1. Vertically and Crosswise: the system of Vedic
Mathematics (BY TARUN GEHLOT)
The system of Vedic Mathematics was rediscovered from ancient Sanskrit texts earlier this
century . The system uses sixteen word-formulae which relate 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.
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.
TARUN GEHLOT

2. 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 threefigures 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
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:
TARUN GEHLOT

3. 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.
ADDITION AND SUBTRACTION OF FRACTIONS
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:
TARUN GEHLOT

4. 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).
EQUATION OF A LINE JOINING TWO POINTS
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.
MULTIPLYING NUMBERS NEAR A BASE
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.
Vertically and Crosswise has a huge range of applications- and remembers it is
just one of sixteen formulae used in Vedic Mathematics!
TARUN GEHLOT

5. The Vedic system is extremely coherent and unified, the methods are so easy they
really amount to mental arithmetic
TARUN GEHLOT