MSC #1 – By One More Than the One Before

As we have done since the holiday season of 2019, we will publish one Math technique daily for 30 days. Each of this post, which shows a different approach to common problems, is taken from our book “30 Master Strategies of Computing” which we hope to print when this pandemic is over.

MSC #1 – By one More than the One Before

“By One More Than the One Before” is the first Vedic Math Sutra or word formula. It is applied in various ways: counting, arithmetic sequences, squaring numbers ending in 5, multiplying complementary numbers, division, divisibility tests, recurring decimal and other applications.

The different applications of this Sutra will be discussed extensively by MATH-Inic Philippines founder Virgilio Prudente in the opening session of the “Inspirational Maths from India, Year 4”webinars scheduled on Dec 4-5 and 11-12, 2021.  

When a number ending in 9 is a divisor or a denominator of a fraction, it would be easier to use its Ekadhika instead. The Ekadhika is “one more than the number before 9”. 2 is the Ekadhika for 19, 3 for 29, 4 for 39 and so on.

If we want to divide a number by 19, for example, we can use its Ekadhika 2 instead. If we want to convert 3/29 into its decimal equivalent, it is simpler to use 4 as divisor. Similarly, if we wish to determine if a number is divisible by 49, we can use an “osculator” of 5.

In our example, we want to get the quotient when 3584 is divided by 29. We will use the 3, the Ekadhila of 29 as the divisor using the following steps:

1. Separate the last digit of the dividend 4 by a remainder bar. Only one digit is separated because the working base of 29 is 30 which has only 1 zero. (We will explain base division in a later post).
2. Divide the first digit of the dividend by our Ekahika, 3 to get 1 with no remainder. 1 is the first quotient digit.
3. Add this 1 to the next figure of the dividend 5 to get 6.
4. Divide 6 by 3 to get the next answer digit 2. Again, there is no remainder.
5. Repeat steps 3 and 4: add 2 to the next dividend digit 8 to get 10; 10 divided by 3 will give 3 remainder 3.
6. Write down 3 as the 3^rd answer digit.
7. Place the remainder 1 before 3 to make it 13.
8. Add this 13 to 4 the last digit of the dividend to get 17, the remainder of the division.

Therefore 3584 ÷ 29 = 123 rem 17.

In order to understand how this method works, let perform the division using the conventional system:

 When we use 3 as the divisor, we are actually dividing by 30. Now, for every 30 we have one 29 and one remainder of 1. Similarly for every 60, we have two 29s and two 1s. Therefore, the remainder is always equal to the number of 29s, which is the quotient figure.

Now we can take only the first digit of 35, and see that we have one 30 or more precisely, one 29 and a remainder of 1. So we add 1 to to the 2^nd digit of 35 to get 6. This is the remainder when 29 is deducted from 35.

With the next dividend 68 or 60 + 8, we can easily see that in 60 there are two 29s and two 1s and one 8. The remainder is thus 8 + 2 or 10. In the final dividend 104, we have 100 + 4. Now if we divide 10 by 3 we will get 3 remainder 1. This is actually dividing 100 by 30, getting a quotient of 3 and a remainder of 10. This is the reason we added 13 to 4 in the last step of our Vedic Division.