclear weapon state despite the refusal of the United States and others to accept it as one,” says prime minister Vajpaee. “Let me repeat, India is a nuclear weapon state. Definition constraints reflecting a three-decade old situation need to come to terms with this reality.”

26 May: Defence minister Fernandes says moratorium (on nuclear tests, declared some days after the May 13 nuclear tests) will not affect nuclear weaponisation.

27 May: Prime minister Vajpaee strongly defends Indian nuclear weapons option and BJP’s hawkish posture towards Kashmir in the Lok Sabha, where many opposition leaders term India’s bomb as “BJP bomb.” At midnight, Pakistan fears an air-strike on its nuclear installations and, in order to prevent it, contacts Permanent Members of the Security Council and warns India of dire consequences.



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Swami Ayyappa and Sabarimala Pilgrimage |




|| Swami Ayyappa and Sabarimala Pilgrimage ||

:: Pilgrimages that deepen Awareness ::

Our ancient masters have inculcated mind and body purification practices to deepen Awareness through enjoyable rituals and pilgrimages. One such yearly pilgrimage, is Sabarimala in Kerala. Ayyappa is the deity in Sabarimala and this temple is visited by men of all ages but restricted to women of age group of 12 to 55. One of the reason is that women at this age group has to take care of the family when men observe this 41 day ritual every year.

:: What is the need of Sabarimala pilgrimage? ::

The importance of pilgrimage is to transcend to Awareness state, the Source of Healing and all other creativeness. The mind because of its wandering nature, tries often to be away from its pure Awareness state. Through rituals and practices (Sadhana) the wandering mind is brought to the state of awareness and finally realize Consciousness, the ultimate aim of life.

To achieve this the mind and body should be purified and finally the Ego has to transcend to Awareness state and realize the pure Consciousness. The various practices and rituals to purify mind and body and the practice of abstinence to transcend from the realm of Ego to the Atma is all beautifully integrated in this pilgrimage. This should be clearly understood, otherwise it will remain as a meaningless torture to the body and mind.

:: What does the form of Ayyappa symbolize? ::

The very form of Ayyappa is very symbolic and denotes the principles of Vedas.
The whole body (foundation) rest on his feet that represents perfection attained by the strict discipline of mind. In Vedic culture, importance is given to the feet of a great person. Feet represents the principle on which the great person stands. It is the principle (Dharma) that had made the person great and he stands on that great principle. We are not prostrating before the mortal body of the person, but we are actually worshipping the highest principle on which that great person stands. Same way the feet of Lord Ayyappa represents that great Dharma obtained by the strict discipline of mind, on which Ayyappa stands.

His sitting posture reflects that of a Yogi, one who has mastered the Self. A black belt around his legs anchorages the wandering mind and the color black represents complete with drawl or absorption to inwards. The color of an object is due to its ability to absorb one or many colors in light spectrum and reflect the rest. Black represents complete absorption of all colors.

Hands represent action and his hands are in a state of inaction (which means witnessing) and one hand in fearless posture, shows a mind that is free from the clutches of the Ego. Ego controls the mind by inculcating different types of fear.

When there is no fear and the mind is drawn internal, there is Awareness of the thoughts. This Awareness then witness everything as shown in the picture below. This witnessing finally transcends to the bliss of pure Consciousness.

Thus there are three aspects, the Consciousness (Sath) and Awareness (chit) and the bliss (Ananda) and therefore Ayyappa is called as Sath-Chit-Ananda or Sacchidananda. He represents all these three states of Consciousness, Awareness and Ananda.

:: Symbolism of Ayyappa ::

Ayyappa represents the state of a Yogi in Sath-chith-ananda. Why Ayyappa is called the son of Vishnu and Shiva? Vishnu is the witnessing Consciousness (sath) and shiva is the acting awareness (chith). Witnessing is static witnessing, while Awareness is in dynamic action.

Ayyappa is in the sitting posture to denote the two aspects of life, action (dynamic) and inaction (silent static witnessing), as told in Bhagavad Gita. Sitting posture is neither in action (Awareness) nor in-action (Consciousness). Our true nature is witnessing Consciousness, but for Consciousness to witness to happen, Awareness (action), takes a form of a body (matter). But in the body form we are always think that we are just the body and mind. We never realize that Awareness is behind all the action of life and our true nature (Atma) is that witnessing Consciousness.

We have taken a human form to experience our true nature that is Consciousness. Unfortunately we always identify with the body form and forget that our true nature is Consciousness. This ignorance of our own real self, causes us to discard the old body (death) and take new body form (birth) again and again. This endless birth and death is called Samsara. An exit out of this roller coast ride happens, when we realize our true Self, which is Brahman or Consciousness. This exit is called Moksha or liberation, salvation etc.

~ Om Swamiye Saranam Ayyappa ~

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Bengali wedding rituals



Bengali wedding rituals
The blowing of the conch shells along with ululation by the women are some characteristics of a Bengali wedding. Laced with elaborate rituals and colorful customs, Bengali weddings are an occasion of great revelry and jolly celebrations. A certain somberness and intellectual dignity differentiates Bengali weddings from the rest. Let us delve a little deeper to understand the Bengali biye.
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PRE-WEDDING RITUALS
Adan pradan: This ritual involves the prospective bride and groom along with the elders of their family sitting with the priest who after establishing that the couple are not close relatives or of the same lineage, sets the date of the wedding.

Aashirbaad/Patri patra: Performed a few days before the wedding, it is a confirmation of the marriage alliance formally. In the holy presence of Lord Narayana, the groom's family (excluding the groom) visits the bride and blesses her with gifts. The same ceremony is reciprocated by the bride's family thereafter.

Ai buddo bhaat: This ritual refers to the last maiden meal that the girl partakes at her own house before she is married off. Giving her company are close relatives and friends of the the bride. Simultaneously, the ritual also takes place at the groom's house.

Vridhi puja: The vridhi puja is conducted to honor all the ancestors of both the bride and groom. It is performed a day before the wedding by the paternal uncle of the bride/groom. All the required items are arranged in a baran dala (a silver plate) which is decorated with the 'shri' symbol.

Dodhi mongol: At the crack of dawn on the wedding day, ten married women from both sides take the bride/groom separately to a nearby lake/pond to formally invite Goddess Ganga. They also carry back a pitcher of water to bathe the bride/groom. Post the bath, the couple are served a lavish meal of fried fish, rice, curd and chiruya (poha), which is their last meal till they are married.

Piris: This involves the wedding piris (wooden seats) being brought to the venue amidst the blowing of conch shells and ululations. They are specially crafted and decorated by close friends and relatives.

Gae holud tattva: This refers to the haldi ceremony where turmeric paste is applied to the groom. The same paste is then sent to the bride's place along with the tattva (gifts) for the bride and her family members. The arrival of the tattva is met with the blowing of the conch shells. The bride's haldi ceremony then commences.

Adhibas tattva: Adhibas tattva refers to the gifts that the bride's family sends for the groom's side. Placed on kasar thala (a brass plate), the gifts include saris, sweets, curd and fish.

Kubi patta: It is a short ceremony to revere Saint Kuber, held at both the bride's and groom's houses.

Snan: This ceremony involves bathing the bride and groom individually which takes place in the afternoon of the wedding day. Married women apply turmeric and oil to the couple, who then go for a bath. They change into new clothes given by their respective in-laws. Their old clothes are given away to a barber.

Sankha porana: The bride in her maternal home wears the sankha porana (conch shell bangles) that have been dipped in turmeric water in the company of seven married women and the priest chanting mantras.
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WEDDING RITUALS
Bor jatri: This refers to the groom and his kinsmen undertaking the journey to the wedding venue.

Bor boron: When the bor jatri arrives at the venue, the bride's mother along with other members come out to welcome the groom with the traditional aarti, sprinkling trefoil and husked rice placed on a bamboo winnow. Generally, the bride's maternal uncle or brother lifts up the groom and escorts him to the altar.

Potto bostro: After the groom is seated at the wedding altar, he is offered a set of new clothes by the person who is to do the sampradan, most often the father of the bride.

Saat paak: The bride seated on the piri is lifted by her brothers and is taken around the groom in seven complete circles while her face remains covered by two betel leaves.

Shubho drishti: This ritual marks the moment when the bride and groom look at each other for the first time during the wedding.

Mala bodol: After having exchanged loving glances, the couple now exchanges garlands of fragrant flowers thrice.

Sampradan: The couple then sits at the altar where the bride is given away to the groom by the father of the bride (or any male member). A sacred thread is tied to the couple while mantras are being chanted.

Yagna: The couple sits in front of the fire, chanting mantras after the priest and invokes the blessings of Agni, the fire god. They encircle around the fire thereby solemnizing the occasion.

Anjali: The bride's brother puts puffed rice into her hands. The groom holds her from the back and together, they make an offering to the fire.

Sindoor daan and Ghomta: The groom now puts sindoor from a small pot on the bride's hair parting. The bride then covers her head with a new sari offered by the groom as ghomta (veil).
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POST-WEDDING RITUALS
Bashor ghor: According to Bengali customs, the groom spends the night at the bride's place where night-long fun and merriment continues.

Bashi biye: Not much in vogue these days, the groom puts sindoor on the bride, visits the mandap and prays to Sun god.

Bidaai: The bride finally is ready to leave her maternal home. The newlyweds are usually escorted by the groom's father/uncle/brother.

Bou boron: This refers to the formal welcome of the newlyweds to the groom's house. The bride is made to dip her feet into a plate of alta (colored red dye) and walk into the house. Amidst ringing of bells and conch shells, the groom puts an iron bangle onto the bride's left arm.

Bou bhaat: Bou bhaat signifies the first time that the bride serves food (usually rice preparations) to her in-laws. The groom gifts her sari at this point. It is generally followed by the reception in the evening.

Kaal ratri: This ritual takes place on the second night after the wedding where the newlyweds are not allowed to even look at each other.

Phool shojja: The last of the wedding ceremonies, the bride and groom are adorned in new clothes. Their nuptial bedroom is decorated with fresh flowers and they are left alone to enjoy conjugal bliss. Generally, the clothes and flowers arrive as gifts from the bride's house.

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Bride of Bengal



The Women of Bengal are very charming and beautiful with lovely long black hair.
The Women of Bengal are very charming and beautiful with lovely long black hair. The unique style of draping their sarees makes them look homely and traditionally awesome. The Bengalis have a very pleasing way of pampering the brides-to-be. The Bengalis are very artistic and hence the brides are decorated with lovely ornaments and showered with gifts and sarees.

The Bengalis are very artistic and hence the brides are decorated with lovely ornaments and showered with gifts and sarees.

In a series of ceremony the Bengali bride receives sarees and sweet-meats from her husband-to-be’s family. In the Haldi and oil ceremony the bride is smeared with oil and haldi so that her skin will glow and she shall look more beautiful and gorgeous on her wedding day. Even sandal wood paste is used to give her a flawless skin.

The Bengali brides wear red, maroon, pink color saree on their wedding day.
Red is the color that Bengalis love and flaunt it on every special occasion. The Bengali brides wear red, maroon, pink color saree on their wedding day. The banarasi silk in widely used and is common and most popular. The traditional saree is generally in white with red or maroon or pink border. The bengali bride might prefer a completely bright red silk saree with gold zari or buta work on it. The sarees with kantha work also look amazing on the brides. The silk sarees that are available for brides are hand crafted beautifully with various ancient prints and designs.


The bengali bride looks no less than a princess on her wedding day, the whole focus of the wedding is almost the bride alone and hence her friends and relatives try their best and make her look fabulous.

bengali bride and groom
The bindi that adorns the face of the bengali bride is quite huge. On the bride’s fore head small white and red dots are decorated to make her look even more cute and pretty. The bride might choose any design she likes. The decoration on the forehead just above the eye-brows ends near the end of the brows or it may even extend a little.


auspicious

The hugely popular banarasi silk that the bengali bride wears for her wedding is of high quality and very heavy. Now a days the brides also prefer the kanjeevarams, paithani , etc as even they are elegant , royal and spectacular when draped. The sarees are also available in poly-silk and chiffons and georgettes which are quite popular too.

auspicious
As marriage is the happiest and auspicious day of their life they spend hugely on the sarees and ornaments. The alta is a must for Bengali brides. They apply generous amounts of alta on hands and feet of the bengali bride and decorate her with a particular design. The ornaments are huge and very attractive and very pleasing. They are made in gold which is considered to be auspicious and also wearing ornaments of gold are compulsory for the bride. The lipsticks that they wear are in different shades of bright reds.


The white saree with red border is worn by the bengalis mainly during puja days.And also during durga puja there is the ceremony of sindoor daan on dashami.That day is like traditionally reserved for wearing the white garad silk saree with red border.

Every Bengali wife is supposed to wear an iron bangle along with her white shankha(shell bangle) and red pola(coral bangle) apart from the usual sindoor that every Hindu wife has to wear.

and we Bengalis have like mainly two different set of traditions among us. Its like one set belongs to East Bengal ( erstwhile Bangladesh) and the other part belongs to West Bengal. This happened when there was like united Bengal before partition.nowadays its actually no big thing but yes there is difference in eating habits, yearly traditions and in wedding rituals too.


Bengali wedding rituals

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The Shiva Lingam - most misunderstood motifs of Hinduism




“Lingam” is one of the most misunderstood motifs of Hinduism. It has been subject to such a bad smear campaigning by Westerners, especially missionaries, which even Merriam-Webster dictionary defines it as a stylized phallic symbol.
The Shiva Lingam

The expression ‘linga’ in the Agama context signifies ‘symbol’ (chinha). Derived from the root ‘ligi gatyau’, it refers to movement, and words having been movement as their etymological meaning have also connotations of knowledge (‘sarve gatyarthah jnanarthah’). Linga therefore means that by which the Divine is cognized or approached (‘lingyate jnayate anena iti lingam’).

The Agama texts also bring out another valid explanation for the word ‘linga’: linga in its primary sense is broken up into ‘ling’ (to dissolve, to get merged, to destroy) and ‘ga’ (to emerge, to go out). Linga is so called because all phenomena are dissolved in Siva at the time of cosmic dissolution, and it emerges from Siva once again at the time of creation. (Ajitagama, 3, 16-17).

Thus Shiva Linga represents the mark of the cosmos.

The Shiva Lingam
Swami Vivekananda gave by far the best rebuttal to Western claims that it might be a symbol of phallic worship, by giving proof from the vedas.

Swami Vivekananda gave a lecture at the Paris Congress of the History of Religions in 1900 during which he refuted the statements of some Western scholars that referred to Shiva linga as phallic worship.

Vivekananda’s words at the congress were in connection with the paper read by Mr.Gustav Oppert, a German Orientalist, who tried to trace the origin of the Shalagrama-Shila and the Shiva-Linga to phallicism. To this Vivekananda objected, adducing proof from the Vedas, and particularly the Atharva-Veda Samhita, to the effect that the Shiva-Linga had its origin in the idea of the Yupa-Stambha or Skambha—the sacrificial post, idealized in Vedic ritual as the symbol of the Eternal Brahman.

The Shiva Lingam
Swami Sivananda, also explains why equating Siva Lingam with the phallus is a mistake. According to him, “This is not only a serious mistake, but also a grave blunder. In the post-Vedic period, the Linga became symbolical of the generative power of the Lord Siva. Linga is the differentiating mark. It is certainly not the sex-mark. You will find in the Linga Purana:

“Pradhanam prakritir yadahur-lingamuttamam; Gandhavarnarasairhinam sabda-sparsadi-varjitam”
—The foremost Linga which is primary and is devoid of smell, colour, taste, hearing, touch, etc., is spoken of as Prakriti (Nature).”

Bana lingam (One kind of Shiva linga )
The Bana Lingam is a most Sacred Symbol and Divine Energy Tool, both in the ancient and in this modern world. The Bana Lingas are Swayambhu Shiva Lingas that have taken shape in the Sacred Narmada River, in the Central Western part of India. This is why the Bana Lingams are also known as the Narmada Banalingas or Narmadeshwar Shiva Lingas.The Narmadha Bana Lingas became very famous throughout the world, after the film “Indiana Jones and the Temple of Doom” was screened : this is the very same Sacred Stone that they were searching for .
The story:There is a story narrated in Aparajita-pariprchchha (205, 1-26) about the origin of the bana-lingas and their association with the Narmada river. Siva wanted to destroy the ‘tri-pura’, which had been obtained as a boon by the arrogant demon Banasura, and he let go a fiery dart from his great bow ‘pinaka’. The dart broke the three ‘puras’ into tiny bits, which fell on three spots:

The Shiva Lingam
1, on the hills in Sri-kshetra (of unknown identity),
2, on the peaks of Amarakantaka in the Vindhya ranges, and
3, on the banks of the holy river Narmada. The bits that fell in these places soon multiplied into crores,. each bit becoming a linga. As they formed part of the possession of Banasura, they were called Bana-Lingas.

Science:
The Shiva Lingam It is the considered view of many researchers and geologists that the unique composition of the Narmadha Shiva Lingas was due to the impregnation of it’s rocky river-sides and the rocks in the river bed, 14 million years ago by a large meteorite that crashed into the Narmada River. The fusion of the Meteorite and the Earthly Minerals has spawned a new and unique type of Crystalline Rock with extraordinary energetic qualities - the Narmada Bana Lingam. The bana Lingas contain Crypto Crystalline Quartz (masses made up of either fibrous or granular aggregates of tiny, microscopic Quartz Crystals) and a Gemstone material called Chalcedony (with an iron oxide and geothite inclusion) alongwith Basalt and Agate - this unique composition coupled with elliptical shape has a precise resonance in alignment with our Energy Centers or Chakras and are used for thousands of years as Divine Energy Generators for Cleansing, Healing and for Meditation. The Narmada Bana Lingas are quite strong and the hardness is a 7 on the Moe’s Scale, one of the highest frequency vibration rates of all stones on earth. The vibration of bana linga is said to be perfect for purification purpose.

The Shiva Lingam
Thus Matsya-Purana (Chap. 165-169), truly said drinking the water from this river (Narmada) and worshipping Siva (Bana Lingam) will secure freedom from all states of misery.


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कितने सारे भगवान ? - How many Gods?




ओम् नमो भगवते वासुदेवायः

कितने सारे भगवान ? यह लोग हमेशा कहते है | कुछ लोग जो या तो अज्ञानी हैं या जिनकी मंशा सनातन धर्मं के प्रति अच्छी नहीं है, इस वाक्य का अत्यधिक प्रयोग करते दिख जायेंगे |

उपरी लोकों में देवताओं का वास है जिसे मुर्ख लोग कई भगवान कहकर पुकारते हैं | देवता मनुष्य से ज्यादा उन्नत प्राणी हैं | इस ब्रम्हांड का निर्माण करते समय श्री ब्रम्हा ने १४ लोकों में जीवन की स्थापना की और विभिन्न प्रकार के प्राणियों के निर्माण किया | देवता उन प्राणियों में सबसे उत्तम हैं | देवताओं के अलावा भी कई और उन्नत प्राणी उपरी लोकों में हैं जैसे यक्ष, गन्धर्व इत्यादि | प्राणियों की यह विविधता इस पृथ्वी पर भी स्पस्ट देखा जा सकती है | लाखों प्रकार के जीव इस पृथ्वी पर निवास करते हैं |

श्रीमद भगवत गीता में परमात्मा श्री कृष्ण ने देवताओ के सृजन एवं मनुष्यों के साथ उनके सामंजस्य का स्पस्ट वर्णन किया है |

कृपया निम्न श्लोकों कों देखे :
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Ch3:Sh10
सहयज्ञाः प्रजाः सृष्टा पुरोवाचप्रजापतिः ।
अनेन प्रसविष्यध्वमेष वोऽस्त्विष्टकामधुक्‌ ॥

भावार्थ : प्रजापति ब्रह्मा ने कल्प के आदि में यज्ञ सहित प्रजाओं को रचकर उनसे कहा कि तुम लोग इस यज्ञ द्वारा वृद्धि को प्राप्त होओ और यह यज्ञ तुम लोगों को इच्छित भोग प्रदान करने वाला हो॥10॥

Ch3:Sh11
देवान्भावयतानेन ते देवा भावयन्तु वः ।
परस्परं भावयन्तः श्रेयः परमवाप्स्यथ ॥

भावार्थ : तुम लोग इस यज्ञ द्वारा देवताओं को उन्नत करो और वे देवता तुम लोगों को उन्नत करें। इस प्रकार निःस्वार्थ भाव से एक-दूसरे को उन्नत करते हुए तुम लोग परम कल्याण को प्राप्त हो जाओगे॥11॥

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ऊपर के श्लोकों से हम निम्न तथ्यों कों अलग कर सकते हैं :
१. श्री ब्रम्हा ने देवता समेत विभिन्न प्राणियों का सृजन ब्रम्हांड के निर्माण के समय किया
२. मनुष्य देवताओं का पूजन करते हैं और देवता मनुष्य एवं दूसरे जीवों के पालन करते हैं |

ऊपर के श्लोकों में स्पस्ट तौर पर लिखा है कि मनुष्य देवतों का पूजन कर उन्हें प्रसन्न करें और देवतागण मनुष्यों के पालन करें | मनुष्य एवं देवता दोनों एक दूसरे कों उन्नत करें | इस प्रकार से देवताओं एवं मनुष्यों का धर्म निर्धारित किया गया है|

इसको थोडा और स्पष्टता के साथ समझने के लिए एक्स उदहारण लेते हैं | हम देखतें हैं कि सभी बड़े देशों में प्रधान मंत्री या राष्ट्रपति देश का प्रमुख होता है | प्रधान मंत्री या राष्ट्रपति के साथ या उसके अधीन कई अन्य मंत्री होते हैं | देश की केन्द्रीय सत्ता फिर छोटे छोटे राज्यों में विभाजित होती है | हर एक राज्य का फिर अपना मुख्य मंत्री या राज्यपाल होता है जो उस राज्य का प्रमुख होता है | इतना ही नहीं राज्य पुनः जिलों या काउंटी में विभाजित होते हैं और हर एक जिले का एक प्रमुख होता है | इस प्रकार सत्ता का पूरा कार्य व्यवस्थित होता है |

मान लें किसी कों आय कर जमा करना है तो वह नजदीक के किसी आयकर ऑफिस में जाकर अपना आयकर जमा करता है | इसी प्रकार किसी अन्य कार्य के लिए भी हम किसी स्थानीय सरकारी कार्यालय में ही जाता हैं, प्रधान मंत्री के पास नहीं जाता | बल्कि बड़े से बड़े कार्य भी हम इसी प्रकार की किसी सरकारी कार्यालय में ही संपन्न होते हैं | प्रधान मंत्री अपने मंत्रियों के साथ देश की उच्च नीति निर्धारण का कार्य करता हैं | मंत्री परिषद की सारी नीतियां फिर ऊपर से नीचे सभी सरकारी निकायों द्वारा लागु की जाती हैं |

यह व्यवस्था सिर्फ सरकारी कार्यों ही नहीं बल्कि हर एक प्रकार कि संस्थायों जैसे बैंक, न्याय पद्धति, कंपनियों आदि में लागु होती हैं | बल्कि इस प्रकार की व्यवस्था सबसे उत्तम है और सार्वभौम भी |

ईश्वर का विधान भी ठीक इस प्रकार से कार्य करता है | ईश्वर ने देवताओं कों इस ब्रम्हांड के सारे कार्यों कों सुचारू रूप से चलाने के लिए निर्मित किया है | देवता अपने नीचे अन्य उच्च प्राणियों की सहायता से परमात्मा श्री विष्णु द्वारा निर्धारित दैविक नियमों कों लागु करते हैं | इन्द्र देव इन देवताओं के राजा हैं |

श्री शिव शंकर, श्री ब्रम्हा एवं श्री विष्णु उसी परम ब्रम्ह के तीन रूप हैं जो इस ब्रम्हांड के तीन प्रमुख कार्यों निर्माण, पालन एवं विनाश का निर्धारण करते हैं | इस प्रकार ब्रम्हांड का पूरा कार्य संपन्न होता है |

मनुष्य देवताओं की पूजा करते हैं और देवता मनुष्यों एवं अन्य जीव कों प्रथ्वी पर जीवन से सम्बंधित सभी साधनों कों उपलब्ध कराते हैं | इस प्रकार देवता श्री विष्णु द्वारा निर्धारित नियमों कों लागु करते हैं | मनुष्यों की समृधि देवताओं का यथा संभव पूजन करके प्राप्त होता है | देवताओं और मनुष्य के इस सम्बन्ध में बाधा होने से पृथ्वी पर समस्याओं का आगमन होता है | मनुष्यों कों हमेशा देवताओं का पूजन कर उनका आशीर्वाद लेना चाहिए |

मुझे उम्मीद है इस सत्य कों जानने के बाद किसी के मन में किसी प्रकार कि शंका नहीं होगी | इस विषय पर और भी अधिक विवरण मैं आने वाले विवेचन में दूँगा |
परमात्मा श्री कृष्ण आपका मार्गदर्शन करें |
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(English)
On namo bhagavate vasudevayah"

How many Gods? Many of us might be coming through phrase like this. Rather this is one of the propaganda used against Hinduism by those who are either ignorant or have ill intensions.

So called multiple Gods are actually Devatas(Demigods or deities). They are wrongly referred to as gods by the Ignorant western translators. These deities were one of those living beings created by Lord Bramha at the time of creation. Devatas are the higher beings who resides in higher planets called Devalokas(Heaven). There are not just Devatas(deities) but other higher beings , to name a few are Gandharvas, Yakshas etc. All of them resides in their respective Lokas (higher planets). The number of such planets where life exists is supposed to be 14. The references of the same can be found in Vedas and Puranas. Shrimad Bhagvat Gita gives a brief description about deities and their significance for human.

For this post I would like to put following references:
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Ch3:Sh10
saha-yajnah prajah srstva purovaca prajapatih
anena prasavisyadhvamesa vo ’stv ista-kama-dhuk

"In the begining , the creator Brahma created being(including deities) with the Yajna(Sacrifice) and said "By Yajna(Sacrifice) you propagate, and grow , let Yajna(Sacrifice) be your wish-fulfilling cow of plenty"

Ch3:Sh11
Te deva bhavayantu vah devan bhavayatanena
parasparam bhavayantahsreyah param avapsyatha

"By this Yajna(religious sacrifice), you please demigods and then those demigods will nurture you. In this way, supporting each way, you shall obtain supreme good"

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From above two shlokas we can summarize following important facts.
1. Lord Bramha created all living beings at the time of creation including Devatas.
2. Deities are responsible for nurturing human and other living beings.

It is very clear from the above that Human should please Devatas(demigods) by worshiping in different way. Devatas in turn will nurture human.

To understand in better way we can draw a parallel with a Governing system. There is a prime minister/President who is the head. He has many ministers under him. Centeral Govt/Federate State is further divided into states/provinces. Each state has one chief minister/Governor who is the head of that state. The states are further divided into smaller governing bodies like districts /counties etc. There are officials at local levels who represent state/center government. This way whole governance is done.

If somebody wants to submit Income tax he is not going to Prime minister for it, rather he goes to local Income tax office to pay the Income tax.

This system works not only in Governing bodies but in almost all types of organizations, like banks, Institutes, Companies etc. Rather hierarchy is the best way to accomplish all complex tasks.

The ministery of God works in same way. Devatas(deities) are the ministry of Lord. The task of all deities is divided. Indra is the king of Devatas(deities). Devatas perform their tasks to nurture human and other living beings as per the divine law Made by Lord Shri Vishnu. Lord Bramha and Lord Shiva and Lord Vishnu manifested from same Parmatma(Lord) to perform the three major tasks of Universe, the creation, maintenance and destruction.

Human worship deities and deities in turn nurture human. This co-relation and coexistence is the best possible way for prosperous existence of human on earth. Deviation from this co-relation is invitation to the troubles as this disturbs the balance between Devatas and Human.

I hope this note will help devotees to remove confusion related with Devatas. All such talks of many gods are due to ignorance or propagated for ill intentions. God has his full ministry and all tasks in this universe is well managed by divine laws.


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What is Karma?



Karma is a term that you may have heard before. It probably brings to mind the idea of “what goes
around, comes around.” To some extent, this is accurate, but the Sanskrit word “karma” simply
means “action” or “deed.”
According to Hindu philosophy, every action (karma) has a reaction or outcome. When an individual’s actions are positive or selfless, and righteous (dharmic), they will experience positive effects or rewards. If their actions, on the other hand, are negative (i.e. lying, stealing, hurting, etc.), the results will be negative. The k arma of an individual’s actions, positive or negative, may be experienced immediately, later in their present life, or possibly in a future life or lives. It is important to remember that an individual’s karma is based on their thoughts, words, and actions and the choices they make.

There are three types of karma :
  • Kriyaman karma - These are actions performed in the current life that may produce results in the same or subsequent life. Some forms of current karma are also known as agami karma .
  • Prarabdha karma - This is k arma whose effects have already begun. It takes longer to manifest, but occurs at some point in an individual’s present lifetime.
  • Sanchita karma - This is accumulation of all past k arma , and the results of this usually occur in a future lifetime.
What is Reincarnation?
Hindu teachings state that every birth is the result of an individual’s unique karmic circumstances. Thus, when a person or living thing dies, their soul is attracted to circumstances that will help balance out their karmic needs and debt in order to advance spiritually. People may be born into circumstances where they suffer in order to reap the consequences of bad decisions from this and previous lives. Others might be born into circumstances in which their suffering is minimal as the reward for following their dharma well in this and previous lives. Hindus also believe that people can be reincarnated as other living things based on their previous actions. This process of reincarnation and the presence of souls in all living things is the basis for respect that Hindus are encouraged to show for all people and forms of life.

Suffering, as understood in Hinduism, is not necessarily just physical or material; it also refers to an  individual’s state of mind. For example, a woman might be born into a wealthy family, but she may have a disposition that never allows her to fully enjoy her prosperity. On the other hand, a kind man who finds himself handicapped after an accident could remain kind, caring, and helpful in spite of his physical limitations. The woman’s wealth, but inability to enjoy it, and the man’s ability to endure physical difficulty, yet continue to live joyfully are both understood to be results of their past and present karma . In other words, karmic needs and debts manifest not only as an individual’s life circumstances, but the way in which they deal with them. On the other hand, it’s also said that the response to circumstances is always open to choice -- karma generates an outcome, but not the response to that outcome. According to Hindu teachings, a person’s response to suffering should be informed by wisdom. Hindus hold that a key component to karma and spiritual advancement i s acting in ways to alleviate the suffering - be it physical, material, or mental - of other living things.

The cycle of reincarnation (birth, death, and rebirth) is called samsara. While the concept of karma acts as a positive motivator for leading a spiritual life, samsara is, in some ways, a negative reinforcer, and Hindus
strive to be free of this cycle.

Many Hindus believe that s amsara is a feature of life based on the individual’s deluded belief (m aya ) that one’s existence is independent of everything and everyone else. With this understanding of m aya , the individual forgets not only his or her own Divine nature, but the presence of that same divine nature in the rest of existence. The mistaken belief in an individual’s own autonomy from the rest of existence is what drives them to act selfishly or in a fashion that generates karma , and this k arma , in turn, keeps the individual tied to the cycle of s amsara . Others define maya as the state of forgetfulness of the complete dependence of existence on the Divine, and as a result, also forgetting the need for all actions to be selfless and in loving devotion of God.


What is Moksha?

Most schools of Hindu philosophy conclude that the ultimate goal is for the individual to work through
their reservoir of karma in order to attain moksha. Hindus typically believe that when a soul completely balances its “karmic bank account,” by reaping the consequences of all actions, good and bad, it is ready to attain moksha (though there are situations in which moksha can be attained beforethis balance). Moksha is, therefore, liberation from the cycle of birth and rebirth (samsara).

For many Hindus, this translates to the perfected ability to live in the present moment, detached from desire and the fruits of action (consequences or rewards), and experience absolute peace and the awakening of pure compassion towards all. As such, m oksha can be achieved in this lifetime through self-realization (a tma-jnana ) or realization of an individual’s true, divine nature. In this state, the soul stops creating the karma that binds it to the physical world, and finds liberation.

For others, the path to moksha is one that is paved by loving devotion to God and selfless service, where every action is viewed as an offering to God. Moksha, in this view, can only be attained upon physical death. It is described as experiencing bliss and closeness to God, the depth of which is dependent on the innate nature of the individual soul and their karma. Since Hindus believe in karma and reincarnation, the concept of heaven and hell as worlds of eternal glory or damnation do not exist for them. Hindus also do not ascribe to the concept of Satan or a devil that is in eternal opposition to God. Some Hindus may believe
in what is described in Hindu scriptures as two planes of existence called svarga and naraka that can be likened to heaven and hell, respectively. Neither svarga nor naraka, however, are either permanent or eternal. Both are intermediary planes of existence in which the soul might exhaust a portion of its karmic debt or surplus before taking physical birth once again to strive for moksha.

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वैदिक गणित के चमत्कारी सूत्र -Vedic Mathematical Formulae



Vedic Mathematical Formulae
(वैदिक गणित के सूत्र)

Sutras
  • 1.    Ekadhikena Purvena (एकाधिकेन पूर्वेण)
पहले से एक अधिक के द्वारा।
‘By one more than the previous one’
Corollary: Anurupyena
  • 2.    Nikhilam navatascaramam Dasatah (निखिलम् नवतश्चरमं दशतः)
सभी नौ में से परन्तु अन्तिम दस में से।
‘All from nine and last from ten’
Corollary: Sisyate Sesasamjnah
  • 3.    Urdhva – tiryagbhyam (ऊर्ध्व तिर्यग्भ्याम्)
सीधे और तिरछे दोनों प्रकार से।
‘Vertically and crosswise’
Corollary: Adyamadyenantyamantyena
  • 4.    Paravartya Yojayet (परावर्त्य योजयेत्)
पक्षान्तरण कर उपयोग में लेना।
‘Transpose and apply’
Corollary: Kevalaih Saptakam Gunyat
  • 5.    Sunyam SamyaSamuccaye (शून्यं साम्य समुच्चये)
समुच्चय समान होने पर शून्य होता हैं।
‘When the samuchayas are same, then it is Zero’
Corollary: Vestanam
  • 6.    Anurupye – Shunyamanyat (आनुरूप्ये शून्यमन्यत्)
अनुरूपता होने पर दूसरा शून्य होता हैं।
‘If one is in ratio, the other one is zero’
Corollary: Vestanam
  • 7.    Sankalana – Vyavakalanabhyam  (संकलन-व्यवकलनाभ्याम्)
जोड़कर और घटाकर।
‘By addition and subtraction’
Corollary: Yavadunam Tavadunikritya Vargancha Yojayet
  • 8.    Puranapuranabhyam (पूरणापूरणाभ्याम्)
अपूर्ण को पूर्ण करके।
‘By completing’
Corollary: Antyayordashake'pi
  • 9.    Chalana - Kalanabhyam (चलन-कलनाभ्याम्)
चलन-कलन के द्वारा।
‘By calculus’
Corollary: Antyayoreva
  • 10.  Yavadunam (यावदूनम्)
जितना कम हो, अर्थात् विचलन।
‘By the deficiency’
Corollary: Samuccayagunitah
  • 11.  Vyastisamastih (व्यष्टिसमष्टिः)
एक को पूर्ण तथा पूर्ण को एक मानते हुए।
‘Whole as one and one as whole’
Corollary: Lopanasthapanabhyam
  • 12.  Sesanyankena charamena (शेषाण्यड्केन चरमेण)
अंतिम अंक से अवशेष को।
‘Reminder by the last digit’
Corollary: Vilokanam
  • 13.  Sopantyadvayamantyam (सोपान्त्यद्वमन्त्यम्)
अन्तिम और उपान्तिम का दुगुना।
‘Ultimate and twice the penultimate’
Corollary: Gunitasamuccayah Samuccayagunitah
  • 14.  Ekanyunena Purvena (एकन्यूनेन पूर्वेण)
पहले से एक कम के द्वारा।
‘By one less than the previous one’
Corollary: Dhwajanka
  • 15.  Gunitasamuchayah (गुणितसमुच्चयः)
गुणितों का समुच्चय।
‘The whole product (The product of the sums)’
Corollary: Dwandwa Yoga
  • 16.  Gunakasamuchayah (गुणकसमुच्चयः)
गुणकों का समुच्चय।
‘Set of multipliers (All the multipliers)’
Corollary: Adyam Antyam Madhyam

Sub-Sutras
  • 1.    Anurupyena (आनुरूप्येण)
अनुपातों से।
‘Proportionality’
  • 2.    Sisyate-Sesasmjnah (शिष्यते शेषसंज्ञः)
एक विशिष्ट अनुपात में भाजक के बढ़ने पर भजनफल उसी अनुपात में कम होता हैं तथा शेषफल अपरिवर्तित रहता हैं।
‘Quotient decreases in same ratio as divisor increases and remainder remain constant’
  • 3.    Adyamadyen –Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)
प्रथम को प्रथम के द्वारा तथा अन्तिम को अन्तिम के द्वारा।
‘The first by the first and the last by the last’
  • 4.    Kevalaih saptakam-Gunyat (केवलैः सप्तकं गुण्यात्)
7 के लिए गुणक 143
‘For 7 the Multiplicand is 143’
  • 5.    Vestanam (वेष्टनम्)
आश्लेषण करके।
‘By ousculation’
  • 6.    Yavadunam Tavadunam (यावदूनम् तावदूनम्)
विचलन घटा करके।
‘Subtract by the deficiency’
  • 7.    Yavadunam Tavadunikrtya Varganca Yojayet (यावदूनम् तावदूनीकृत्य वर्ग च योजयेत्)
संख्या की आधार से जितनी भी न्यूनता हो उतनी न्यूनता और करके उसी न्यूनता का वर्ग भी रखें।
‘What ever the deficiency subtract that deficit from the number and write along side the square of that deficit’
  • 8.    Antyayor Dasake’pi (अन्त्ययोर्दशकेऽपि)
अन्तिम अंकों का योग 10 वाली संख्याओं के लिए।
‘Numbers of which the last digits added up give 10’
  • 9.    Antyayoreva (अन्त्ययोरेव)
अन्तिम पद से ही।
‘Only the last terms’
  • 10.  Samuchayagunitah (समुच्चयगुणितः)
गुणनफल की गुणन संख्याओं का योग।
‘The sum of the products’
  • 11.  LopanaSthapanabhyam (लोपनस्थापनाभयाम्)
विलोपन तथा स्थापना से।
‘By alternate elimination and retention’
  • 12.  Vilokanam (विलोकनम्)
देखकर।
‘By mere observation’
  • 13.  Gunita Samuccayah Samuccaya Gunitah (गुणितसमुच्चयः समुच्चयगुणितः)
गुणनखण्ड़ो की गुणन संख्याओं के योग का गुणनफल गुणनफल की गुणन संख्याओं के योग के समान होता हैं।
‘The product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product (The product of the sum is the sum of the products)’
  • 14.  Dhwajank (ध्वजांक्):
ध्वज लगाकर।
‘On the flag’


Ekadhikena Purvena (एकाधिकेन पूर्वेण)
The Sutra means: “By one more than the previous one”.
पहले से एक अधिक के द्वारा।
1. This sutra is useful to the ‘squaring of numbers ending in 5’.
Example: 252.
For the number 25, the last digit is 5 and the 'previous' digit is 2. According to formula 'One more than the previous one', that is, 2+1=3. The Sutra gives the procedure 'to multiply the previous digit 2 (by one more than itself) by 3. It becomes the L.H.S (left hand side) of the result, that is, 2 X 3 = 6. The R.H.S (right hand side) of the result is 52, that is, 25.
252 = 2 X 3 / 25 = 6/25=625.
152 = 1 X (1+1) /25 =225;
952 = 9 X 10/25 = 9025;
1352 = 13 X 14/25 = 18225;
2. Vulgar fractions whose denominators are numbers ending in Nine
We take examples of 1 / a9, where a = 1, 2………… In the conversion of vulgar fractions into recurring decimals, Ekadhika process can be effectively used both in division and multiplication.

Division Method:  Value of 1 / 19.
The numbers of decimal places before repetition is the difference of numerator and denominator, 19 -1=18 places. For the denominator 19, the purva (previous) is 1. Hence Ekadhikena purva (one more than the previous) is 1 + 1 = 2.
  • 1.      0.10 (Divide numerator 1 by 20, 0 times, 1 remainder)
  • 2.      0.005 (Divide 10 by 2, 5 times, No remainder)
  • 3.      0.0512 (Divide 5 by 2, 2 times, 1 remainder)
  • 4.      0.0526 (Divide 12 or12 by 2, 6 times, No remainder)
  • 5.      0.05263 (Divide 6 by 2, 3 times, No remainder)
  • 6.      0.0526311(Divide 3 by 2, 1 time, 1 remainder)
  • 7.      0.05263115 (Divide 11 or 11 by 25 times, 1 remainder)
  • 8.      0.052631517 (Divide 15 or 15 by 2, 7 times, 1 remainder)
  • 9.      0.0526315718 (Divide 17 or 17 by 2, 8 times, 1 remainder)
  • 10.       0.0526315789 (Divide 18 or 18 by 2, 9 times, No remainder)
  • 11.       0.052631578914 (Divide 9 by 2, 4 times, 1 remainder)
  • 12.       0.052631578947 (Divide 14 or 14 by 2, 7 times, No remainder)
  • 13.       0.05263157894713 (Divide 7 by 2, 3 times, 1 remainder)
  • 14.       0.052631578947316 (Divide 13 or 13 by 2, 6 times, 1 remainder)
  • 15.       0.052631578947368 (Divide 16 or 16 by 2, 8 times, No remainder)
  • 16.       0.0526315789473684 (Divide 8 by 2, 4 times, No remainder)
  • 17.       0.05263157894736842 (Divide 4 by 2, 2 times, No remainder)
  • 18.       0.052631578947368421 (Divide 2 by 2, 1 time, No remainder)



Multiplication Method: Value of 1 / 19
For 1 / 19, 'previous' of 19 is 1. And one more than of it, is 1 + 1 = 2. Therefore 2 is the multiplier for the conversion. We write the last digit in the numerator (अंश) as 1 and follow the steps leftwards.
  • 1.      1
  • 2.      21(multiply 1 by 2, put to left)
  • 3.      421(multiply 2 by 2, put to left)
  • 4.      8421(multiply 4 by 2, put to left)
  • 5.      168421 (multiply 8 by 2=16, 1 carried over, 6 put to left)
  • 6.      1368421 (6 X 2 =12, +1 = 13, 1 carried over, 3 put to left)
  • 7.      7368421 (3 X 2, = 6 +1 = 7, put to left)
  • 8.      147368421 (7 X 2 =14, 1 carried over, 4 put to left)
  • 9.      947368421 (4 X 2, = 8, +1 = 9, put to left)
  • 10.       18947368421(9 X 2 =18, 1 carried over, 8 put to left)
  • 11.       178947368421(8 X 2 =16,+1=17, 1 carried over, 7 put to left)
  • 12.       1578947368421(7 X 2 =14,+1=15, 1 carried over, 5 put to left)
  • 13.       11578947368421(5 X 2 =10,+1=11,1 carried over, 1 put to left)
  • 14.       31578947368421(1 X 2 =2,+1=3, 3 put to left)
  • 15.       631578947368421(3 X 2 =6, 6 put to left)
  • 16.       12631578947368421(6 X 2 =12, 1 carried over, 2 put to left)
  • 17.       52631578947368421(2 X 2 =4,+1=5, 5 put to left)
  • 18.       1052631578947368421(5 X 2 =10, 1 carried over, 0 put to left)
Now from step 18 onwards the same numbers and order towards left continue.
Thus 1 / 19 = 0.052631578947368421

Example: Value of 1 / 7.
1/7 में हर के अंक को 9 बनाने के लिए हर और अंश में 7 से गुणा करते हैं।
(1/7 = 7/49); हर के 49 का पूर्वेण हैं, 4; जिसका एकाधिक 5 हैं। भिन्न के आवृति दशमलव स्वरूप का अन्तिम अंक 7 होगा तथा 7-1 = 6 अंकों के पश्चात् दशमलव अंकों की पुनरावृति होगी।
  • 1.      7
  • 2.      357 (multiply 7 by 5 =35, 3 carried over, 5 put to left)
  • 3.      2857 (5 X 5 =25, +3 = 28, 2 carried over, 8 put to left)
  • 4.      42857 (8 X 5 =40, +2 = 42, 4 carried over, 2 put to left)
  • 5.      142857 (2 X 5 =10, +4 = 14, 1 carried over, 4 put to left)
  • 6.      2142857 (4 X 5 =20, +1 = 21, 2 carried over, 1 put to left)
अतः

Nikhilam Navatas’charamam Dasatah (निखिलम् नवतश्चरमं दशतः)
The formula simply means: “all from 9 and the last from 10
सभी नौ में से परन्तु अन्तिम दस में से।

Example: 1000 - 368 = 632
We subtract each figure of 368 from 9 and the last figure from 10.
हम संख्या 368 के सभी अंकों को 9 में से तथा अन्तिम अंक 8 को 10 में से घटाते हैं।

The formula can be very effectively applied in multiplication of numbers, which are nearer to bases 10, 100, 1000 or to the powers of 10 (for example: 96 x 98 or 102 x 104).
जब दो संख्याएँ आधार 10 , 100 या 10 की धात के निकट हो तो उनका गुणनफल सूत्र निखिलम् द्वारा ज्ञात किया जाता हैं।

  • i.     When both the numbers are lower than the base.
Find 92 X 86. Here base is 100. Now follow the rules, 92 is 8 less than base 100. And 86 is 14 less than the same base 100. Hence 8 and 14 are called deviations from the base.


  • ii.   When both the numbers are higher than the base.
Here the deviation is positive as the numbers are higher then base.


We consider 03x07=21. This is done because, we need to consider two digits in deviation as it the base 100 has two zeros. If the deviation is near 1000 then we need to consider 3 digits in the deviation (e.g., 004 and not just 4).

  • iii. One number is more and the other is less than the base.
In this situation one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted.



12/ (-8) =112 how?
12/ (-8) should be read as 'one two, eight bar'. Here 'one' and ' two ' are in normal form. ‘Eight' is in complement form (परम मित्र रुप में). So, when we bring a carry from normal form to complement form, '12' becomes '11' and 'eight bar' becomes '2'( complement of 8). Hence 12/ (-8) =112.

Anurupyena (आनुरूप्येण)
The upa-Sutra 'Anurupyena' means 'proportionality' or 'similarly'.
अनुपातों से।
इस सूत्र के उपयोग से आनुपातिक गुणन या भाग किया जाता हैं। जब संख्याऐं सैद्धान्तिक आधार 100 से काफी दूर हो तो क्रियात्मक आधार उपयोग में लाया जाता हैं।
This Sutra is highly useful to find products of two numbers when both of them are near the Common bases like 50, 100 etc (multiples of powers of 10).

Example: 47 X 42
As per the previous methods, if we select 100 as base we get
47     -53
42     -58
This is much more difficult. Now by ‘Anurupyena’ we take a different working base through we can solve the problem. Take the nearest higher multiple of 10. In this case it is 50.
Treat it as 100 / 2 = 50.
  • 1.      We choose the working base. Working base is 100 / 2 = 50
  • 2.      Write the numbers one below the other
47
42
__
  • 3.      Write the differences from 50 against each number on right side
47     -03
42     -08
________
  • 4.      Write cross-subtraction or cross- addition as the case may be under the line drawn. Multiply the differences and write the product in the left side of the answer.

  • 5.      Since base is 100 / 2 = 50, 39 in the answer represent 39X50.
Hence divide 39 by 2 (because 50 = 100 / 2). Thus 39 ÷ 2 gives 19½ where 19 is quotient and ½ is remainder. This ½, as Reminder gives 50; making the L.H.S of the answer, 24 + 50 = 74 or (½ x 100 + 28) i.e. R.H.S. 19 and L.H.S. 74 together give the answer 1974.

Urdhva Tiryagbhyam (ऊर्ध्व तिर्यग्भ्याम्)
It means “Vertically and cross wise.”
सीधे और तिरछे दोनो प्रकार से।
Urdhva – tiryagbhyam is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number.

Example: 12 X 13
The symbols are operated from right to left.
  • 1.      Multiply vertically 2X3
  
  • 2.      Multiply crosswise 1X2 and 1X3

  • 3.      Multiply vertically 1X1

= 156
The multiplication of 3 digit number with 3 digit number:



Adyamadyena-Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)
The Sutra “Adyamadyena-Antyamantyena” means “the first by the first and the last by the last”.
प्रथम को प्रथम के द्वारा तथा अन्तिम को अन्तिम के द्वारा।
Area of rectangle:
Example:
Find out the area of a rectangle whose length and breadth are respectively 5 ft.2 inches and 4 ft.5 inches.
Generally we continue the problem like this.
Area    = Length X Breadth
= 5’ 2" X 4’ 5"               (Since 1’ = 12")
= (5 X 12 + 2) (4 X 12 + 5) conversion in to single unit
= 62" X 53" = 3286 Sq. inches.
Since 1 sq. ft. =12 X 12 = 144 sq.inches
We have area
3286 /144 = Quotient is 22 and Remainder is 118.
Area of rectangle is 22 Sq. ft 118 Sq. inches.
Mental argumentation:
It is interesting to know the mental argumentation. It goes in his mind like this
    5’     2"
    4’     5"
First by first: 5’ X 4’ = 20 sq. ft.
Last by last: 2" X 5" = 10 sq. in.
Now cross wise 5 X 5 + 4 x 2 = 25 +8 = 33.
Adjust units  to left as  33 = 2 X 12 +9 , 2 twelve's as 2 square feet make the first 20+2 = 22 sq. ft ; 9 left becomes 9 x 12 square inches and go towards right  9 x 12 = 108 sq. in. gives 108+10= 118 sq.inch.
We got area in some sort of 22 sq ft and 128 sq. inches.
By Vedic principles "the first by first and the last by last"
5’ 2" can be treated as 5a + 2 and 4’ 5" as 4a + 5,
Where a= 1ft. = 12 inch and a2 = 1 sq. ft = 144 sq. inch.
= (5a + 2) (4a + 5)
= 20a2 + 25a + 8a + 10
= 20a2 + 33a + 10
= 20a2 + (24a+9a) + 10
= 20a2+ (2a+9) a + 10     writing 33 = 2X12 +9
= 22a2+ 9a + 10
= 22 sq. ft. + 9X12 sq. inch + 10 sq. inches
= 22 sq. ft. + 108 sq. inch + 10 sq. inches
= 22 sq. ft. + 118 sq. inch

Factorization of quadratics:
By Vedic process two sub-sutras are used to factorizing a quadratic.
(a) Anurupyena   (b) Adyamadyena-Antyamantyena
The usual procedure of factorizing a quadratic is as follows:
= 2 a2 + 9a + 10
= 2 a2 + 4a + 5a + 10
= 2a (a + 2) + 5 (a + 2)
= (a + 2) (2a + 5)
But by mental process, we can get the result immediately. The steps are as follows.
  • 1.      Split the middle coefficient in to two such parts that the ratio of the first coefficient to the first part is the same as the ratio of the second part to the last coefficient. Thus we split the coefficient of middle term of 2 a2 + 9a + 10 i.e. 9 in to two such parts 4 and 5 such that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2:4 and the ratio of the second pat to the last coefficient, i.e. 5: 10 are the same. It is clear that 2:4 = 5:10. Hence such split is valid. Now the ratio 2: 4 = 5: 10 = 1:2 give one factor (a+2). 
  • 2.      Second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of the factor. i.e. the second factor is

2 a2 + 9a + 10 = (a + 2) (2a + 5)


Sankalana – Vyavakalanabhyam (संकलन-व्यवकलनाभ्याम्)
This Sutra means:  by addition and by subtraction.
जोड़ने और घटाने के द्वारा।
This sutra is widely used in solving a simultaneous equation where the coefficients of algebraic value are found interchanged.

Example:
84a + 41b = 166                                                                                           (1)
41a + 84b = 209                                                                                           (2)
With the help of Sankalana – vyavakalanabhyam
Add equation (1) and (2)
125a + 125b = 375
125 (a + b) = 375
a + b =3                                                                                                          (3)
Subtract equation (1) from (2)
43a - 43b = -43
43 (a – b) = -43
a – b = -1                                                                                                        (4)
Adding equation (3) and (4)
2a = 2
a = 1
Subtracting equation (3) from (4)
2b = 4
b = 4 /2
b = 2
Hence a = 1 and b = 2

Example:
7a + 3b = 13                                                                                                  (1)
3a + 7b = 17                                                                                                  (2)
समीकरण (1) (2) को  जोड़ने पर
10a + 10b = 30
a + b =3                                                                                                          (3)
समीकरण (1) (2) को  घटाने पर
4a - 4b = -4
a – b = -1                                                                                                        (4)
समीकरण (3) (4) को  जोड़ने पर
2a = 2     
a =1                                                                                                                (3)
समीकरण (3) (4) को  घटाने पर
2b = 4
b = 2
अतः a = 1 and b = 2

Yavdunam Taavdunikritya Vargancha Yojayet (यावदूनम् तावदूनीकृत्य बर्ग च योजयेत)
This sutra means “What ever the deficiency subtract that deficit from the number and write along side the square of that deficit.
संख्या की आधार से जितनी न्यूनता हो उसमे उतनी न्यूनता और करके उसी न्यूनता का वर्ग भी रखे।
This sutra is used to calculate squares of numbers near (lesser) to powers of 10
Example:  982
  • 1.      The nearest power of 10 to 98 is 100.
  • 2.      We take 100 as our base.
  • 3.      Since 98 is 2 less than 100, hence deficiency is 2.
  • 4.      We decrease the number by an amount equal to the deficiency which is (98 -2) = 96. This is the left side of our answer.
  • 5.      On the right hand side put the square of the deficiency. That is square of 2 = 04.
  • 6.      Hence the answer is 9604.
While calculating step 5, the number of digits in the squared number (04) should be equal to number of zeroes in the base (100). Hence in our case, the base 100 has 2 zeros and hence square of 2 is 04 and not just 4.
Example: 962.
  • 1.    96, 100 के पास हैं।
  • 2.    अतः आधार 100 लेते हैं।
  • 3.    आधार से न्यूनता = 4
  • 4.    96 में से 4 घटाते हैं। (96 - 4 ) = 92 जो उत्तर का बायाँ भाग होगा।
  • 5.    4 का वर्ग करते हैं, 42 = 16 ( 2 digits) जो उत्तर का दायाँ भाग होगा।
  • 6.    पद 4 तथा 5 के परीणाम को साथ में रखने पर 962 = 9216 प्राप्त होता हैं।

Yavadadhikam Taavadhikikritya Vargancha Yojayet (यावद्धिकम् तावद्धिकृत्य वर्ग च योजयेत)
This sutra means “whatever the extent of its surplus, increment it to that very extent; write along side the square of that extent.
संख्या की आधार से जितनी अधिकता हो उसमें उतनी अधिकता और करके उसी अधिकता का वर्ग भी रखें।
This sutra is very useful in calculating the squares of numbers nearer (greater) to powers of 10.
इस सूत्र का उपयोग उन संख्याओं का वर्ग ज्ञात करने में किया जाता हैं, जो 10 की घात से थोड़ी बड़ी हो।

Example: 1092
  • 1.      109, 100 के पास हैं।
  • 2.      हम आधार 100 लेते हैं।
  • 3.      आधार से अधिकता = 9
  • 4.      109 में 9 जोड़ते हैं, (109+9 ) = 118 जो उत्तर का बायाँ भाग होगा।
  • 5.      अब 9 का वर्ग करते हैं, 92 = 81, जो उत्तर का दायाँ भाग होगा।
  • 6.      उत्तर के दायें भाग में उतने ही अंक रखते हैं, जितने आधार में शून्य हो। यदि अंक कम या अधिक हो तो उन्हें समायोजित करते हैं।
  • 7.      पद 4 तथा 5 के परीणाम को साथ में रखने पर 1092 = 11881 प्राप्त होता हैं।

Example: 1042
  • 1.      The nearest power of 10 to 104 is 100.
  • 2.      We take 100 as our base.
  • 3.      Since 104 is 4 more than 100, hence surplus is 2.
  • 4.      We increase the number by an amount equal to the surplus which is (104 +4) = 108. This is the left side of our answer.
  • 5.      On the right hand side put the square of the surplus. That is square of 4 = 16.
  • 6.      Hence the answer is 10816.


Antyayor Dasakepi (अन्त्ययोर्दशकेऽपि)
The Sutra means - numbers of which the last digits added up give 10.
अन्तिम अंकों का योग 10 वाली संख्याओं के लिए।
जिन अंकों के चरम (अन्तिम) अंकों का योग 10 या 10 की घात हो तथा शेष निखिलम् अंक समान हो, उनकी गुणन संक्रिया इस विधि द्वारा की जाती हैं।
15 and 15, 1 is common and 5 + 5 = 10
57 and 53, 5 is common and 7 + 3 = 10
82 and 88, 8 is common and 2 + 8 = 10
126 and 124, 12 is common and 6 + 4 = 10
425 and 475, 4 is common and 25 + 75 = 100

Example: 32 X 38
32 X 38
= 3 x 4 / 2 x 8
= 12 /16
= 1216
  • 1.      Sum of last digits is 2+8 =10,
  • 2.      Remaining digits =3 are same in both numbers.
  • 3.      RHS 2x8 =16,
  • 4.      LHS 3 x (3+1) =12

Example: 83 X 87
83 X 87
= 8 x 9 / 3 x 7
= 72 /21
= 7221
  • 1.      चरमं अंकों का योग = 3+7 =10,
  • 2.      शेष निखिलम् अंक समान =8
  • 3.      दायाँ पक्ष = 3 x 7 =21,
  • 4.      बायाँ पक्ष = 8 x (8+1) =72


Lopana Sthapanabhyam (लोपनस्थापनाभ्याम्)
The sutra means 'by alternate elimination and retention’.
विलोपन तथा स्थापना से।
This sutra is used to factorizing a quadratic equation of type ax2+by2+cz2+dxy+eyz+fzx. It is a homogeneous equation of second degree with three variables x, y, z.
इस सूत्र का उपयोग द्विघात समीकरण के गुणनखण्ड़ करने में किया जाता हैं।
Example: 3a2 + 6b2+ c2 +11ab + 5bc + 4ac
  • 1.    सर्वप्रथम c=0; रखकर c को विलोपित करते हैं, तथा आद्यमाद्येन अन्त्यमन्त्येन सूत्र की सहायता से गुणनखण्ड़ करते हैं।
3a2 +11ab + 6b2
= 3a2 +9ab +2ab + 6b2
= (3a + 2b) (a + 3b)      
  • 2.    अब b=0; रखकर b को विलोपित करते हैं, तथा आद्यमाद्येन अन्त्यमन्त्येन सूत्र की सहायता से गुणनखण्ड़ करते हैं।
3a2 + 4ac + c2
= 3a2 +3ac + ac + c2
= (3a + c) (a + c)   
  • 3.    इन दो गुणनखण्ड़ समूह की सहायता से विलोपन के कारण आई रिक्तियों की पूर्ति करते हैं।
= (3a + 2b +c) (a + 3b +c)

Example: 2a2 + 6b2+ c2 +7ab + 5bc + 3ac
  • 1.      Eliminate c and retain a, b; factorize
2a2 + 7ab + 6b2 = (2a + 3b) (a + 2b)
  • 2.      Eliminate b and retain a, c; factorize
2a2 + 3ac + c2 = (2a + c) (a + c)
  • 3.      Fill the gaps, the given expression
= (2a + 3b + c) (a + 2b + c)


Gunita Samuccayah - Samuccaya Gunitah (गुणितसमुच्चयः समुच्चयगुणितः)
The product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product (i.e., the product of the sum is the sum of the products).
गुणनखण्ड़ों की गुणन संख्याओं के योग का गुणनफल, गुणनफल की गुणन संख्याओं के योग के समान होता हैं।
This sutra is useful for the factorization of quadratic expressions.
Example: (2a + 1) (3a + 5) = 6a2 + 13a + 5
Here:  (2 + 1) (3 + 5) = (6 + 13 + 5)
= 24: Thus verified.
Example: (x + 5) (x + 7) (x - 2) = x3 + 10x2 + 11x – 70
(1 + 5) (1 + 7) (1 - 2) = 1 + 10 + 11 – 70
6 x 8 x -1 = 22 – 70
– 48 = – 48 Verified


Paravartya-yojayet (परावर्त्य योजयेत्)
‘Paravartya – Yojayet’ means 'transpose and apply'
चिन्ह परिवर्तित कीजिये तथा संक्रिया प्रारम्भ कीजिये।

Simple division of algebra
Example: Divide (a3 – 3a2 + 10a – 4) by (a – 5)

  • 1.    (a3 / a) gives a2, 1 the first coefficient in the Quotient.
  • 2.    Multiply 1 by + 5, (obtained after reversing the sign of second term in the Quotient) and add to the next coefficient in the dividend. It gives 1 X (+5) = +5, adding to the next coefficient, i.e., –3 + 5 = 2, this is second coefficient in Quotient.
  • 3.    Multiply 2 by +5, i.e., 2 X +5 =10, add to the next coefficient 10 + 10 = 20. This is third coefficient in Quotient.
  • 4.     Thus Quotient is a2 + 2a + 20
  • 5.    Now multiply 20 by + 5 =100. Add to the next (last) term, 100 + (-4) = 96, which becomes R, i.e., R =96.

Example: Divide (2a5 + a3 – 3a + 7) by (a3 + 2a – 3)
We treat the dividend as (2a5 + 0a4 + 1a3+ 0a2 – 3a + 7) and divisor as (a3 + 0a2 + 2a - 3).

क्रियाविधिः
  • 1.    भाजक के प्रथम पद को छोड़कर शेष पदों के गुणकों के चिन्ह बदलकर परावर्तित अंक प्राप्त करते हैं, जो क्रमशः 0, -2, +3 हैं।
  • 2.    2a5 में a3 का भाग देने पर 2a2 आता हैं, अतः भागफल का प्रथम गुणक =2;
  • 3.    भागफल का प्रथम गुणक अंक (2) X परावर्तित अंक (0) = 0, मध्य खण्ड़ के +0a4 के नीचे 0 लिखते हैं, (2) X (-2) = -4, मध्य खण्ड़ के +1a3 के नीचे -4 लिखते हैं, तथा (2) X (+3) = +6, तृतीय खण्ड़ के +0a2 के नीचे +6 लिखते हैं।
  • 4.    भाजक के द्वितीय पद का योग = 0 + 0 = 0, अतः भागफल का दूसरा गुणक = 0;
  • 5.    भागफल का दूसरा गुणक अंक (0) X परावर्तित अंक  (0) = 0, मध्य खण्ड़ के +1a3 के नीचे 0 लिखते हैं, (0) X (-2) = 0, तृतीय खण्ड़ के 0a2 के नीचे 0 लिखते हैं, तथा (0) X (+3) = 0, तृतीय खणड़ के -3a के नीचे 0 लिखते हैं।
  • 6.    योग = 1 - 4 + 0 = -3, अतः भागफल का तीसरा गुणक = -3;
  • 7.    भागफल का तीसरा गुणक अंक (-3) X परावर्तित अंक  (0) = 0, तृतीय खण्ड़ के 0a2 के नीचे 0 लिखते हैं, (-3) X (-2) = +6, तृतीय खण्ड़ के -3a के नीचे +6 लिखते हैं, तथा (-3) X (-3) = +9, तृतीय खणड़ के +7  के नीचे +9 लिखते हैं।
  • 8.    अतः भागफल = 2a2 - 3; शेषफल = + 6a2 + 3a -2

Paravartya in solving simple equations:
'Paravartya yojayet' means 'transpose and apply'. According to the rule invariable change its sign with every change of side from left to right, (+) becomes (-) and; and (X) becomes (÷). Further it can be extended from numerator to denominator in the concerned problems.
प्रत्येक पक्षान्तरण में गणितीय राशियों के चिन्ह परिवर्ति‍त होते हैं। इस प्रकार (+) चिन्ह (-) हो जाता हैं व (-) चिन्ह (+) हो जाता हैं, (x) का (÷) (÷) का (x) हो जाता हैं।

Application 1: If ax + b = cx + d.
By paravartya, we get-


Example: 4x + 3 = 2x + 9
Here a =4, b = 3, c = 2, d = 9


Application 2: If (x + a) (x +b) = (x +c) (x +d).
By paravartya, we get -


Example: (x + 7) (x + 9) = (x - 8) (x - 11).
Here a =7, b = 9, c = - 8, d = -11


Application 3: If

By paravartya, we get-


Example:


Application (4): If

By paravartya we get-


Example:


Application (5): If


Example:


Application (6): If


Example:


Simple equations:
By Paravartya sutra we can derive the values of x and y. which are given in two simple equations.

Example:
2x + 3y = 13,
4x + 5y = 23.
  • 1.      x का मान ज्ञात करने के लिए दोनों समीकरण के y के गुणक तथा अचर राशियों का बज्र गुणा करते हैं, तथा बज्र गुणा से प्राप्त राशियों को घटाते हैं, प्राप्त संख्या x के लिए अंश के रुप में प्रयुक्त होती हैं।
2x + 3y = 13
4x + 5y = 23
“X” के लिए अंश
= 3 x 23 – 5 x 13
= 69 – 65 = 4
  • 2.      दोनों समीकरणों के x तथा y के गुणक का बज्र गुणा कर घटाने पर प्राप्त संख्या x के लिए हर के रुप में प्रयुक्त होती हैं।
“X” के लिए हर
= (3 x 4)(2 x 5)
= 12 – 10 = 2       
अतः X = 4 ÷ 2 = 2
  • 3.      y का मान ज्ञात करने के लिए दोनों समीकरण के x के गुणक तथा अचर राशियों का बज्र गुणा करते हैं, तथा बज्र गुणा से प्राप्त राशियों को घटाते हैं, प्राप्त संख्या y के लिए अंश के रुप में प्रयुक्त होती हैं।
“Y” के लिए अंश
= (13 x 4)(23 x 2)
= 52 – 46 = 6
  • 4.      y के लिए हर = 2; जो पद 2 से प्राप्त हुआ।
अतः y = 6÷2 = 3
अतः समीकरण में, x = 2 तथा y = 3

Sunyam Samyasamuccaye (शून्यं साम्यसमुच्चये)
The Sutra 'Sunyam Samyasamuccaye' means 'Samuccaya is the same, that Samuccaya is Zero.' The term 'Samuccaya' has several meanings under different contexts.
'जब समुच्चय एक समान हो तो उस समुच्चय का मान शून्य होता हैं'। भिन्न-भिन्न परिस्थितियों में समुच्चय के अर्थ भिन्न-भिन्न होते हैं।

Situation 1: “Samuccaya” as a term which occurs as a common factor in all the terms concerned and proceed as follows.
यदि समीकरण के प्रत्येक पद में x एक सर्वनिष्ट खण्ड़ हो तो x = 0 होगा।

Example: The equation 12x + 3x = 4x + 5x has the same factor “x” in all its terms. Hence according to the sutra it is zero.
12x + 3x = 4x + 5x
x = 0

Example: In 2(x+1) = 7(x+1)
(x + 1) is Common Samuccaya
Hence   (x + 1) = 0
x = -1

Situation 2: If the product of independent terms in a equations like (x+a) (x+b) = (x+c) (x+d), is same then x = 0;
यदि समीकरण के दोनों पक्षों में अचर राशियों रहित (स्वतंत्र) पद समान हो तो x का मान शून्य  (0) होगा।

Example: (x + 3) (2x + 5) = -3(x - 5)
Samuccaya is 3 x 5 = 15 = -3 x -5
Since it is same, Hence x = 0

Situation 3: “Samuccaya” as the sum of the denominators of two fractions having the same numerical numerator.
यदि समीकरण में दो भिन्नों के अंश परस्पर समान हो तो उनके हरों का योग शून्य रखने पर चर राशि का मान प्राप्त होता हैं।
Example:

Numerator are same = j;
Hence according to sutra, sum of the denominators is zero.
(2x + 1 + 3x + 4) = 0
= (5x + 5) = 0
5x = -5
x = -1
Situation 4: If the sum of the numerators and the sum of the denominators are same, then that sum = 0.
यदि समीकरण में दोनों पक्षों के अंशों तथा हरों का योग परस्पर समान या एक निश्चित अनुपात में हो तो उनके अंशों तथा हरों का योग शून्य रखने पर चर राशि का मान प्राप्त होता हैं।

Example:

Sum of numerator = 2x+3+2x+5 = 4x + 8
Sum of denominator =2x+5+2x+3 = 4x + 8
Hence according to sutra, sum is zero
(4x + 8) = 0
4x = -8
x = -2

Example:

दोनों पक्षों के अंशों का योग = 3x+4+x+1 = 4x+5
दोनों पक्षों के हरों का योग = 6x+7+2x+3 = 8x+10
दोनों पक्षों का अनुपात = 1 : 2
सूत्र के अनुसार
(4x +5) = 0
4x = -5
x = -5/4

Situation 5: If the differences of the numerators and denominators of each side are same, then that difference = 0.
यदि समीकरण में एक पक्ष के अंश तथा हर का अन्तर दूसरे पक्ष के अंश तथा हर के अन्तर के समान या एक निश्चित अनुपात में हो तो किसी भी अन्तर का मान शून्य रखने पर चर राशि का मान प्राप्त होता हैं।

Example:

बायें पक्ष के अंश व हर का अन्तर = (3x+6) – (6x+3)
= - 3x + 3 = - (3x-3)
दायें पक्ष के अंश व हर का अन्तर = (5x+4) – (2x+7)
= 3x - 3
सूत्र के अनुसार
3x - 3 = 0
3x = 3
x = 1

Situation 6: ‘Samuccaya’ with the same sense but in a different context and application.
यदि समीकरण के प्रत्येक पक्ष में दो पद हो और पद का प्रत्येक अंश परस्पर समान हो तथा दोनों पक्षों के हरों का योग समान हो तो योग को शून्य रखने पर चर राशि का मान प्राप्त होता हैं।
Example:

बायें पक्ष के हरों का योग = x+2+x+6 = 2x+8
दायें पक्ष के हरों का योग = x+1+x+7 = 2x+8
सूत्र के अनूसार
(2x + 8) = 0
2x = -8
x = -4

Sunyam Samyasamuccaye in Cubes:

Example: (x – 6)3 + (x – 8)3 = 2 (x – 7)3
Traditional method,
(x – 6)3 + (x – 8)3 = 2 (x – 7)3
x3 – 18x2 + 108x – 216 + x3 – 24x2 + 192x – 512
= 2 (x3 – 21x2 + 147x – 343)
2x3 – 42x2 + 300x – 728 = 2x3 – 42x2 + 294x – 686
300x – 728 = 294x – 686
300x – 194x = 728 – 686
6x = 42
x = 42 / 6 = 7
Vedic method,
We have (x – 6) + (x – 8) = 2x – 14. Taking out the numerical factor 2, we have (x – 7) = 0
According to “Sunyam Samyasamuccaye” (x – 7) = 0.
Hence x = 7


Anurupye– Shunyamanyat (आनुरूप्ये शून्यमन्यत्)
The Sutra means: 'If one is in ratio, the other one is zero'.
अनुरूपता होने पर दूसरा शून्य होता हैं।
This Sutra in used to solve simultaneous simple equations in which the coefficients of 'one' variable are in the same ratio to each other as the independent terms are to each other. In such a case the Sutra says the 'other' variable is zero.
यदि किसी युगपत् समीकरण के किसी एक चर का अनुपात अचर राशियों के अनुपात के समान हो तो दूसरा चर शून्य होगा।

Example:
3x + 4y = 1 
4x + 12y = 3
The ratio of y-coefficients is 4:12 = 1:3, which is same as the ratio of independent terms is = 1:3.
Hence the other variable x = 0
4y = 1 or 12y = 3
y = ¼

Example:
175x + 140y = 350 
350x + 324y = 700
X के गुणकों का अनुपात = 175:350 =1:2 तथा
अचर पदों का अनुपात =350:700 = 1:2 समान हैं,
अतः y = 0
(175x =350) or (350x =700)
x = 2


Puranapuranabhyam (पूरणापूरणाभ्याम्)
The Sutra can be taken as Puranapuranabhyam which means by the completion or non - completion. We use it to solve the roots of quadratic equation.
अपूर्ण को पूर्ण करके।
इस सूत्र का प्रयोग द्विघात तथा त्रिघात समीकरणों को हल करने में किया जाता हैं।
ax2 + bx + c = 0
x2 + (b/a)x + c/a  =  0    ( dividing by a )
x2 + (b/a)x = - c/a
Completing the square (purana) on L.H.S
[x2 + (b/a)x + (b2/4a2)] = -c/a + (b2/4a2)
[x + (b/2a)] 2 = (b2 - 4ac) / 4a2

Example: x3 + 6x2 + 11 x + 6 = 0.
क्योंकि (x + 2) 3 = x3 + 6x2 + 12x + 8
दोनों पक्षों में (x + 2) जोड़ते हैं।
x3 + 6x2 + 11x + 6 + (x + 2) = x + 2
x3 + 6x2 + 12x + 8 = x + 2
(x + 2) 3 = (x + 2)
a3 = a                               for (a = x + 2)
a = 0, a = 1, a = - 1
x + 2 = 0, 1,-1
x = -2,-1,-3


Chalana - Kalanabhyam (चलन-कलनाभ्याम्)
The Sutra means 'Sequential motion' or “By calculus”.
चलन कलन के द्वारा।

Application 1: It is used to find the roots of a quadratic equation
(x2 – 3x + 1) = 0.
Now by calculus formula:

2x–3 = ±√5
x = 3±√5 / 2
Every Quadratic can thus be broken down into two binomial factors.

Application 2: Gunak samuccaye: यदि द्विघात समीकरण ax2 + bx + c, किन्ही दो पदों का गुणनफल हैं, तो इसकी प्रथम अवकलन गुणन संख्या दोनों गुणनखण्ड़ों का योग होती हैं।             
ax2 + bx +c
= (x + d) (x +e)
By calculus formula:

Sutra says 2ax + b = (x + d) + (x +e)
Example:
x2 + 5x + 4 =(x+4) (x+1)
(2x + 5) =(x+4) + (x+1)

Ekanyunena Purvena (एकन्यूनेन पूर्वेण)
This Sutra is a Sub-sutra to Nikhilam which means 'by one less than the previous one’.
पहले से एक कम के द्वारा।
This sutra is used to multiply a number by 9, 99, 999...
दो गुणन संख्याओं में जब एक संख्या के सभी अंक 9 हो तो एकन्यूनेन पूर्वेण विधि द्वारा गुणा किया जाता हैं। जिस संख्या के सभी अंक 9 हो उसे गुणक तथा दूसरी संख्या को गुण्य कहते हैं।

Example: 11 x 99
11 X 99
= 11-1 / 99-10
= 10 / 89
= 1089
  • 1.      बायाँ पक्ष= 11 - 1 = 10
  • 2.      दायाँ पक्ष = 99-10 = 89

Example: 125 X 9
  • 1.      Divide the multiplicand (125) of by a Vertical line or by the Sign (:) put as many digits as the multiplier into a right hand portion.
125 has to be written as 12/5 or 12:5

  • 2.      Subtract from the multiplicand one more than the whole excess portion on the left.
Left portion of multiplicand is 12. One more than it 12 + 1 = 13,
Now subtract this from multiplicand,         

  • 3.      Subtract the R.H.S. part of the multiplicand by nikhilam process.
R.H.S of multiplicand is 5 its nikhilam is 5.
It gives the R.H.S of the product
Answer is 11: 2: 5 = 1125



Antyayoreva (अन्त्ययोरेव)
'Antyayoreva' means 'only the last terms'.
अन्तिम पद से ही।
This is useful in solving simple equations. The type of equations are those whose numerator and denominator on the L.H.S. leaving the independent terms stand in the same ratio to each other as the entire numerator and the entire denominator of the R.H.S. stand to each other.
यह सूत्र उन समीकरणों को सरल करने में प्रयुक्त होता हैं, जिनके बायें पक्ष में अन्तिम पदों ( स्वतंत्र पदों) को छोड़कर अंश तथा हर का अनुपात वही होता है, जो दायें पक्ष के पूरे अंश तथा हर का होता हैं। इसे अन्त्ययोरेव अर्थात् अन्तिम पदों के अनुपात द्वारा आसानी से हल किया जा सकता हैं।
Example:

Example: (a + 2) (a + 3) (a + 11) = (a + 4) (a + 5) (a + 7)
Sum of the binomials on both the L.H.S. and R.H.S = 3a + 16 are same. Hence antyayoreva can be applied. Adjusting we get



Vilokanam (विलोकनम्)
The Sutra 'Vilokanam' means 'Observation'.
देखकर।

Example: x + (1/x) = 10/3
उक्त समीकरण में विलोकनम् के अनुसार बायाँ पक्ष दो व्युत्क्रमों (x तथा 1/x) का योग हैं। साथ ही दायाँ पक्ष भी दो व्युत्क्रमों (3 तथा 1/3) का योग हैं।
अतः x = 3, 1/3.

Example:


Simultaneous Quadratic Equations:

Example: 5x – y = 7 and xy = 6.
xy = 6 gives
x = 6, y = 1;
x = 1, y = 6;
x = 2, y = 3;
x = 3, y = 2 and of course negatives of all these.
Observe that
For x = 6, y = 1; 5x – y = 5 (6) – 1 = 30 – 1 ≠ 7.
For x = 1, y = 6; 5x – y = 5 (1) – 6 = 5 – 6 ≠ 7.
For x = 3, y = 2; 5x – y = 5 (3) – 2 = 15 – 2 ≠ 7.
These set are not solutions because they do not satisfy the equation 5x – y = 7.
But for x = 2, y = 3; 5x – y = 5 (2) – 3 = 10 – 3 = 7
Hence x = 2, y = 3 is a solution.
Negative values of the above are also not the solutions. Thus one set of the solutions is x = 2, y = 3.

Partial Fractions:

Example:

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