TriNetre - Archive for August 09, 2003

(no longer updated)

After going through the basic introduction on Vedas and Vedic mathematics, let us look at multiplication this week. There are a couple of Sutras that help in multiplication. Here we look at the first of these Sutras "Nikhilam Navatascaramam Dasatah" which roughly translates to "all from 9 and last from 10". As we will see later on, this Sutra is more suited for a particular set of numbers and may becomes tedious in certain cases. Not to despair, a more general and more powerful and generic Sutra will be introduced in later weeks.

Let us take a simple example to illustrate the Sutra's working - 6 * 9.

• Step 1 - Decide on a base, a number that is a power of ten and that which is nearest to the numbers to be multiplied. In this case it's simple - 10.
  
  
• Step 2 - Put the numbers 9 and 6 above and below:
  9
  6
  
• Step 3- Subtract each of the numbers from the base (10) and write the remainder on the left hand side, separated by a "-" sign.
  9 - 1
  6 - 4
  
• Step 4 - Now, the final answer (the product) consists of two parts, one of the left hand side and one on the right hand side, separated by a vertical line.
• Step 5 - There are four ways to find the left hand side of the answer.
  1. Subtract the base (10) from the sum of the given two numbers (9,6). So, left = (9+6)-10 = 5
  2. Subtract the sum of the deficiencies (1,4) from the base. Left = 10 - (1+4) = 5
  3. Cross subtract deficiency on second row from the original number on first. Left = 9 - 4 = 5
  4. Cross subtract deficiency on first row from the original number on the second. Left = 6 - 1 = 5
• Step 6 - Vertically multiply the "deficiency" numbers to get the right hand side of the answer. However, one has to note that the number of digits allowed in the right hand side is equal to power of 10 that the base is. In our case, the base is 10, which is the first (1) power of 10. So we can have 1 digit in the right. If we have more than that, just carry it to the left hand side and add to the digits in the left hand side. So, right = 1*4 = 4

So answer:
(10)
9 - 1
6 - 4
------
5 / 4

---

Let us look at another example - 6*4

Step 1 - Base = 10

Step 2 -
6
4

Step 3 -
6 - 4
4 - 6

Step 5 -
(1)Left = (6 + 4) - 10 =0
(2)Left = 10 - (4 + 6) = 0
(3) Left = 6 - 6 = 0
(4)Left = 4 - 4 = 0

Step 6 - Right = 24, of which 2 will be carried to the left hand side and added to it. So Left = 0 + 2 = 2, Right = 4

Summing up
(10)
6 - 4
4 - 6
----------------
0 / (2)4 = 2/4
----------------

---

Let us take a larger number 73 * 98

Step 1 - Base = 100

Step 5 (1) Left = 73+98 - 100 = 71
Step 5 (2) Left = 100 - (27+2) = 71
Step 5 (3) Left = 73 - 2 = 71
Step 5 (4) Left = 98 - 27 = 71
In this case, Step 5 (c) seemed to be easiest to me.

Step 6 - Right = 27 * 2 = 54

So summing up:
(100)
73 - 27
98 - 2
-------
71/54

---

More examples - 88 * 75

(100)
88 - 12
75 - 25
--------------
63/300 = 66/00

98 * 75
(100)
98 - 2
75 - 25
-------
73/50


93 * 97
(100)
93 - 7
97 - 3
------
90/21

Home work And you thought homeworks were over in the school? :)Rather than solve a couple more problems, it would be more interesting to see how this Sutra actually works, using algebraic notiations and expressions. Game for it? I will post the solution next week.

Also next week - Special cases of Nikhilam multiplication

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