As a small boy it was fascinating for me to sit for hours with absorbing interest in a carpenter’s workshop next door marveling his dexterity in using the tools, often trying to drill a hole using the hand drill in vain. It needed a lot of skill and physical strength too. Now the same workshop is being run by the younger generation. What a sea change there! What a variety of furniture they make using modern tools! Hand drills have been replaced by electric drills. Things are done more easily and quickly now. They have earned a fortune now. They deserve it.
A mathematician is a great craftsman and perhaps the greatest of them all. As we know mathematics is the queen of all sciences. It is also the tool which all the branches of science and technology make use of for their better performance. So a mathematician should always sharpen his tools, improve them, update them and keep them ready for his own use and for those in other fields. He must also know how to use them judiciously so as to obtain optimum results easily, quickly and accurately like a good craftsman.
Speed and accuracy are the two eyes of mathematics. A mathematician should always be on the lookout for shortcuts. Identities such as (a+b)2=a2+2ab+b2 or algorithms for arithmetic operations are nothing but shortcuts. Mathematicians all over the world are constantly at work to find such shortcuts. They will be ever refining them. Indians have contributed a lot for finding such shortcuts.
The invention of place value in mathematical notation is perhaps the greatest invention in the whole history of mathematics. We know how cumbersome it is to represent 1492 or add 732 and 499 in Roman or any other non-Indian number system. Even electronic computers can work only with a number system with place value is another story. But the invention of place value and the corresponding algorithms for operations are the greatest shortcuts ever known.
Coming to formulae, ancient Indian mathematics is replete with them (giving both accurate and approximate values.) for finding out lengths, areas, volumes, etc. Bhaskara II’s Leelavathy, for example is a gold mine of such beautiful formulae. Neelakanta’s rule for circular arc, Brahmagupta’s rule for volume of a frustrum, approximations for Indian trigonometric functions Sines and Versines (jhya. Kodijhya, ukramajhya, etc.), innumerable formulae in Indian Astronomy, etc. are only a few others to quote. There have always been attempts to refine them. The same attempts were on in the western world also. Newton’s calculus is the culmination of such refinements started two thousand years before him by Archimedes. In Logic too such refinements have taken place.
During the renaissance, there was an explosion of knowledge. For Astronomy, navigation and several other fields, scientists had to do thousands of complicated calculations and it took months to solve a single problem. Now arose a morning star in the mathematical horizon. John Napier, an Englishman discovered logarithms. It was nothing but an old tool which he sharpened to an unbelievable degree. Theory of indices was known even to school children. With great pains he prepared a table of logarithms and another of anti-logarithms. Using these he was able to convert any number as a power of 10 and conversely (approximately) and thereby do any big calculations with the help of the theory of indices. For about four centuries the scientific community and students all over the world depended on this shortcut method until the advent of electronic calculator which are the most advanced tools available today.
Needless to say all these tools must be put to optimum use for our quick progress. Our students must be trained to handle them using the right tool at the right place to get optimum results. But are we using them? An emphatic “No” is the answer. Before substantiating this statement with examples, the author would like to discuss about a more important thing.
Induction and deduction are two important tools of mathematics, Induction is a process by which we make a general statement looking at some particular cases. It is nothing but an intuitive and intelligent guess, a conjuncture. Whether our guess will prove to be true or not depends upon the number of particular cases we have scanned. If our generalization is based on only a few cases, it is more likely to fail. But those having penetrating minds can scan large vistas of the pattern at hand quickly. Such people can arrive at more accurate conclusions, though these conclusions must be reinforced by logical deductions before final acceptance. Logical deduction is a product of the Western world. Aristotle’s Formal Logic led to paradoxes. Russel and Whitehead refined it by getting rid of the paradoxes and proved that mathematics is logic. Gauss, Einstein’s guru introduced rigour in logical proof. So unless a general statement is proved logically with rigour, it cannot be accepted as true. Western mathematics is very adament in this.
But Eastern and especially Indian mathematics is fundamentally inductive in nature. Logical approach is a slow process It has set rules which can easily be followed by a person of average intelligence. But induction is a quick process and it demands high intellectual capacity, a penetrating mind which can quickly perceive the inherent structure of the pattern at hand, It can also make numerous theorems quickly. Ramanujan was induction (conjecture) personified. He wrote and wrote thousands of theorems in his “Notebooks” without giving any proof. He was so sure of them that he never cared about proof. But later when his theorems were put to logical tests by Westerners, only a few turned to be incorrect. But his output was enormous. Not only Ramanujan, but Indian mathematicians in general were endowed with this capacity of induction. That is why some people tend to believe that Indian mathematics is revealed by God and not discovered by men. But the fact is that Indians were good at inductive reasoning because their training was such. They had a system of learning in which they were subjected to rigourous training for long years under the feet of their great masters. They had to learn by heart all the wisdom of the past enshrined as capsules of verses, aphorisms, ‘Katapaya’, and Vedic sutras and the like. – all formulae, shortcuts, algorithms and mathematical facts. They were at their finger tips and they learnt what when and how to use them. So whenever a problem was given, they came with a ready answer, because they could do a lot of calculations mentally and by shortcuts. This is the secret of their power of intuition and induction.
Lastly let us have a bird’s eye view of Vedic Mathematics. What we have about it is a book written by Swamy Bharathi Krishna Thirtha which gives 16 simple shortcuts by which a variety of problems can be solved. Though the origin of these Sutras is disputed, some of the claims of the proponents of Vedic Mathematics are worth considering. They say that anything fixed naturally becomes monotonous and boring and what they call “Maths Anxiety” is created. Vedic Mathematics on the other hand offers a new and comprehensive approach based on pattern recognition. It encourages constant expression of a student’s creativity. For any problem there is one general technique applicable to all cases. There are also a number of special techniques for special pattern problems. The element of choice and the flexibility of each step keeps the mind lively and alert and develops clarity of mind and intuition and thereby holistic development of the human brain.
Mathematics taught in our schools – the syllabus and the textbooks give only fixed and right methods. The teachers and examiners make them more rigid and sanctified. There is no freedom of choice and flexibility to the students. They need not be alert to the situation in hand and apply a special method or a shortcut. They have to traverse through long, tedious, monotonous and beaten tracks. They need not have presence of mind. But questions asked in many competitive tests such as GRE, SAT, GMAT, etc. call for presence of mind. In those tests students have to observe keenly and perceive quickly some speciality of the problem and strike the nail on the head. Otherwise they will have to beat around the bush and lose the battle. The textbooks do not give such kinds of training to our students.
Now let us consider some examples. In chapter II of Leelavathy, (stanzas 18-19) Bhaskaracharya gives three rules for squaring a number. One of them is, “Twice the product of two parts added to the sum of the squares of the parts. For example, 2972= (200+97)2= 2x200x97+2002+972. But this can be continued to find 972 as 972= (90+7)2= 2x90x7+902+ 72.
So, at one stretch, 2972= 38800+40000+8100+1260+49= 88209. This is a general rule applicable to any number. But the third rule he gives is a special one which goes like this, “Product of the sum of the squares of the number and an assumed quantity. added to the square of the assumed quantity is its square” This is based on the identity a2= (a +b) (a-b) +b2. This method is a special (easy) method to find the square of a number very near to powers or multiples of ten. As examples he gives 297 and 10005 to be squared. Thus 2972= (297+3)(297-3)+32=300x294+9=88209 and 100052= (10005+5)(10005-5)+52=10010x10000+25=100100025.
In Vedic Mathematics Swamiji gives this third method in a different style to find the squares of numbers very near to powers of ten and multiples of ten. He uses the Sutra “Yavathoonam Thavathoonam” by which he means “Add whatever is in excess and subtract whatever is in deficit. Thus 9942= 988036 (because 994 is 6 less than 1000 and therefore 994-6=988, and using three spaces for 62, we write as 988036. Similarly 10042= 1004016, everything done mentally in one step. To find the square of a number (generally) Swamiji gives the Sutra “Urdhva Thiryagbhyam”. Our students must be well acquainted with these methods There is a problem in std IX (State Board) book on circles. “Find the equation of the circle passing through the points (1,0), (0,1) and (0, -1)” The method given in the book (as an example) is as follows: Let the equation to the circle be x2+y2+2gx+2fy+c=0 Since the three points lie on the circle, we get three equations in g, f and c. By solving these we can get the numerical values of g, f and c. Substituting these we can get the equation to the circle as x2+y2=1. The whole thing comes to a page. But an intelligent boy will give the answer in one line because to him this is a special case in which OA=OB=OC=1, and the equation to the circle with the origin as centre and radius 1 is x2+y2=1. The authors of the book may argue that the students must know the most general method. Well, in that case the points given by them should not yield any special method.
Many text books of standard IX and X contain problems in heights and distances. The solutions given in 30o, 45o and 90o is a special problem and not a general one. Students know the ratios of sides in such triangles. So by using ratio and proportion they can solve them in just one or two lines. Then why confuse them with one page?
We can give a hundred other examples where problems can be solved in two or three lines, but students are compelled to solve them using long steps. The more the number of steps, the more the number of marks awarded. If it were a proposition to be proved logically, then we have to necessarily follow Gausian rigour, and the author will take his hat off. But if it is a problem of finding the value of something, an intelligent man will do it by the simplest and quickest possible method, like the good carpenter who made a good fortune. Shortcuts and special methods must be encouraged by teachers and teachers must be encouraged by the valuation system in our country.
Solving equations such as 2x+3y=1 and 3x+y=4 is a problem in Std.VII and they will do it in 7 or 8 steps. A nice formula has been derived in Std. IX (State Board) book to find x and y in two lines. But unfortunately the textbook writers have not used this formula anywhere in the innumerable places where such equations are solved. The same case with many other formulae. Then why should the derive those formulae at all?
Lastly a word about the use of tables and calculators. A large variety of statistical and other tables are used in industry and trade. Reading and interpreting tables is very important. Questions on them are asked in competitive tests also. Calculator is a boon to us. Our students must make the best use of it. If we follow these, our syllabus which is dubbed as loaded will become light to our students and we can even think of enriching it still further. We belong to India, the land of inductive mathematics; we must retain this honor. We belong to the land of the great Ramanujan.
Let us be worthy of them. We are in the computer age when everyone wants to do things quickly. Let us prepare our students for the computer age and for the 21st century and not for the stone age. Let us make use of all the modern tools available to us. Let us not dub our students as dullards any more. They are really bright They want to do things fast. Let us not restrain them.