Thursday, 27 September 2012

Vertically and Crosswise :The system of Vedic Mathematics

The system of Vedic Mathematics was rediscovered from ancient Sanskrit texts earlier this century and has now been fully structured for use in schools. The system uses sixteen word-formulae whichrelate to the way in which we use our mind.
Vertically and Crosswise is one of these formulae. Its application in multiplying numbers is fairly well known now but in fact its range of application is very great- as shown in this article.
If you are not aware of its use in multiplication here is an example.
Suppose we want to multiply 33 by 44:
Multiplying vertically on the right we get 3×4 = 12, so we put down 2 and carry 1 (written 12 above).
Then we multiply crosswise and add the two results: 3×4 + 3×4 = 24. Adding the carried 1 gives 25 so we put 5 and carry 2 (25).
Finally we multiply vertically on the left, get 3×4 = 12 and add the carried 2 to get 14 which we put down.
The simple pattern used makes the method easy to remember and it is very satisfying to get the answer in one line. It is also easy to see why it works: the three steps find the number of units, number of tens and number of hundreds in the answer.
This multiplication can also be carried out from left to right, and this has many advantages. Let us find 33 × 44 from left to right:
Vertically on the left, 3×4 = 12, put 1 and carry 2 to the right (12 above).
Crosswise we get 3×4 + 3×4 = 24 (as before), add the carried 2, as 20, to get 44 and put down 44.
Finally, vertically on the right 3×4 = 12, add the carried 4, as 40, to get 52which we put down.
We always add a zero to the carried figure as shown because the first product here, for example, is really 30×40 = 1200 and the 200 is 20 tens. So when we are gathering up the tens we add on 20 more. This does not seem so strange when you realise that a similar thing occurs when calculating from right to left: when we started the first calculation above with 3×4 = 12 the 1 in 12 was counted as 1 in the next column even though its value is 10. 
Although the first method above is useful for mental multiplication the second method is better because we write and pronounce numbers from left to right and so it is easier to get our answers the same way. This method can be extended to products of numbers of any size. Another advantage of calculating from left to right is that we may only want the first one, two or three figures of an answer, but working from the right we must do the whole sum and get the most significant figure last. In the Vedic system all operations can be carried out from left to right (right to left is not excluded though) and this means we can combine operations: add two products for example. We can extend this further to the calculation of sines, cosines, tangents and their inverses and the solution of polynomial and transcendental equations (Nicholas et al, 1999).
The same vertical and crosswise method can be used for algebraic multiplication's. For example (2x + 5)(3x + 1):
Either method will do. From the left we have
The above left to right method can be simply reversed to give us a one line division method.
Suppose we want to divide 1452 by 44. This means we want to find a number which, when multiplied by 44 gives 1452, or in other words we want a and b in the multiplication sum:
Since we know that the vertical product on the left must account for the 14 on the left of 1452, or most of it, we see that a must be 3.
This accounts for 1200 of the 1400 and so there is a remainder of 200. A subscript 2 is therefore placed as shown.
Next we look at the crosswise step: this must account for the 25 (25), or most of it. One crosswise step gives: 3×4 = 12 and this can be taken from the 25 to leave 13 for the other crosswise step, b×4. Clearly b is 3 and there is a remainder of 1:
We now have 12 in the last place and this is exactly accounted for by the last, vertical, product on the right. So the answer is exactly 33.
It is not possible in this short article to describe all the variations but the method is easily extended for
a) dealing with remainders,
b) dividing any two numbers,
c) continuing the division (if there is a remainder) to any number of figures,
d) dividing polynomial expressions.
The multiplication method described here simplifies when the numbers being multiplied are the same, i.e. for squaring numbers. And this squaring method can also be easily reversed to provide one line square roots: easy to do, easy to understand.
The usual method using common denominators is cumbersome and difficult to learn. By contrast the Vedic method allows the answer to be written straight down.
We multiply crosswise and add to get the numerator of the answer and we multiply the denominators to get the denominator of the answer.
This looks like "horizontally and crosswise" rather than "vertically and crosswise" but fractions can also be written: 2/3 + 4/7, in which case we have:
in which we see "vertically and crosswise".
Subtraction is similar, we cross-multiply and subtract:
When the denominators are not relatively prime we may divide out the common factor and cross-multiply with these reduced figures (see Williams & Gaskell 1997).
Find the equation of the line joining (5, 3) and (2, 7).
By conventional methods we need to know or look up the appropriate formula: 
We substitute the four values, simplify, remove the fraction, open the brackets and rearrange the equation to finally get 3y = -4x + 29.
Or, by the one-line Vedic method:
By vertically and crosswise:
we subtract vertically in the first column to get the y-coefficient, 5 - 2 = 3,
we subtract vertically in the second column to get the x-coefficient, 3 - 7 =-4,
and we cross-multiply and subtract to get the absolute term, 5×7 - 3×2 =29.
We can also solve all sorts of problems in coordinate geometry, transformations, trigonometry etc. and there are more advanced applications in 3-dimensional work, trigonometrical equations, differential equations, complex numbers, simple harmonic motion and so on.
In addition to the general methods described above the Vedic system offers many special methods which can be used when certain conditions are satisfied. These are often extremely effective and powerful. The final example is a special method. 
To multiply, say, 88 by 98 we observe that these numbers are close to the base of 100 and once again we obtain the answer by one line mental arithmetic:
We see that 88 is 12 below 100 and 98 is 2 below, as shown.
Cross-subtracting we get 88-2 = 86 (or 98-12 = 86) for the first part of the answer,
and multiplying vertically we get 12×2 = 24 for the second part.
So 88 × 98 = 8624.
There are a large number of variations of this method which render it applicable in many other situations (Williams & Gaskell, 1997).
Vertically and Crosswise has a huge range of applications- and remember it is just one of sixteen formulae used in Vedic Mathematics!
The Vedic system is extremely coherent and unified, the methods are so easy they really amount to mental arithmetic. A full Vedic Mathematics course covering the National Curriculum for Key Stage 3 is now available and is being successfully used in some schools. (Williams & Gaskell, 1998)
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