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.

Sutra 2 - All from 9 and Last from 10

With this method you only ever need multiplication tables up to 5 times 5. It is one of many ancient Indian sutras and this one involves a cross subtraction method which, according to old historical traditions, is responsible for the acceptance of the ´ mark as the sign of multiplication. Here is a very simple example of the method. Can you give a good explanation of WHY it works?

Suppose we want to multiply 9 by 7. We subtract each number from 10 and, using these differences (or deficiencies), write:

9-1

7-3

6 3

The product has two parts, left and right.
To get the right part (or units digit) multiply the deficiencies (1×3)
The left hand digit (tens digit) of the answer can be found in four different ways. Why do they all give the same answer?

1. Subtract 10 from the sum of the two given numbers (9+7=16, 16-10=6)
2. Subtract the sum of the two deficiencies (1+3=4) from 10 and you get 6.
3. Cross subtract (9-3=6)
4. Cross subtract (7-1=6)

This gives the answer 63.

Here are some more examples. Try some of your own.

9-1	8-2	9-1	8-2
6-4	7-3	9-1	5-5
5 4	5 6	8 1	4 0	Note: Here you have to express 5 times 2 as 1 ten and 0 units.

Sutra 1 - By one more than the one before

By one more than the one before

"Ekādhikena Pūrveṇa" is the Sanskrit term for "[by] One more than the previous one". It provides a simple way of calculating values like 1/x9 (e.g: 1/19, 1/29, etc). The sūtra can be used for multiplying as well as dividing algorithms.

Example: let's calculate 1/19. In this case, x = 1 . For the multiplication algorithm (working from right to left), the method is to start by denoting the dividend, 1, as the first (rightmost) digit of the result. Then multiply that digit by 2 (i.e.: x + 1 ), and denote that next digit to its left. If the result of this multiplication was greater than 10, denote (value – 10) and keep the "1" as "carry over" which you'll add to the next digit directly after multiplying.

The preposition "by" means the operations this formula concerns are either multiplication or division. [In case of addition/subtraction preposition "to" or "from" is used.] Thus this formula is used for either multiplication or division. It turns out that it is applicable in both operations.

Note: This sūtra can also be applied to multiplication of numbers with the same first digit and the sum of their last unit digits is 10.

An interesting sub-application of this formula is in computing squares of numbers ending in five. Examples:

35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25

or by the sūtra, multiply "by one more than the previous one."

35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25

The latter portion is multiplied by itself (5 by 5) and the previous portion is square of first digit or first two digit (3×3) or (12×12) and adding the same digit in that figure (3or12) resulting in the answer 1225.

(Proof) This is a simple application of (a + b)² = a² + 2ab + b² when a = 10c and b = 5, i.e.

It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Examples:

37 × 33 = (3 × 4),7 × 3 = 12,21

29 × 21 = (2 × 3),9 × 1 = 6,09

This uses (a + b)(a − b) = a² − b² twice combined with the previous result to produce:

(10c + 5 + d)(10c + 5 − d) = (10c + 5)² − d² = 100c(c + 1) + 25 − d² = 100c(c + 1) + (5 + d)(5 − d).

We illustrate this sūtra by its application to conversion of fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.

Method 1: example: using multiplication to calculate 1/19

For 1/19, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating decimal. (If the denominator contains only factors 2 and 5, the result is a purely non-circulating decimal, else it is a mixture of the two: a short non-circulating sequence of digits, followed by an endless repetition.) Each factor of 2 or 5 or 10 in the denominator gives one fixed decimal digit.

So we start with the last digit of the result, being the dividend:

1

Multiply this by "one more", that is, 2 (this is the "key" digit from 'Ekādhikena')

21

Multiplying 2 by 2, followed by multiplying 4 by 2

421 → 8421

Now, multiplying 8 by 2, sixteen

68421

1 ← carry

multiplying 6 by 2 is 12 plus 1 carry gives 13

368421

1 ← carry

Continuing

7368421 → 47368421 → 947368421

1

Now we have 9 digits of the answer. There are a total of 18 digits (= denominator − numerator) in the answer computed by complementing the lower half (with its complement from nine):

052631578

947368421

Thus the result is 1/19 = 0.052631578,947368421 repeating.

1
21
421
8421
68421 (carry 1) – we got 16, so we keep 6 and carry 1
368421 (carry 1) – we get 6*2 + carry 1 = 13, so we keep 3 and carry one

do this to eighteen digits (19–1. If you picked up 1/29,
you'll have to do it till 28 digits). You'll get the following
1/19 = 052631578947368421
     10100111101011000

Run this on your favorite calculator and check the result!

Method 2: example: using division to calculate 1/19

The earlier process can also be done using division instead of multiplication. We start again with 1 (dividend of "1/x9"), dividing by 2 (" x + 1 "). We divide 1 by 2, answer is 0 with remainder 1

result .0

Next 10 divided by 2 is five

.05

Next 5 divided by 2 is 2 with remainder 1

.052

next 12 (remainder,2) divided by 2 is 6

.0526

and so on.

Other fractions can sometimes be converted into the format of "d/x9"; as another example, consider 1/7, this is the same as 7/49 which has 9 as the last digit of the denominator. The previous digit is 4, by one more is 5. So we multiply (or divide) by 5, that is:

…7 → 57 → 857 → 2857 → 42857 → 142857 → .142,857 (stop after 7 − 1 digits)

Carry overs - 3 2 4 1 2

Next step into Vedic Mathematics...

Having known about Vedic Mathematics how about getting to know some of the basics of it... So here they are... Don't worry if u do not make a head or tail of them... We'll look into the details one by one in the later posts...

Below is a list of the sūtras, translated from Sanskrit into English: "By one more than the previous one" "All from 9 and the last from 10" "Vertically and crosswise (multiplications)" "Transpose and apply" "Transpose and adjust (the coefficient)" "If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero" By the Parāvartya rule "If one is in ratio, the other one is zero." "By addition and by subtraction." By the completion or non-completion (of the square, the cube, the fourth power, etc.) Differential calculus By the deficiency Specific and general The remainders by the last digit "The ultimate (binomial) and twice the penultimate (binomial) (equals zero)," "Only the last terms," By one less than the one before The product of the sum All the multipliers Subsūtras or corollaries "Proportionately" The remainder remains constant "The first by the first and the last by the last" For 7 the multiplicand is 143 By osculation Lessen by the deficiency "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of the deficiency)" "By one more than the previous one" "Last totaling ten" The sum of the products "By (alternative) elimination and retention (of the highest and lowest powers)" By mere observation The product of the sum is the sum of the products On the flag

Vedic Mathematics - Number rules the universe

Hi...

Afraid of Mathematics??? or Want to master it?? Try Vedic Mathematics - get rid of the fear and master the subject... It is a really cool concept consisting of a list of 16 basic sutras, or aphorisms. The 16 sutras are cryptic in the beginning, but practise unleashes their true power...

You need to see Vedic Mathematics in action to appreciate it fully the many special aspects and features. The main points are:

1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example Vertically and Crosswise and By One More than the One Before. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on.

2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system.

3) Vedic Mathematics is a system of mental mathematics (though it can also be written down).

Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots).

This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along.