The knowledge of Vedic Mathematics had been lost in subsequent years, but the late Shankaracharya of Jagannath Puri rediscovered it from the Atharva Veda. According to him, if children are taught these fast procedures, memorization of the multiplication table up to 5 x 5 would be sufficient. The noted mathematical genius, Trachtenberg, who independently worked out some of these methods, was also of the same opinion.
Another argument in favour of the above view is that the multiples of a number are 'latent' in it and can be released almost instantly In the case of numbers not exceeding two digits. This concept leads to a new method of last multiplication by multipliers of two digits-the 'Sarala' method. This helps to widen the scope of the answer-at-sight method.
Views of other Mathematicians
Meyer, the celebrated mathematics writer, goes into ecstasies and becomes almost lyrical when he speaks of how the decimal system invented by the Hindu mathematicians enabled all the other sciences to progress and gave us the comforts and luxuries of modern life.
Another popular mathemetics and science writer, Isaac Astmov says: "As a matter of fact, the lack of a proper system of writing numbers held back the advance of Greek mathematics. If the greatest of the Greek mathematicians, Archimedes, had only had the number system, he would have invented calculus, (which he nearly invented any way), and it would not have had to wait 1800 years for Newton to invent it."
But neither Meyer nor Asimov realised that even fuller use could be made of this system, in fact had been made and wonderfully fast methods developed by the ancient Hindu mathematicians for all arithmetical computations.
When we enter this magic wonderland we sometimes get so excited that we stumble upon some minor discoveries ourselves. One such discovery is that it is enough to memorise the table upto 5 x 5. It can be realized that the multiples of a number are 'latent' in it and in the case of numbers not exceeding two digits, these can be 'released' by a fast and simple operation. This also leads to a new method of last multiplication, which, however, is limited to multipliers of two digits. The nurnber multiplied can have any numher of digits.
Views of Shankaracharya & Trachtenberg
But first let us hear what the illustrious Shankaracharya has to say about the multiplication table:
"We may also draw the attention of all students and teachers of mathematics to the well-known and universal fact that, in respect of arithmetical multiplications, the usual present- day procedure everywhere is for the children in the primary classes to be told to get by heart the multiplication tables, up to 16 x16, or 20 x 20 and so on. But according to the Vedic multiplication system, the multiplication tables are not really required above 5 x 5. And a pupil who knows simple addition and subtraction of single digit numbers and the multiplication table up to 5 x 5 can improvise all the necessary multiplication tables for himself at any time very easily and speedily, nay practically immediately."
The Shankaracharya then goes on to describe the answer-at-sight methods for multiplication which, especially for numbers up to 20, are as fast as recalling from memory and do not require a knowledge of the multiplication table beyond 5 x 5.
Can it be a mere coincidence that this is exactly the opinion of Trachtenberg, whose system of arithmetical operations is now being widely taught in Europe and America? This is what he writes in the opening paragraph of the first chapter of his book 'The Trachtenberg speed system of basic mathematics':
"The first item on the agenda is a new way to do basic multiplication: We shall multiply without using any memorized multiplication tables. Does this sound impossible? It is not only possible, it is easy. Most people know the tables pretty well, in fact perfectly, except for a few doubtful spots.
Eight times seven or six times nine are a little uncertain to most of us, but the smaller numbers like four times five are at the command of everyone. We approve of using this hard-won knowledge. Now wish to do some multiplying without using the multiplication table."
What both these great thinkers are pointing out is that the table is useful in its own way but, with superfast methods at our disposal, it is no longer essential beyond 5 x 5.
Pros & Cons of this method
Let us consider the pros and cons for retaining the multiplication table to the extent now in use, or for reducing it to a mere 5 x 5 as advised by the Shankaracharya and Trachtenberg. It is a time-honored ready reckoner which, once etched upon our memory, remains with us throughout lift. As memory training, it is excellent and many teachers have told me that most children enjoy memorizing and repeating the table in class. But others have pointed out that, for an appreciable number, it is a stumbling block and that the aversion for mathematics begins to develop at this stage.
Psychologists are of the opinion that for those children who find it difficult to memorize the table, the real obstacle is not a weak memory, but the nervous fear of forgetting. The knowledge that there is an easy alternative they can fall back upon when memory fails, will, therefore, go a long way to remove their nervousness and make it easier for them to get the table by heart.
Learning the Method of Multiplication
Actually there are two alternatives, since as mentioned earlier, the multiples of all numbers not exceeding two digits can be obtained very fast by a simple procedure.
Sufficient here to say that we regard the multiplier numbers in their relationship to the 'nearest multiple of ten'. Thus 9 is regarded as 10-1, 8 as 10-2, 17 as 20-3, 36 as 40-4, etc. Further if we want to say 28 x 37, we begin with 28 x 40 and remove 3 multiples of 28. But this is done in a fast and elegant manner without having to find out what 28 x 3 is.
Now consider the following examples:
The first illustration will be with 9, which is 10-1. The multiples of 9 from 90 downwards can be got by 'operating' or 'playing' upon 9 as follows:
From 9 subtract 1 (you get 8), and put that 1 on the right side of 8. You get 81, which is 9 x 9
From 9 subtract 2 (you get 7) and put that 2 on the right. You get 72 which is 9 x 8
Repeating the operation with 3 gives you 63 i.e., 9 x 7 and so on and finally with 8 will give you 18
Operating upon 9 gave us the multiples of 9 from 90 downwards.
18 is equal to 9 x 2 and doing likewise with 18 will give us the multiples of 9 from 180 downwards.
From 18 subtract 1 (we get 17) and put that 1 on the right and you get 171 which is 9 x 19.
Repeating with 2 digits gives us 162 which is 9 x 18 so on.
If we want to get any higher multiple of 9, says 9 x 67.
Now 67 is between 60 and 70 and so we take up for operation 9 x 7 which is 63.
Since 67 is 3 less than 70, we subtract 3 from 63 (we get 60) and putting that 3 on the right we get 603 which equals 9 x 67.
We can use this method for 8, 7 etc. by noting their relation to 10. As 8 is equal to 10- 2 and 7 is equal to 10-3.
So when multiplying by 8, for every 1 you subtract, you put 2 on the right and in the case of 7, for every 1 you remove, you put 3 on the right.
Thus, 8 x 9 will be (8-1) i.e. 7 with 2 on the right or 72.
Similarly, 8 x 8 will be (8-2) i.e. 6 with 4 on the right or 64.
Likewise to get 7 x 9, we subtract 1 from 7 and putting 3 on the right we get 63.
On the same lines 7 x 8 would be (7-2) and putting 6 on the right or 56.
The advantages of this procedure are more fully seen when multiplying by slightly bigger numbers like 16, 17, 18, 19 etc.
Let us 'play' with 19 and see how its multiples are released. Remember how we dealt with 9.
Since 19 is equal to 20-1, for every 2 we subtract, we put 1 on the right side. Subtracting 2 from 19(you get 17) and putting 1 on the right you get 171, which is 19 x 9.
From 19 subtract 4 and put 2 on the right and you get 152 which is 19 x 8 and so on.
Suppose we want 19 x 73.
Since 73 is between 70 and 80, we begin with 19 x 8, which as we saw above is 152.
As 73 is 7 less than 80, we subtract 7 units of 2 i.e.14 from 152 getting 138. Putting 7 on the right gives us 1387 which is equal to 19 x 73.
Coming to 18 and 17, we note that 18 is equal to 20-2 and 17 is equal to 204.
So when finding out the multiples of 18, for every 2 we subtract, we put a 2 on the right and in the case of 17 for every 2 we subtract, we put a 3 on the right.
This last illustration will show how this method can be used to multiply a long number by a number not exceeding two digits.
We shall multiply 3456 by 19.
Since in this method we take the nearest higher multiple of 10 and subtract the necessary multiples, we begin with 19 x 4. The subsequent stages will be 19 x 35, then 19 x 346 and finally 19 x 3456.
The working will be as follows:
19 x 4 is equal to 76.
Next is 19 x 35. Since 35 is 5 less than 40, we subtract 10 from 76 and put 5 on the right side, getting 665.
Now we go to 19 x 346. As 346 is 4 less than 350 we will get (665-8)/4 or 6574.
Finally going to 19 x 3456 we have (6574-8)/4 or 65664.
Instead of 19, our multiplier has all along been 2, a much smaller number.
Using this method we can multiply by numbers like 93 or 79 mentally and in one step, which widens considerably the scope of the answer-at-sight method in which the two numbers to be multiplied have to be very close as the differences from the base number must be small enough for the child to multiply mentally.
Thus formerly we could use the answer- at-sight device for 109 x 106 but not for 109 x 178 as in the latter case the right hand part of the answer, 9 x 76, cannot be worked out mentally and quickly. As the 'Sarala' method enables the child to do this, he can now use the answer at sight procedure in a larger number of cases.
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