The Later Sulbha Sutras
These include first 'use' of irrational numbers, quadratic equations of the form a x2 = c and ax2 + bx = c, unarguable evidence of the use of Pythagoras theorem and Pythagorean triples, predating Pythagoras (c 572 - 497 BC), and evidence of a number of geometrical proofs. This is of great interest as proof is a concept thought to be completely lacking in Indian mathematics.
Example: Pythagoras theorem and Pythagorean triples, as found in the Sulba Sutras.
The rope stretched along the length of the diagonal of a rectangle makes an area which the, vertical and horizontal sides make together.
In other words:
& nbsp; a2 = b2 + c2
Examples of Pythagorean triples given as the sides of right angled triangles:
5, 12, 13
8, 15, 17
12, 16, 20
12, 35, 37
Of the Sulvas so far 'uncovered' the four major and most mathematically significant are those composed by Baudhayana, Manava, Apastamba and Katyayana (perhaps least 'important' of the Sutras, by the time it was composed the Vedic religion was becoming less predominant). However in a paper written 20 years ago S Sinha claims that there are a further three Sutras, 'composed' by Maitrayana, Varaha and Vadhula (SS1, P 76). I have yet to come across any other references to these three 'extra' sutras. These men were not mathematicians in the modern sense but they are significant none the less in that they were the first mentioned 'individual' composers. E Robertson and J O'Connor have suggested that they were Vedic priests (and skilled craftsmen).
It is thought that the Sulvas were intended to supplement the Kalpa (the sixth Vedanga), and their primary content remained instructions for the construction of sacrificial altars. The name Sulvasutra means 'rule of chords' which is another name for geometry.
This is not particularly compelling evidence but does suggest that the composers of the sulba-sutras may have had a greater depth of knowledge than is generally thought.
Many suggestions for the value of p are found within the sutras. They cover a surprisingly wide range of values, from 2.99 to 3.2022.
Pythagoras's theorem and Pythagorean triples arose as the result of geometric rules. It is first found in the Baudhayana sutra - so was hence known from around 800 BC. It is also implied in the later work of Apastamba, and Pythagorean triples are found in his rules for altar construction. Altar construction also led to the discovery of irrational numbers, a remarkable estimation of sqrt2 in found in three of the sutras. The method for approximating the value of sqrt2 gives the following result:
sqrt2 = 1 + 1/3 + 1/3.4 - 1/3.4.34
This is equal to 1.412156..., which is correct to 5 decimal places.
It has been argued by scholars seemingly attempting to deprive Indian mathematics of due credit, that Indians believed that sqrt2 = 1 + 1/3 + 1/3.4 - 1/3.4.34 exactly, which would not indicate knowledge of the concept of irrationality. Elsewhere in Indian works however it is stated that various square root values cannot be exactly determined, which strongly suggests an initial knowledge of irrationality.
Indeed an early method for calculating square roots can be found in some Sutras, the method involves repeated application of the formula: sqrtA = sqrt(a2 + r) = a + r/2a, r being small.
Example: Application of formula for calculating square roots.
If A=10, take a = 9 and r = 1.
Thus sqrt10 = sqrt(32 + 1) = a + r/2a = 3 + 1/6 = 3.16667 in (modern) decimal notation.
sqrt10 = 3.162278 to six decimal places when calculated on a calculator. Thus, after only one application of the formula, a moderately accurate value has been calculated.
Many of the Vedic contributions to mathematics have been neglected or worse. When it first became apparent that there was geometry contained within works that was not of Greek origin, historians and mathematical commentators went to great lengths to try and claim that this geometry was Greek influenced (to a greater or lesser extent).
It is undeniable that none of the methods of Greek geometry are discernible in Vedic geometry, but this merely serves to support arguments that it is independently developed and not in some way borrowed from Greek sources.
In light of recent evidence and more accurate dating it has been even more strongly claimed by A Seidenberg (in S Kak) that:
...Indian geometry and mathematics pre-dates Babylonian and Greek mathematics. [SK1, P 338]
This may be a somewhat extreme standpoint, and it seems likely that there was traffic of ideas in all directions of the Ancient world, but there is little doubt that the vast majority of Indian work is original to its writers. It may lack the cold logic and truly abstract character of modern mathematics but this observation further helps to identify it as uniquely Indian. Of all the mathematics contained in the Vedangas it is the definite appearance of decimal symbols for numerals and a place value system that should perhaps be considered the most phenomenal.
Before the period of the Sulbasutras was at an end the Brahmi numerals had definitely begun to appear (c. 300BC) and the similarity with modern day numerals is clear to see (see Figures 7.1 and 7.3). More importantly even still was the development of the concept of decimal place value. M Pandit in a recent paper (discussed in RG2) has shown certain rules given by the famous Indian grammarian Panini (c. 500 BC) imply the concept of the mathematical zero. Further to this there is a small amount of evidence of the use of symbols for numbers even earlier in the Harrapan culture. My evidence comes primarily from a paper by S Kak, which analyses some of Panini's work, and there is further support from a paper by S Sinha. B Datta and A Singh also give evidence of an early emergence of numerical forms and the decimal place value system.
Home Schooling Your Child In Vedic Maths
If you enjoyed maths at school - or since leaving school - you will probably already use mathematical language comfortably with your children and find that they understand basic concepts such as the counting numbers and simple fractions at a young age. Unfortunately, a large section of the population found school maths so confusing - or even distressing - that they avoid anything remotely mathematical. Fear of maths can put parents off the whole idea of home education. Yet parents who know little about history or geography don't find this off-putting, as they usually expect to learn from books as they go along. Parents who are unmusical, or don't know any foreign languages have few worries about educating their children themselves. But fear of maths is somehow over-riding, and becomes almost irrational.
If this describes you, your fear may be passed on to your children even if they are in school. If they see you looking in horror at their maths homework, they're unlikely to be inspired and confident! So whether or not you're considering home education for your children, it's worth re-thinking your whole attitude to maths.
What is maths anyway? It's the underlying structure of the world, which we see in patterns, shapes, quantities and intelligent guesses. Why do we need it? We need to understand the concepts of numbers and quantities when we bake cakes, or decorate a room. We see patterns in art and music. Businesses need to make intelligent guesses (or ‘estimates') of how much something is going to cost, how many people need their products, how fast it is going to sell. We need to keep track of our bank accounts, and ensure we do not spend more than our income.
Unfortunately the modern system of arithmetic does not stress on learning mathematics by pattern recognition and hence it becomes uninteresting and painful to children.
So How would you like if suddenly all your math worries are taken care of and your children not only adore math but they sleep, eat and breathe math? Wouldn't it be amazing if such a system exists and they be able to master and learn it easily?
The solution to all the worries and boredom lies in ancient Indian scriptures called Vedas. Well you must be wondering What on earth has ancient Indian scriptures got to do with your high school kid struggling with math? A lot if you read carefully the next few lines.
The ancient Indian scriptures called Vedas are a storehouse of knowledge of every field including mathematics and science. Through this System of Mathematics every mathematical problem no matter how difficult can be solved faster and easily without much effort. The system is called High Speed Vedic Mathematics and this is the secret to the distinguished edge of the Indians in Mathematics over the centuries. It is a worldwide fact that the zero was invented by the Indians and the numbers we use today form a part of the Hindu System of Numbers.
A big part of this method is learning the recognition of patterns in numbers , letters or pictures. Once the student has learnt efficiently how to find symmetry and pattern of numbers or objects, learning mathematics becomes fun and simple! It arouses interest in mathematics and the children suddenly want to discover more and more of this beautiful method cause it gives them speed and accuracy which they desire. Vedic Maths is a zero-error technique and this take care of the very root of the phobia of mathematics.You may want to home school your child into Vedic Mathematics and get him the edge which he deserves. Any teenager can learn Vedic Mathematics and enhance his numerical abilities. It helps the students not only inside the class room or traditional academics, but it also enhances the IQ by enhancing the analytical skills and thought processes. This happens cause the system is very coherent and intuitive and uses both sides of the brain thereby giving the student the winning edge.
Some interesting notes on Zero Evolution.
India: 458 A.D. (debated)
The final independent invention of the zero was in India. However, the time and the independence of this invention has been debated. Some say that Babylonian astronomy, with its zero, was passed on to Hindu astronomers but there is no absolute proof of this, so most scholars give the Hindus credit for coming up with zero on their own.
The reason the date of the Hindu zero is in question is because of how it came to be.
Most existing ancient Indian mathematical texts are really copies that are at most a few hundred years old. And these copies are copies of copies of copies passed through the ages. But the transcriptions are error free...can you imagine copying a math book without making any errors? Were the Hindus very good proofreaders? They had a trick.
Math problems were written in verse and could be easily memorised, chanted, or sung. Each word in the verse corresponded to a number. For example,
viya dambar akasasa sunya yama rama veda
sky (0) atmosphere (0) space (0) void (0) primordial couple (2) Rama (3) Veda (4)
0 0 0 0 2 3 4
Indian place notation moved from left to right with ones place coming first. So the phrase above translates to 4,230,000.
Using a vocabulary of symbolic words to note zero is known from the 458 AD cosmology text Lokavibhaga. But as a more traditional numeral--a dot or an open circle--there is no record until 628, though it is recorded as if well-understood at that time so it's likely zero as a symbol was used before 628.
Which it probably was, considering that 30 years previously, an inscription of a date using a zero symbol in the Hindu manner was made in Cambodia.
A striking note about the Hindu zero is that, unlike the Babylonian and Mayan zero, the Hindu zero symbol came to be understood as meaning "nothing." This is probably because of the use of number words that preceded the symbolic zero.
Vedic Maths: Mental Maths?
Numbers have fascinated man since their inception. They demand very fine sense of discrimination, alertness, and power of reasoning. There is no doubt in the fact that dealing with numbers calls for the best the human mind can give. The numbers have become inseparable part of our curriculum and our daily life.
Mental Arithmetic is a form of training which deals with carrying out arithmetic operations with out the use of paper/pencil or even calculator. There are Chinese method of mental arithmetic which is based on 'Abacus' i.e. a Chinese calculator, consisting of beads. The concept of mental arithmetic based on Abacus is useful for children as it is easy for them in the brain development stage to practice on Abacus and after some time they don't think about numbers but they convert numbers in to Abacus beads and perform mental operations and again convert beads back to number and give answer with amazing speed and accuracy. However this mental arithmetic has been found effective for the children of the age group of 6 to 13 years only. For students who have surpassed this age bracket Abacus Mental arithmetic is not very useful.
in most of the competitive examinations conducted by IIT's, IIM's, other engineering and management course entrance examinations, the ability of candidate with familiarity with numbers and fundamentals is tested against time. It has been observed that most of the students who are selected for these prestigious competitions have very keen understanding of number, which traditionally comes after a lot of practice and memorizing tables, squares, cubes of numbers up to two or three digits. This is a very tedious and time consuming task, a student who is weak in mathematics is either ill prepared in this subject or his/her other subjects get ignored.
Vedic method of Mental arithmetic is a well researched subject, by His Holiness Jagadguru Shankaracharaya. Though the subject is quite vast and universal in nature, here we teach the techniques of Vedic mathematics which are simple and do not involve memorizing tables beyond 9. Arithmetic problems, which usually take 15 to 20 steps to solve, can be solved in a few lines. The course is quite useful for the students in competitive examinations, because it gives them powerful tool of analysis, logic, speed and accuracy in solving problems.
VEDIC MATHEMETICS BASED MENTAL MATHS. Vedic Mathematics is based on simplification of all arithmetic operations through some highly researched Sutras. The aim of Vedic Mathematics is to ensure that student can carry out any type of arithmetic operations with much ease and accuracy and speed. However for understanding Mental Arithmetic based on Vedic Mathematics, the age group of student is 13 years upwards. To understand Mental Arithmetic based on Vedic Mathematics Students trained to understand and quickly select the most efficient technique to solve the problem in the least time. This technique has following advantages: -
Very high speed of calculations
Analytical faculty of student becomes vary sharp
The decision making skill are developed
A natural flair for maths will develop
A tremendous sense of self confidence is developed in the student
This technique is very useful for competitive examinations i.e. IIT's, IIM's, IAS and other competitive exams where speed and accuracy are the prime requirements. With adequate practise through extensive exposure in the class room and regular 25 minutes practise every day the student master the scientific art of mastering numbers.
contemplating infinity philosophically
This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.
We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.
Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.
It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.
Vedic mathematical Research done by Dr. S. K. Kapoor
The organisational format of Srimad Bhagwad Gita emerges to be of prime importance as it is parallel to the organising power of the knowledge content of the scripture. This format tallies with the format of human body on the one hand and the sun on the other hand as of real 6-space. The Trinity of Gods namely Brahma, Mahesh, Vishnu are the overlords of real 4, 5 & 6 spaces respectively. The geometrical continuum expressed as manifestation layers of 4 folds of consecutive dimensional spaces contents. These folds of the nth manifestation layer can be represented as under:
| First Fold | Second Fold | Third Fold | Fourth Fold |
| Dimension | Frame | Domain | Origin |
| (n-2) space content | (n-1) space content | n-space content | (n+1) space content |
The transcendence from one manifestation layer to another manifestation layer giving rise to the following (five steps) chain reaction or five steps which are possible within the setup of Panch Mahabhut.
| Manifestation Layer | Dimension | Frame | Domain | Origin |
| nth | Space fold (n-2) | Space fold (n-1) | Space fold n | Space fold (n+1) |
| (n+1)th | Space fold (n-1) | Space fold n | Space fold (n+1) | Space fold (n+2) |
Transition from one space to another space is to be had in terms of unlocking of the seals of the origin points of all the four folds of the manifestation. The modern mathematical models of transition from straight line to plane deserve serious reexamination. In particular the axioms of space filling curves and the axioms of ‘one’ without a predecessor deserve close scrutiny as their rationale emerges to be without basis.
The role of real numbers additive group (R,+) and real numbers field (R,+,×) with reference to straight line deserve to be differentiated.
The plane deserves to be studied as four geometrically distinct quarters. One faced plane and two faced plane are two distinct geometrical setups and they deserve to be taken up as such.
The concepts of origin and dimension are two concepts with respect to which the modern geometrical models are not up to date. These two concepts deserves to be studied in detail as transcendence to the higher dimensional spaces is possible only in terms of their understanding.
Human body is a compactified phenomenon of multi-layer physiological existence. The start with state of existence is that of waking state which is parallel to the expression of 1-space as dimension into 3-space domain. Sequentially, the existence phenomenon unfolds until seventh state of consciousness which would be corresponding to the 7-space as dimension into 9 space domain. The origin point of the 6-space, being the 7-space setup, the human body, geometrically, turns out to be hypercube-6 and this would explain how the primordial sound, the planetary effects, the Yajna oblations etc. operate and precisely influence the individual existence patterns.
Srimad Bhagwad Gita is one such scripture whose organisational format precisely workout for us the structural set up and frames of the 6-space. The study zone of Srimad Bhagad Gita can be worked as under:
The organisational setups of Ganita Sutras, Maheshwara Sutras, Saraswati Mantras, Gyatri Mantra and Om formulation deserve interdisciplinary explorations.
Sankhay Nistntha and Yoga Nistha are complementary and supplementary of each other and as such their complementary nature and supplementary nature deserve to be distinguished well. Non-differentiation of the same is bound to deprive us of most of the results in specific forms.
Vedic Maths FAQ's: The Frequently Asked Questions
Q1.What is Vedic Mathematics?
A1. Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas.
Q2. Who discovered these Vedic Methods and from where?
A2. This ancient system of Mathematics was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krisna Tirthaji.
Q3. What is the most significant feature of the Vedic Mathematics?
A3. The most striking feature of the Vedic system is its coherence. Instead of a collection of unrelated techniques the whole system is beautifully interrelated and unified. These related techniques are all easily understood. This unifying quality is very fulfilling, and makes mathematics easy and enjoyable and encourages innovation.
Q4. What is the single most basic advantage of teaching Vedic Methods to students?
A4. In the Vedic system 'difficult' problems or large, complicated sums can often be solved immediately by using simple Vedic methods. These striking methods are just a part of a complete system of mathematics which is far more systematic than the modern 'system'. Vedic Mathematics has a coherent and unified structure and the methods are complementary, direct and easy.There are many advantages in using a flexible, mental system. Students can invent their own methods; they are not limited to the one 'correct' method. This leads to more creativity.
Q5. What kinds of Arithmetic problems find application in Vedic Maths?
A5. Interest in the Vedic system is growing in education where mathematics teachers are looking for something better and finding the Vedic system is the answer. Initially, students learn the Vedic methods of doing Multiplication, Division, finding Squares, Square Roots and Cube Roots. Students who have learnt to apply Vedic Methods in his / her day to day calculation for a minimum duration of 90 days can be taught further applications of Vedic Methods, such as derivatives of the basic formulae and there after; its applications in Algebra, Geometry, Trigonometry and Calculus. Yes, Vedic Methods find applications, in all branches of Mathematics.
Q6. On what principles is Vedic Maths based?
A6. All of Vedic mathematics is based on sixteen Sutras, or word-formulae. For example, 'Vertically and Crosswise’ is one of these Sutras. These formulae describe the way the mind naturally works and therefore help in directing the student towards the appropriate method of solving a problem.
Q7. Does one need to memorize numbers and other complicated computations, in Vedic Maths?
A7. There is no need for students to memorize tables up to 20 times 20 or remember all the prime numbers up to 100, and such other facts. Vedic mathematics has its own set of systematic and scientific principles which when applied make the computation very simple and easy.The simplicity of Vedic Mathematics means that calculations can be carried out mentally.
Q8. How relevant is Vedic Maths in the current school curriculum?
A8. By using Vedic methods students comes to know the answer in few seconds. They find the principles of the Vedic system systematic, simple and methodical. Research is being carried out in many areas including the effects of learning Vedic Maths on children; developing new, powerful yet easy applications of the Vedic Sutras in geometry, calculus, computing etc. It is perhaps, the most refined and efficient mathematical system possible.
Q9. The students have been used to learning a different method of solving the arithmetic problems in school. Won’t the Vedic methods be confusing, if they are taught to them?
A9. No, absolutely not. By using Vedic methods students will be able to solve the problems promptly and will be able to apply appropriate Vedic methods to solve different problems. Once they have been able to understand the application of the various ‘sutras’ or principles, they themselves will want to apply them in their day to day work, as they will find these methods simpler and faster as compared to standard computation and problem solving methods, in use currently.
Q10. With devices like calculators currently being used widely, why should there be any need for learning Vedic methods?
A10. A calculator cannot think for students. They must understand the basic principles involved in the calculations. Some times a person gets varied answers while using a calculator. For example while solving a problem like: “Two plus three times four equals…..?” By using a calculator a student may get “20” as an answer, others may get an answer of “14”. Therefore simple computational tools are not enough. An understanding of the problem and the correct application of a method are both very essential.
Q11. What are the advantages of applying Vedic Maths technique as against using other methods/ tools; such as the Abacus?
A11. No doubt, Abacus is a nice tool for teaching Mathematics that has originated in China. However, there are certain shortcomings. Firstly, students cannot learn it without the use of an Abacus, an instrument which contains beads. Secondly, it takes a long time to learn its application for addition and subtraction and thereafter using these applications in Multiplication and Division. As against this, a student can learn the basic principles and methods of Vedic Mathematics, and apply them in problems relating to Addition, Subtraction, Multiplication, Division, Square, Square Root and Cube Root, in as little as 12 sessions. Besides, there is no dependency on instruments such as the calculator or an abacus.
Q12. How does Vedic Mathematics help a student in the present scenario?
A12. Many times when a student secures less marks in Mathematic, and is ask the reason for it, he / she says “I could not complete the paper in time” or “I could complete the paper, but could not get enough time to correct my mistakes.” By using Vedic methods students can complete their work in a shorter period of time.
Vedic Maths & its advantages
• It encourages mental calculations. It is easy, simple, direct and straightforward.
• Maths, A dreadful subject is converted into a playful and blissful subject, which we keep on learning with smiles on the face and joy in the heart.
• Vedic Maths enriches our knowledge and understanding of maths, which shows clear links and continuity between different branches of maths.
• We are living in the age of competitions. Vedic Mathematics methods come to us as a boon for all competitions. Present maths requires much effort in learning.Vedic Maths being most natural way of working can be learnt and mastered with very little efforts and in a very short time.
• Vedic Maths system also provides us with a set of checking procedures for independent crosschecking of whatever we do. If you make the habit of applying the simple and quick checks at different stages of working. We move on confidently, and keep on smiling at every stage, after confirming the correctness of work.
• The element of choice and flexibility at each stage keeps the mind lively and alert and develops clarity of mind and intuition by integrated training of the two hemispheres of brain and there by Holistic development of the human brain
automatically takes place through Vedic Mathematics multidimensional thinking.
• Vedic Mathematics system at a very subtle level helps us in development of the spiritual part of personality.
Vedic Maths: Multiplication using Nikhilam
Write the two numbers, one below the other, on the left hand side. Subtract 10 from each of them, and write the difference on the right hand side as follows
9 -1
7 -3Now multiply the numbers on the right hand side (-1 and -3 ) to get 3 as the last digit of the product. Add the two numbers on the left (9 and 7) to get 16 and subtract the nearest power of 10 (10 in this case) from it to get the next digit (to get 6). So
9 -1
7 -3
--------
6 3
This method is called 'Nikhilam'. In the above case, the numbers are close to 10. 10 is referred to as the base. Another way of obtaining the left hand side of the product is by cross-addition. In the above case 6 can be obtained by cross addition of 9 and -3 or 7 and -1.
Let's try some more examples
- 8 x 7
- 9 x 9
Now, you may wonder why do we need special methods for multiplying such small numbers? What about big numbers? The same method can be applied to multiply numbers, which are near to any power of 10. Thus the base can be any power of 10.
Vedic Maths & Result verification using Navasesh
Here is a method for checking your answers after multiplication. But before getting into the method you should know what navasesh means and how to obtain the navasesh of a number.
Navasesh of a number is "the remainder obtained when the number is divided by 9".
For example,
navasesh of 24 is 6
navasesh of 63 is 0
navasesh of 110 is 2
Now, it is easier to calculate the navasesh of smaller numbers by dividing the number by 9. In the case of larger numbers this may be a time consuming process. So here is a method of finding the navasesh of a number without dividing the number.
All you've to do is to sum the digits of the number to a single digit. This single digit is the navasesh of the number.
For example,
navasesh of 24 is 2+4=6
navasesh of 233 is 2+3+3=8
Looks much simpler! Doesn't it? This process can be carried out mentally. Now,there may be occasions while carrying out the addition process , where the sum equals or exceeds 9. If the sum equals 9, treat it as 0, and continue adding the other digits. If it exceeds 9, then add the individual digits of the sum, and continue adding the rest of the digits.
For example consider finding the navasesh of 4578 mentally.
The sum 4+5=9 (equals 9). So the sum is 0. Continue adding 0 with 7. 0+7=7.
Again when 7+8=15,reduce the sum to 1+5=6.
6 is the navasesh of 4578
Now that you are familiar with finding navasesh of a given number here comes the method of checking your answers after multiplication.
Consider the multiplication of 789 by 67. If you multiply, you'll get the answer as 52863. We can check this result as follows:
Find the navaseshs of the two multiplicands.
Navasesh of 789 is 6 and that of 67 is 4.
Find the product of the two navaseshs.
6 x 4 =24.
Now find the navasesh of this product.
Navasesh of 24 is 6.
Let us call this 6 as the product navasesh.
Now find the navasesh of your answer. If it is not equal to the product navasesh then your answer is wrong.
Navasesh of 52863 is 6, which is equal to product navasesh.
Warning:-
It is possible that you arrive at a wrong answer whose Navsesh is that of the actual answer. For example,
the navasesh of 52863 and that of 51963 are equal! You may be fooled and you think the wrong answer is right. But the possibility of this occuring in an actual multiplication is very low. But, at the same time, if the Navaseshs are not equal the method assures you that your answer is definitely wrong!
Vedic geometry
As a result of the mathematics required for the construction of these altars, many rules and developments of geometry are found in Vedic works. These include:
- Use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
- Equivalence through numbers and area.
- Squaring the circle and visa-versa.
- Pythagorean triples discovered algebraically.
- Statements of the Pythagorean theorem and a numerical proof.
- Computations of π, with the closest being correct to 2 decimal places.
Lagadha (circa 1350- 1200) was probably the earliest known mathematician to have used geometry and trigonometry for astronomy.
Yajnavalkya ( 9th century BC) composed the Shatapatha Brahmana, which contains geometric aspects, including several computations of π, with the closest being correct to 2 decimal places (the most accurate value of π upto that time), and gives a rule implying knowledge of the Pythagorean theorem.
The Sulba Sutras ("Rule of Chords" in Vedic Sanskrit), which is another name for geometry, were composed between 800 BC and 500 BC and were appendices to the Vedas giving rules for the construction of religious altars. The Sulba Sutras contain the first use of irrational numbers, quadratic equations of the form a x2 = c and ax2 + bx = c, the use of the Pythagorean theorem and a list of Pythagorean triples discovered algebraically predating Pythagoras, geometric solutions of linear equations, and a number of geometrical proofs. These discoveries are mostly a result of altar construction, which also led to the first known calculations for the square root of 2, which were correct to a remarkable 5 decimal places.
Baudhayana (circa 800 BC) composed the Baudhayana Sulba Sutra, which contains a statement of the Pythagorean theorem, geometric solutions of a linear equation in a single unknown, several approximations of π (the closest value being 3.114), along with the first use of irrational numbers and quadratic equations of the forms ax 2 = c and ax2 + bx = c, and a computation for the square root of 2, which was correct to a remarkable five decimal places.
Manava (circa 750 BC) composed the Manava Sulba Sutra, which contains approximate constructions of circles from rectangles, and squares from circles, which give approximate values of π, with the closest value being 3.125.
Apastamba (circa 600 BC) composed the Apastamba Sulba Sutra, which contains the method of squaring the circle, considers the problem of dividing a segment into 7 equal parts, calculates the square root of 2 correct to five decimal places, solves the general linear equation, and also contains a numerical proof of the Pythagorean theorem, using an area computation. The historian Albert Burk claims this was the original proof of the theorem which Pythagoras copied on his visit to India.
Vedic Number Representation
Vedic knowledge is in the form of slokas or poems in Sanskrit verse. A number was encoded using consonant groups of the Sanskrit alphabet, and vowels were provided as additional latitude to the author in poetic composition. The coding key is given as Kaadi nav, taadi nav, paadi panchak, yaadashtak ta ksha shunyam
Translated as below
- letter "ka" and the following eight letters
- letter "ta" and the following eight letters
- letter "pa" and the following four letters
- letter "ya" and the following seven letters, and
- letter "ksha" for zero.
In other words,
- ka, ta, pa, ya = 1
- kha, tha, pha, ra = 2
- ga, da, ba, la = 3
- gha, dha, bha, va = 4
- gna, na, ma, scha = 5
- cha, ta, sha = 6
- chha, tha, sa = 7
- ja, da, ha = 8
- jha, dha = 9
- ksha = 0
For those of you who don't know or remember the varnmala, here it is:
ka kha ga gha gna
cha chha ja jha inya
Ta Tha Rda Dha Rna
ta tha da dha na
pa pha ba bha ma
ya ra la va scha
sha sa ha chjha tra gna
Thus pa pa is 11, ma ra is 52. Words kapa, tapa , papa, and yapa all mean the same that is 11. It was upto the author to choose one that fit the meaning of the verse well.
An interesting example of this is a hymn below in the praise of God Krishna that gives the value of Pi to the 32 decimal places as .3141592653589793238462643 3832792.
Gopi bhaagya madhu vraata
Shrngisho dadhisandhiga
Khalajivita khaataava
Galahaataarasandhara
Vedic Algorithms in Digital Signal Processing
The implementation of Vedic multiplication on 8085/8086 microprocessors and comparing it with conventional mathematics methods clearly indicates the computational advantages offered by Vedic methods.Therefore, such approaches are extremely beneficial in digital signal processing applications. There is an overwhelming need to explore Vedic algorithms in detail so as to verify its applicability in different domains of engineering.
Vedic algorithms implementations on specially designed BCD architecture will also help to enhance processor throughput.An awareness of Vedic mathematics can be effectively increased if it is included in engineering education. In future, all the major universities may set up appropriate research centres to promote research works in Vedic mathematics.
Vedic Philosophy and Mathematics
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars are divided about how early the symbol for zero came to be used in numeric notation in India, (Ifrah arguing that the use of zero is already implied in Aryabhatta) tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period. Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions (such as plus, minus, square root etc) eventually became the foundation stones of modern mathematical notation.
Vedic maths quick tips!
To multiply a number by 11, first put down the digit in the units place in the units place for the answer. Then add the digit in the units place with digit in the tens place for the, substitute the units place digit of the number so obtained the for the tens place digit of the answer, and treat the tens place digit as carry for the addition between tens place and hundreds place of the original number. eg: 23*11 = 2 (2+3) 3 = 253; 765*11 = (7+1) (7+6+1) (6+5) 5 = 8415; 29043*11 = (2+1) (2+9) (9+0) (0+4) (4+3) 3 = 319473.
Also taking analogy from above, to multiply a number by 22,33,44... first multiply the number by 11 and by 2,3,4...
To find the square of a number between 26 and 49, first subtract the difference of the number from 50, from itself. Then divide the number so obtained by 2. Now jot down the square of the difference of the number from 50 in the end of the result of division, allowing for only two places ( not more not less ) and treating the hundreds place digit as carry over, if any. eg: 46^2 = (46 - 4)/2 4^2 = 42/2 16 = 2116; 39^2 = (39 - 11)/2 11^2 = 28/2 121 = 1 (4+1) 21 = 1521.
Drawing analogy from above, to find the square of a number from 51 to 74, add the difference instead of subtracting. eg: 56^2 = (56 + 6)/2 6^2 = 62/2 36 = 3136; 70^2 = (70 + 20)/2 20^2 = 90/2 400 = 4 (5+4) 00 = 4900.
To multiply a number by 125, first add 3 zero's at the end of the number, then divide by 8. eg: 394*125 = 394 * (125*8) /8 = 394000/8 = 49250.
To know the remainder when a number is divided by 3 or 9, first add the digits of the number and then divide the sum by 3 or 9 as the case maybe and take the remainder of this division; it gives the same result. eg: to find remainder when 480275996 is divided by 3 - sum of digits = 50 / 3 = 16 2/3, so remainder is 2; for division by 9 : 50/9 = 5 5/9, so remainder is 5.
This above does not even cover the tip of the tip of the iceberg of knowledge that can be gleaned from Vedas.
Permutations and Combinations in Vedic Mathematics
The story of Pythagoras' theorem
A perfect example of this sort of misattribution involves the so-called Pythagoras theorem, the well-known theorem which was attributed to Pythagoras who lived around 500 B.C.E., but which was first proven in Greek sources in Euclid's Geometry, written centuries later. What is not well understood is that this theorem was known to the authors of the Vedas, and was proved in Baudhayana's Shulva Sutra, which was composed several centuries before Pythagoras, and which might have been a source for Greek geometry, transmitted via the Persians who traded both with the Greeks and the Indians.
Math was not the only science in which the ancient Indians excelled. Various sciences, including mathematics, astronomy, linguistics and grammar were considered to be Vedangas, literally limbs or branches of the Vedas, that is, the knowledge which was necessary for the proper performance of the Vedic rites. One thing we might point out here is that the division between religion and science is not applicable to the Vedic context, wherein the two are seen as natural and necessary complements. Indeed, the very word Veda, derived from the verbal root vid, 'to know', with alternative meanings of 'to find or discover' and 'to be', can be literally translated as most generally "knowledge," or, more specifically, 'science', a word which is likewise derived from the Latin verb 'scire,' 'to know'. The Vedas, in short, include everything that their authors, writing thousands of years ago, considered worth knowing.
Knowledge of mathematics, and geometry in particular, was necessary for the precise construction of the complex Vedic altars, and mathematics was thus one of the topics covered in the brahmanas. This knowledge was further elaborated in the kalpa sutras, which gave more detailed instructions concerning Vedic ritual. Several of these treat the topic of altar construction. The oldest and most complete of these is the previously mentioned Shulva Sutra of Baudhayana, which proved Pythagoras theorem several centuries before Pythagoras.
The ancient Indians did not stop with geometry, but continued to develop advanced mathematical techniques. Aryabhata, for example, developed and solved in the fifth century C.E. complex algebraic and trigonometric problems which were neither conceived nor solved in Europe until over a thousand years later. The European developments, in turn, were dependent upon Indian works such as Aryabhata's Aryabhatiya, which was transmitted by the Arabs to Europe, and translated into Latin in the thirteenth century. Such cultural debts of Europe to India, like the Pythagoras theorem, are not widely acknowledged, not due to any lack on the part of Indian scholarship but rather due to a lack on the side of European scholars, who were blinded by the cultural chauvinism characteristic of the colonial period. This is a darkness from which the West is only now beginning to awake.
The Man who knew Infinity: a life of the genius Ramanujan by Robert Kanigel
You might think: how would anyone understand the psyche and the drives behind a person who was born into a demon-haunted late 19th century and was so enamoured with mathematics that he left to go to a completely alien place where even the food was not palatable.
You might think: perhaps this biography of Ramanujan will ignore his own story to concentrate on the more accessible lives of the famous Cambridge mathematicians like Hardy.
You might think: how can anyone make us understand why Ramanujan eventually died at an early age succumbing to tuberculosis; remaining a vegetarian in war-torn England even when he was consumed by malnutrition.
Well, think again.
This book is a tour-de-force in science writing. It is amazingly detailed in every aspect and Kanigel could not have done a better job if he was channeling Ramanujan himself. Kanigel is obviously fond of Ramanujan having spent so much time documenting his life, but he also has the necessary external point of view in many places which makes you thankful that this is not a mere hagiographic survey.
The math is dumbed down a bit as is necessary for a mass market book like this. However, the explanations of Ramanujan's math exploits are usually done well. At least interesting stories are not eliminated altogether because the math was be too hard to explain.
Here is one (non-mathematical) story from the book:
"Even the prevalence of body odours among the English mystified him -- until, the story goes, one day he was enlightened about it at a tea party. A woman was complaining that the problem with the working classes was that they failed to bathe enough, sometimes not even once a week. Seeing disgust writ large on Ramanujan's face, she moved to reassure him that the Englishmen he met were sure to bathe daily. "You mean," he asked, "you bathe only once a day?"
Vedic maths and the problems of the "western" Maths approach
Well known problems of modern mathematics may be cited as:
1. Everywhere continuous but nowhere differentiable functions
2. Hypercubes 1 to 7 increase but hypercube 8 onwards decrease
3. Space Filling Curves
4. Riemann Hypothesis
5. Goldbach's conjecture
6. Fermat's Last Theorem
Isn't it that these problems are there because of the axioms accepted by the modern mathematics and its approach of taking only a linear approach in terms of dimensions?
And then follows a question as to whether Vedic mathematics is in a position to help the modern mathematics to come out of its mental block and to un-tie its logical knots and to solve the problems? &nbs p;
The Vedic geometric concepts worked out by reacent researchers promise us geometric comprehensions of our existence phenomenon transcending our existing three space format. The real four and higher spaces formats of Vedic comprehensions are new wonderful worlds of very rich mathematics which may ensure us powerful technologies and much potentialised disciplines of knowledge. The basic comprehension pointed out is the way the cosmic surface constitutes and binds the solid granules as synthetic solids manifesting in the cosmos.
Modern researchers are attempting to reconstruct the discipline of geometry as a discipline based on Vedic concepts. They have designated this discipline as Vedic Geometry as opposed to vedic maths. Their results has added a new dimension to the dialogue initiated with the interpretation of the Ganita Sutras and their potentialities brought to focus by Swami Bharti Krisna Tirthaji Maharaj in his path beaking book regarding vedic mathematics.
"Vedic" contributions to Civil engineering
From complex Harappan towns to Delhi's Qutub Minar, India's indigenous technologies were very sophisticated. They included the design and planning of water supply, traffic flow, natural air conditioning, complex stone work, and construction engineering. It is but natural to think there must have been some form of mathematics because without some form of rational calculation method, these achievements would have been simply impossible. Hence, despite all the controversy regarding vedic maths, we can safely assume that there indeed was some form of mathematics whether vedic or not!
Most students learn about the ancient cities of the Middle East and China. How many have even a basic understanding of the world's oldest and most advanced civilization the Harappan or Indus-Sarasvati Valley Civilization in India? The Indus-Sarasvati Civilization was the world's first to build planned towns with underground drainage, civil sanitation n, hydraulic engineering, and air-cooling architecture. While the other ancient civilizations of the world were small towns with one central complex, this civilization had the distinction of being spread across many towns, covering a region about half the size of Europe. Weights and linguistic symbols were standardized across this vast geography, for a period of over 1,000 years, from around 3,000 BCE to 1500 BCE. Oven-baked bricks were invented in India in approximately 4,000 BCE. Over 900 of the 1,500 known settlement sites discovered so far are in India.
Since the Indus-Sarasvati script is yet to be decoded, it remains a mystery as to how these people could have achieved such high levels of sophistication and uniformity in a dispersed complex and with no visible signs of centralized power. Also the precision of man made things tells us about the system of mathematics which is today broadly called vedic maths.
For instance, all bricks in this civilization are of the ratio 1:2:4 regardless of their size, location or period of construction. There are many pioneering items of civil engineering, such as drainage systems for water (open and closed), irrigation systems, river dams, water storage tanks carved out of rock, moats, middle-class style homes with private bathrooms and drainage, and even a dockyard; there is evidence of stairs for multiple-storied buildings; many towns have separate citadels, upper and lower towns, and fortified sections; there are separate worker quarters near copper furnaces; granaries have ducts and platforms; and archeologists have found geometric compasses, linear scales made of ivory. Indians also pioneered many engineering tools for construction, surgery, warfare, etc. These included the hollow drill, the true saw, and the needle with the hole on its pointed end.
The Vedic Maths controversy
Some reasearchers who have been trying to show that vedic maths is not maths at all, interviewed well-placed persons working in banks, industries and so on. Most of them said that when Vedic Mathematics was introduced they came to know about it through their children or friends. A section of them said that they were able to teach the contents of the book to their children without any difficulty because the standard was only primary school level.
They said it was recreational and fun, but there was no relevance in calling it as Vedic Mathematics. We are not able to understand why it should be called Vedic Mathematics and we see no Vedas ingrained in it. The sutras are just phrases, they seem to have no mathematical flavour. This book could have been titled “Shortcut to Simple Arithmetical Calculations” and nothing more. Some of them said an amateur must have written the book! Few people felt that the Swamiji would have created these phrases and called them sutras; then he would have sought some help from others and made them ghost-write for him. Whatever the reality what stands in black and white is that the material in the book is of no mathematical value or Vedic value!
We see a lot of controversery regarding the authenticity of the sutras and if they are really found in the vedas. But the point is, whichever way, it seems to work!
Astronomy in ancient India
"In India I found a race of mortals living upon the Earth. but not adhering to it. Inhabiting cities, but not being fixed to them, possessing everything but possessed by nothing".
Astronomy is one area which has fascinated all mankind from the beginnings of history. In India the first references to astronomy are to be found in the Rig Veda which is dated around 2000 B.C. Vedic Aryans in fact deified the Sun, Stars and Comets. Astronomy was then interwoven with astrology and since ancient times Indians have involved the planets (called Grahas) with the determination of human fortunes. The planets Shani, i.e. Saturn and Mangal i.e. Mars were considered inauspicious.
In the working out of horoscopes (called Janmakundali), the position of the Navagrahas, nine planets plus Rahu and Ketu (mythical demons, evil forces) was considered. The Janmakundali was a complex mixture of science and dogma. But the concept was born out of astronomical observations and perception based on astronomical phenomenon. In ancient times personalities like Aryabhatta and Varahamihira were associated with Indian astronomy.
It would be surprising for us to know today that this science had advanced to such an extent in ancient India that ancient Indian astronomers had recognised that stars are same as the sun, that the sun is center of the universe (solar system) and that the circumference of the earth is 5000 Yojanas. One Yojana being 7.2 kms., the ancient Indian
In Indian languages, the science of Astronomy is today called Khagola-shastra. The word Khagola perhaps is derived from the famous astronomical observatory at the University of Nalanda which was called Khagola. It was at Khagola that the famous 5th century Indian Astronomer Aryabhatta studied and extended the subject. Aryabhatta is said to have been born in 476 A.D. at a town called Ashmaka in today's Indian state of Kerala. When he was still a young boy he had been sent to the University of Nalanda to study astronomy. He made significant contributions to the field of astronomy. He also propounded the Heliocentric theory of gravitation, thus predating Copernicus by almost one thousand years.
But considering that Aryabhatta discovered these facts 1500 years ago, and 1000 years before Copernicus and Galileo makes him a pioneer in this area too. Aryabhatta's methods of astronomical calculations expounded in his Aryabhatta-siddhanta were reliable for practical purposes of fixing the Panchanga (Hindu calendar). Thus in ancient India, eclipses were also forecast and their true nature was perceived at least by the astronomers. The lack of a telescope hindered further advancement of ancient Indian astronomy. Though it should be admitted that with their unaided observations with crude instruments, the astronomers in ancient India were able to arrive at near perfect measurement of astronomical movements and predict eclipses.
Indian astronomers also propounded the theory that the earth was a sphere. Aryabhatta was the first one to have propounded this theory in the 5th century. Another Indian astronomer, Brahmagupta estimated in the 7th century that the circumference of the earth was 5000 yojanas. A yojana is around 7.2 kms. Calculating on this basis we see that the estimate of 36,000 kms as the earth's circumference comes quite close to the actual circumference known today.
Credit for Vedic contribution to science
Our understanding of the contributions to science by ancient Indians has improved considerably during last few decades. For example, Seidenberg discovered that the "Pythagoras" theorem was known to ancient Indians centuries before the Greeks, and is described in The Shatapatha Brahmana. Similarly the contribution of ancient Indians to mathematics, music, grammar, computing science, astronomy and cosmology are being recognized.
The use of binary numbers forms the basis for the operation of digital computers. B. van Nooten of the University of California, Berkeley, describes his discovery of binary numbers in Pingala's "Chandahshastra" ;, an ancient Indian text on music. In order to classify the meters, Pingala constructs a "Prastara" or a matrix of binary numbers. Pingala also describes how to find the binary equivalent of a decimal number.
The hashing technique is used in computer science to retrieve a record from a table. scholars discussed the similarity of "The Katapayadi Scheme" to modern hashing techniques. Indians devised ways to represent numbers in the form of text. Each letter was assigned a specific numerical value. A verse from "Sadratnamala" in fact represents the value of pi up to sixteen decimal places! A vast body of scientific information is hidden in ancient Hindu scriptures and Sanskrit texts. Some scholars have explained the astonishing discovery of speed of light in a medieval text by Sayana. Sayana comments on a verse in Rigveda that Sun traverses 2,202 yojanas in half a nimesha. Yojana is an ancient Indian unit of length and nimesa is the unit of time. Upon conversion in modern units, this yields the value of 186,000 miles per second. Now it is well known that this is the velocity of light. Why would Sayana call this the velocity of Sun? It turns out that Sayana was following the ancient Indian tradition of codifying the knowledge. In this code Sun represents light.
Vedic Maths article
From an article in the Hindu, 2001, By SUDHANSHU RANADE
IT is all very well to be proud of our traditions; but before succumbing to the temptation one must first take the trouble to find out what those traditions were.
Take for instance the claims: that the Indian astronomers Aryabhatta and Bhaskaracharya were "quite familiar" with the gravitational force long before Newton; that electricity, magnetism, sound and ether were all well defined in Vedic times; even the concept of nuclear fusion was known; that the Egyptians built their pyramids by means of Indian arithmetic.
So far as the first two of these are concerned, all I can say is that, if true, it proves that we (some of us) were then far ahead of their times. But it must be remembered at the same time that being far ahead of your times is both painful and unproductive.
Because you would not have at your disposal the means required to elaborate on, and test, your discoveries; leave alone the means to convert others to your view; or science into technology. As for the pyramids, all I can say is that someone has got his history very mixed up; or his geography.
But let me not pursue these points further; I do not have with me the material on the basis of which such claims are being made. I am, however, in a position to say something about "Vedic Mathematics".
Someone sent me some clippings about this so that I could see for myself, and then attempt to convince others, what a wonderful thing it was. The clippings turned out to contain a number of clever ideas; but I did not find in them anything resembling a system, or a well worked out body of ideas.
When I pointed this out, promptly, by return of post, I was sent three books; two of them authored by James T. Glover, "head of mathematics at St. James Independent Schools in London" (Vedic Mathematics for Schools, Books I and II) and prefaced by Dr. L. M. Singhvi, former High Commissioner for India in the UK.
I enjoyed the books; they do indeed make many sorts of calculation simpler; and they do constitute a system. Since Mr. Glover's books took "their inspiration from the pioneering work of the late Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja", a former Shankaracharya of Puri, I next turned my attention to the third book on Vedic Mathematics that had been sent to me; the book by the Shankaracharya himself.
This book too made fascinating reading. But, it is incorrect to call this "Vedic Mathematics". For two reasons; one, as stated in the preface to this book itself, its material is not to be found in the Vedas. Second, mathematics in ancient India, trigonometry for instance, must have progressed far beyond this level; otherwise it would not have been possible for them to delve so deeply into, say, astronomy.
That brings us around full circle. Clearly, the pioneers of ancient India were far ahead of their times. But, oddly enough, those who seek to lead the renaissance today, those who point to the glorious past with the greatest insistence; how does it happen, that they themselves have got left so very far behind.
SUDHANSHU RANADE
"Vedic" contributions to Mathematics
"The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages."
Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that "the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization."
Great & Famous Mathematicians of India
Aryabhata (475 A.D. -550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of 
as 3.1416, claiming, for the first time, that it was an approximation. (He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832.) He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata.
Brahmagupta (598 A.D. -665 A.D.) is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He gave the formula for the area of a cyclic quadrilateral as 
where s is the semi perimeter. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as "Numerical Analysis".
After Brahmagupta, the mathematician of some consequence was Sridhara, who wrote Patiganita Sara, a book on algebra, in 750 A.D. Even Bhaskara refers to his works. After Sridhara, the most celebrated mathematician was Mahaviracharaya or.....
Mahavira. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). The Greeks, by contrast, had studied conic sections in great detail.
Bhaskara (1114 A.D. -1185 A.D.) or Bhaskaracharaya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills. He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He is famous for his book Siddhanta Siromani (1150 A.D.). It is divided into four sections -Leelavati (a book on arithmetic), Bijaganita (algebra), Goladhayaya (chapter on sphere -celestial globe), and Grahaganita (mathematics of the planets). Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it "inverse cyclic". Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's theorem". Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject. As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).
After this period, India was repeatedly raided by muslims and other rulers and there was a lull in scientific research. Industrial revolution and Renaissance passed India by. Before Ramanujan, the only noteworthy mathematician was Sawai Jai Singh II, who founded the present city of Jaipur in 1727 A.D. This Hindu king was a great patron of mathematicians and astronomers. He is known for building observatories (Jantar Mantar) at Delhi, Jaipur, Ujjain, Varanasi and Mathura. Among the instruments he designed himself are Samrat Yantra, Ram Yantra and Jai Parkash.
Famous Indian mathematicians of the 20th century:
Srinivasa Aaiyangar Ramanujan is undoubtedly the most celebrated Indian Mathematical genius. He was born in a poor family at Erode in Tamil Nadu on December 22, 1887. Largely self taught, he feasted on Loney's Trigonometry at the age of 13, and at the age of 15, his senior friends gave him Synopsis of Elementary Results in Pure and Applied Mathematics by George Carr. He used to write his ideas and results on loose sheets. His three filled notebooks are now famous as Ramanujan's Frayed Notebooks. Though he had no qualifying degree, the University of Madras granted him a monthly scholarship of Rs. 75 in 1913. A few months earlier, he had sent a letter to great mathematician G.H. Hardy, in which he mentioned 120 theorems and formulae. Hardy and his colleague at Cambridge University, J.E. Littlewood immediately recognised his genius. Ramanujan sailed for Britain on March 17, 1914. Between 1914 and 1917, Ramanujan published 21 papers, some in collaboration with Hardy. His achievements include Hardy-Ramanujan-Littlewoo d circle method in number theory, Roger-Ramanujan's identities in partition of numbers, work on algebra of inequalities, elliptic functions, continued fractions, partial sums and products of hypergeometric series, etc. He was the second Indian to be elected Fellow of the Royal Society in February, 1918. Later that year, he became the first Indian to be elected Fellow of Trinity College, Cambridge. Ramanujan had an intimate familiarity with numbers. During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=1³+12³=9³+10³.
Unfortunately, Ramanujan's health deteriorated due to tuberculosis, and he returnted to India in 1919. He died in Madras on April 26, 1920.
P.C. Mahalanobis : He founded the Indian Statistical Research Institute in Calcutta. In 1958, he started the National Sample Surveys which gained international fame. He died in 1972 at the age of 79.
C.R. Rao : A well known statistician, famous for his "theory of estimation"(1945). His formulae and theory include "Cramer -Rao inequality", "Fischer -Rao theorem" and "Rao - Blackwellisation".
D.R. Kaprekar (1905-1988) : Fond of numbers. Well known for "Kaprekar Constant" 6174. Take any four digit number in which all digits are not alike. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself.
Harish Chandra (1923-1983) : Greatly developed the branch of higher mathematics known as the infinite dimensional group representation theory.
Narendra Karmarkar : India born Narendra Karmarkar, working at Bell Labs USA, stunned the world in 1984 with his new algorithm to solve linear programming problems. This made the complex calculations much faster, and had immediate applications in airports, warehouses, communication networks etc.
Vedic maths formula for multiplication
"Vedic mathematics" gained popularity primarily through the work of late Sankaracharya (Bharti Krisna Tirtha) of Puri (1884 -1960). Swamiji's "Vedic Mathematics" and the practical demonstrations of Sixteen Sutras (120 words!) stunned the world with their originality and simplicity. The four Vedas (Rigveda, Samaveda, Yajuraveda, Atharvaveda), the four Upvedas, six Vedangas and numerous commentaries on them over the centuries are storehouse of great knowledge. However, many scholars dispute that these Sutras are found in Vedas.
Here we shall describe only one Sutra out of sixteen -the general formula for multiplication. After learning this, you will never take out calculators for multiplication.
URDHVA-TIRYAK SUTRA
Till now, you were multiplying like this:
Question: Multiply 432 by 617.
Answer:
432
x 617
3024
432
2592
266544
More the number of digits in the numbers, more lines and time you consume. No more! Using the Sutra "Vertically and Crosswise", you have
Step 1 (mentally, don't write on notebook) : vertically (last digits) :
2x7=14; write 4 carry 1
Step 2 (mentally) : crosswise (last two digits) :
3x7 +2x1 = 23 +carry 1 = 24; write 4 carry 2
Step 3 : vertically and crosswise (three digits) :
4x7 + 3x1 +2x6 = 43 +carry 2 = 45; write 5 carry 4
Step 4 : (move left; first two digits) :
4x1 +3x6 = 22 +carry 4 = 26; write 6 carry 2
Step 5 : (move left; first digit of each number) :
4x6 = 24 +carry 2 = 26. End.
Write answer : 266544
This is how it appears on notebook :
432
x 617
266544
No matter how big the numbers are, you will need to write only the final answer. All other steps are easily carried out mentally. If the two numbers have different number of digits, write smaller number below the other and pad it on left side with zeros. The theory behind above example is :
ax² +bx +c
dx² +ex
adx4 +(ae+bd)x³ +(af+be+cd)x² +(bf+ce)x +cf
Observe that coefficient of x (units digit) is cf, which is obtained by multiplying last two coefficients (vertically). The coefficient of x1 (tens digit) is bf+ce, which is obtained by crosswise multiplication of last two coefficients. The coefficient of x² (hundreds digit) is af+be+cd, which is obtained by crosswise and vertical multiplication of last three coefficients. Now as all coefficients are used up, we leave last coefficients and use the remaining, and so on.
Here are a few more examples:
Multiply 92 by 67
92
x 67
(Mentally) 2x7 is 14; write 4 carry 1;
9x7 +2x6 = 75 +carry 1 = 76; write 6, carry 7
9x6 is 54, add carry 7 to get 61 -so answer is 6164
Multiply 2376 by 4060
2376
x 4060
6x0 = 0; write 0;
7x0 +6x6 = 36; write 6 carry 3;
3x0 +7x6 +6x0 = 42 +carry 3 = 45; write 5 carry 4
2x0 +3x6 +7x0 +6x4 = 42 +carry 4 = 46; write 6 carry 4
2x6 +3x0 +7x4 = 40 +carry 4 = 44; write 4 carry 4
2x0 +3x4 = 12 +carry 4 = 16; write 6 carry 1
2x4 = 8 +carry 1 = 9; write 9. End. Answer is 9646560
Note that all the calculations can be easily done in mind; you just go on writing a digit of answer one at a time (from right to left). So on notebook you will just write:
2376
x 4060
9646560
Please do the following multiplications by Sutra "Vertically and Crosswise" :
32 x 54?
50 x 98?
123 x 987?
654 x 84?
749 x 302?
3112 x 8735?
3022 x 7004?
This is just a "trailer" from Vedic Mathematics. If you found it useful or interesting (or both), we strongly recommend the book "Vedic Mathematics" by late Sankaracharya (Bharti Krisna Tirtha) of Puri.
The story of Zero
The ancient Indians represented zero as a circle with a dot inside. In Sanskrit, it was called "soonya". This and the decimal number system fascinated Arab scholars who came to India. Arab mathematician Al-Khowarizmi (790 AD - 850 AD) wrote Hisab-al-Jabr wa-al-Muqabala (Calculation of Integration and Equation) which made Indian numbers popular. "Soonya" became "al-sifr" or "sifr". The impact of this book can be judged by the fact that "al-jabr" became "Algebra" of today. An Italian Leonardo Fibonacci (1170 AD - 1230 AD) took this number system to Europe.
The Arabic "sifr" was called "zephirum" in Latin, and acquired many local names in Europe including "cypher". In the beginning, the merchants used to Roman numbers found the decimal system a new idea, and referred to these numbers as "infidel numbers", as the Arabs were called infidels because they had invaded the holy land of Palestine. However, nowadays this system is called Hindu-Arabic System. This positional system of representing integers revolutionised the mathematical calculations and also helped in Astronomy and accurate navigation. The use of positional system to indicate fractions was introduced around 1579 AD by Francois Viete. The dot for a decimal point came to be used a few years later, but did not become popular until its use by Napier.
Modern computers are based on binary system - which uses only two bits - 0 and 1.
The Indian Sulbasutras
The Sulbasutras are appendices to the Vedas which give rules for constructing altars. If the ritual sacrifice was to be successful then the altar had to conform to very precise measurements. The people made sacrifices to their gods so that the gods might be pleased and give the people plenty food, good fortune, good health, long life, and lots of other material benefits. For the gods to be pleased everything had to be carried out with a very precise formula, so mathematical accuracy was seen to be of the utmost importance. We should also note that there were two types of sacrificial rites, one being a large public gathering while the other was a small family affair. Different types of altars were necessary for the two different types of ceremony.
All that is known of Vedic mathematics is contained in the Sulbasutras. This in itself gives us a problem, for we do not know if these people undertook mathematical investigations for their own sake, as for example the ancient Greeks did, or whether they only studied mathematics to solve problems necessary for their religious rites. Some historians have argued that mathematics, in particular geometry, must have also existed to support astronomical work being undertaken around the same period.
Certainly the Sulbasutras do not contain any proofs of the rules which they describe. Some of the rules, such as the method of constructing a square of area equal to a given rectangle, are exact. Others, such as constructing a square of area equal to that of a given circle, are approximations. We shall look at both of these examples below but the point we wish to make here is that the Sulbasutras make no distinction between the two. Did the writers of the Sulbasutras know which methods were exact and which were approximations?
Vedic Mathematics and the Spiritual Dimension
In the valley of the Indus River of India, the world's oldest civilization had developed its own system of mathematics. The Vedic Shulba Sutras (fifth to eighth century B.C.E.), meaning "codes of the rope," show that the earliest geometrical and mathematical investigations among the Indians arose from certain requirements of their religious rituals. When the poetic vision of the Vedic seers was externalized in symbols, rituals requiring altars and precise measurement became manifest, providing a means to the attainment of the unmanifest world of consciousness. "Shulba Sutras" is the name given to those portions or supplements of the Kalpasutras, which deal with the measurement and construction of the different altars or arenas for religious rites. The word shulba refers to the ropes used to make these measurements.
Although Vedic mathematicians are known primarily for their computational genius in arithmetic and algebra, the basis and inspiration for the whole of Indian mathematics is geometry. Evidence of geometrical drawing instruments from as early as 2500 B.C.E. has been found in the Indus Valley.1 The beginnings of algebra can be traced to the constructional geometry of the Vedic priests, which are preserved in the Shulba Sutras. Exact measurements, orientations, and different geometrical shapes for the altars and arenas used for the religious functions (yajnas), which occupy an important part of the Vedic religious culture, are described in the Shulba Sutras. Many of these calculations employ the geometrical formula known as the Pythagorean theorem. This theorem (c. 540 B.C.E.), equating the square of the hypotenuse of a right angle triangle with the sum of the squares of the other two sides, was utilized in the earliest Shulba Sutra (the Baudhayana) prior to the eighth century B.C.E. Thus, widespread use of this famous mathematical theorem in India several centuries before its being popularized by Pythagoras has been documented. The exact wording of the theorem as presented in the Sulba Sutras is: "The diagonal chord of the rectangle makes both the squares that the horizontal and vertical sides make separately."2 The proof of this fundamentally important theorem is well known from Euclid's time until the present for its excessively tedious and cumbersome nature; yet the Vedas present five different extremely simple proofs for this theorem. One historian, Needham, has stated, "Future research on the history of science and technology in Asia will in fact reveal that the achievements of these peoples contribute far more in all pre-Renaissance periods to the development of world science than has yet been realized."3
The Shulba Sutras have preserved only that part of Vedic mathematics which was used for constructing the altars and for computing the calendar to regulate the performance of religious rituals. After the Shulba Sutra period, the main developments in Vedic mathematics arose from needs in the field of astronomy. The Jyotisha, science of the luminaries, utilizes all branches of mathematics.
The need to determine the right time for their religious rituals gave the first impetus for astronomical observations. With this desire in mind, the priests would spend night after night watching the advance of the moon through the circle of the nakshatras (lunar mansions), and day after day the alternate progress of the sun towards the north and the south. However, the priests were interested in mathematical rules only as far as they were of practical use. These truths were therefore expressed in the simplest and most practical manner. Elaborate proofs were not presented, nor were they desired.
