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Thursday, October 15, 2009

Vedic Mathematics Lesson 24: Simple Osculation

I am sure many of you are wondering if I made some kind of spelling mistake in the title of this lesson. Osculation is not exactly a common word, but it is not misspelt either. Osculation is actually a very useful mathematical operation, and I will explain why shortly. Moreover, in this lesson, we will just use the term "osculation" instead of the full "simple osculation". We will learn about other types of osculation in subsequent lessons, and in the meantime, we will use "osculation" to mean just the simple kind.

You can find all the previous posts about Vedic Mathematics below:

Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
10's Complements
Multiplication Part 1
Multiplication Part 2
Multiplication Part 3
Multiplication Part 4
Multiplication Part 5
Multiplication Special Case 1
Multiplication Special Case 2

Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Subtraction
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method
Digital Roots
Straight Division I
Straight Division II
Vinculums
Divisibility Rules

In the previous lesson, while discussing the divisibility rules for 13 (and then 17 and 19) that the divisibility rules bear similarities to that for 7. I promised to explain why in a later lesson. This lesson will contain that explanation. I promise!

But, in the meantime, let me first explain what exactly osculation is.

Osculation is the mathematical process of adding or subtracting the product of a whole number and the remainder of dividing a number by 10, to or from the whole number result of dividing a number by 10. Think back to the divisibility rules for 7. The first of these is that if the sum of five times the last digit of the number added to the number obtained by removing the last digit is divisible by 7, then the original number is divisible by 7. The number obtained by removing the last digit of the original number is actually the whole number result of dividing it by 10. Adding five times the last digit to this is essentially the process of positively osculating the original number by 5.

Similarly, the second divisibility rule was that if twice the last digit subtracted from the number left over after removing the last digit is divisible by 7, then the original number is divisible by 7. Thus, we could have simply said that if the result of negative osculation of a number by 2 is divisible by 7, then the number is divisible by 7.

A number can thus be osculated by any other number. And the osculation can be negative or positive. Positive osculation can be performed indefinitely on any number, and the results will start repeating themselves after some time. Negative osculation can be performed only until the result becomes zero or negative.

Let us try some osculations first to familiarize ourselves with the process. After that, we will examine their significance in the process of testing divisibility by numbers, especially for which most people don't even know divisibility rules exist.

56 positively osculated by 2 gives us 5 + 6x2 = 17. Continuing the osculation by 2, you get 1 + 7x2 = 15, then after that 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, and finally 17. As you can see, at that point, the results start repeating in a cycle.

Note that single digit numbers can also be osculated simply by padding it with a zero to the left. The right-most digit becomes the number itself, and the rest of the number left after the removal of the last digit is zero.

123 negatively osculated by 4 gives us 12 - 3x4 = 0. No further osculation is possible. Negative osculation of a single-digit number will always result in a negative result.

Note that repeatedly osculating a number positively by 1 will eventually give us the digital root of the number since the process is the same as adding each successive digit to the previous ones until we are left with a single-digit number which then repeats endlessly when the process is continued.

For instance, 287 positively osculated by 1 first gives us 28 + 7x1 = 35. This then gives us 3 + 5x1 = 8. Note that not only is 8 the digital root of 287, it is also the result of repeatedly osculating 8 positively by 1.

In Vedic Mathematics, there exists the concept of natural osculators for a number. Osculating a number by its natural osculators is the key to determining divisibility by numbers like 7, 11, 13, etc. So, the question now becomes, what are a number's natural osculators, and how do we determine them.

In Vedic Mathematics, the natural osculators of a number are called Vestanas. Every number that ends in 1, 3, 7 or 9 has two well-defined vestanas.

To find the natural positive osculator (positive vestana), follow the procedure below:

Step 1
  • For numbers ending in 1, multiply them by 9
  • For numbers ending in 3, multiply them by 3
  • For numbers ending in 7, multiply them by 7
  • Nothing needs to be done for numbers ending in 9

As you can see, the application of step 1 results in a number that ends in 9.

Step 2
  • Remove the 9 from the end of the number.
  • Increase the number left behind by 1. This is the natural positive osculator of the number

This is another application of the Ekadhikena Purvena sutra (By One More Than The Previous One).

Thus, for 13, step 1 gives us 39 and the natural positive osculator would be 4 (one more than the number left behind after removing the 9 at the end of the 39). Similarly, the natural positive osculator for 7 is 5 (step 1 results in 49, and application of step 2 gives us 5). Note that the natural positive osculator for 9 is 1, and the natural positive osculator for 3 is also 1.

It is immediately clear to the astute reader that one of the divisibility rules that were introduced for 7, 13, 17 and 19 in the previous lesson was actually positive osculation by the natural positive osculator of the number. Thus, a simple test for divisibility by any number that ends in 1, 3, 7 or 9 is to derive the natural positive osculator of the divisor, and then positively osculate the dividend by this osculator. If the result is divisible by the divisor, then the original dividend is divisible by the divisor.

Note also, that testing divisibility by using the digital root for 3 and 9 is actually a special case of testing divisibility using their natural positive osculator. As we have mentioned before, the natural positive osculator of 3 and 9 is 1, and positive osculation of a number by 1 eventually results in the digital root of the number.

To find the natural negative osculator of a number (negative vestana), follow the procedure below:

Step 1
  • Nothing needs to be done for numbers ending in 1
  • For numbers ending in 3, multiply them by 7
  • For numbers ending in 7, multiply them by 3
  • For numbers ending in 9, multiply them by 9

As you can see, the application of step 1 results in a number that ends in 1.

Step 2
  • Remove the 1 from the end of the number
  • The number left behind is the natural negative osculator of the number

Thus, for 13, step 1 gives us 91, and the natural negative osculator would be 9 (remove the 1 from the end of the number). Similarly, the natural negative osculator of 7 is 2 (step 1 gives us 21 and step 2 gives us 2).

The astute reader will once again have figured out that one of the divisibility rules introduced for 7, 13, 17 and 19 in the previous lesson was actually negative osculation by the natural negative osculator of the number. Thus, another simple test for divisibility by any number that ends in 1, 3, 7 or 9 is to derive the natural negative osculator of the divisor, and then negatively osculate the dividend by this osculator. If the result is divisible by the divisor, then the original dividend is divisible by the divisor.

If you calculate the natural osculators of many numbers, you will also notice that the sum of the natural positive and negative osculators of a number is the number itself.

The natural positive osculator of 7 is 5. Its natural negative osculator is 2. 2 + 5 = 7. Similarly, for 11, the two natural osculators are 1 and 10. For 13, the two natural osculators are 4 and 9. For 3, they are 2 and 1. For 9, they are 1 and 8, and so on. Thus, if you know one of the natural osculators of a number, you don't need to perform the 2-step procedure to derive the other natural osculator. Simply, subtract the known osculator from the number, and the result is the other osculator.

Going beyond 20, we can now derive divisibility rules for any numbers that end in 1, 3, 7 or 9 using this lesson. Take 47 for instance. The natural negative osculator of this number is 14. Thus, negative osculation of any number by 14 is the divisibility test for 47. Similarly, the natural positive osculator of 47 is 33. Thus, positive osculation by 33 is another divisibility test for 47.

Now you know (or can derive) at least two divisibility rules for every number that ends in 1, 3, 7 or 9. Believe me, people will be amazed when you derive these divisibility rules in your head and then tell it to them as if they should have known all along! The look on their faces is truly priceless!! The possibilities are endless!!!

You may think that being able to use osculation to test for divisibility is somewhat limiting because it applies only to divisors ending in 1, 3, 7 or 9. However, note that, apart from 2 and 5, every prime number ends in 1, 3, 7 or 9. Thus, osculation will help you derive divisibility rules for pretty much any prime number that exists. All you have to do for composite numbers is find a set of co-prime factors and test divisibility by them to determine divisibility by the composite number.

As the title of the lesson suggests, what we have learnt about in this lesson is called simple osculation. Obviously, as you can see from the example above, the natural positive and negative osculators of large numbers tend to get pretty large, and osculation by these large numbers may appear to be quite inconvenient (even though it would still be a lot more convenient that actually performing the division in most cases). We will extend this method further into something called complex multiplex osculation in subsequent lessons so that testing divisibility by larger numbers does not necessarily seem as inconvenient (though, obviously, nothing can make working with large numbers as convenient as working with smaller numbers).

In the meantime, be sure to practice osculation, and the derivation of the natural osculators of numbers thoroughly, so that you can do it in your head even for large numbers just as if you are deriving their digital roots. This will make for even more amazed audiences as you can work out not just the divisibility rules, but also the divisibility itself in your head! Good luck, and happy computing!!

2 comments:

Unknown said...

Wt if the unit place is 0?

Unknown said...

Wt if the unit place is 0?

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