Sutra 3- Vertically and crosswise

Long time, since we touched upon the subject of vedic mathematics. So, here it is.. the third sutra of subject


Vertically and crosswise:
Its application in multiplying numbers is fairly well known now but in fact its range of application is very great


MULTIPLICATION


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).
DIVISION
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.