Vedic Mathematics Newsletter No. 132
A warm welcome to our new subscribers.
This issue’s article is by Kenneth Williams, and is titled “Osculation Magic”.
“We can also jump forwards or backwards through the osculators, getting every 2nd, 3rd, 4th etc. one. And we can get specific osculators as required, or find the number of osculators in the cycle (which is also the number of digits in the recurring decimal for n/D). There are many other beautiful and astounding properties within these osculation cycles.”
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NEWS
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8TH ONLINE VEDIC MATHEMATICS CONFERENCE
This is scheduled for 12th, 13th March 2022. Organised by the IAVM it features original research papers, ancient Indian mathematics, Vedic Maths in education plus workshops for teachers and students.
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or visit the IAVM website at https://instavm.org/
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Math2Shine is a Singapore-based Ed-Tech company co-founded by Lokesh Tayal and Kenneth Williams. It is promoting Vedic Mathematics globally.
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We will start taking registrations shortly.
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ARTICLE FOR NEWSLETTER 132
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OSCULATION MAGIC
Tirthaji’s osculation technique has many applications: in divisibility testing, converting fractions to decimals, multiplications, divisions, checking calculations, continued fractions, number bases, substituting etc.
The osculators are explained in Tirthaji’s book (chapters 29 and 30) and here we look at how to generate the osculators in sequence from the first one.
In many of the applications we need to select a suitable osculator from the many that can be used.
For example, if we need to test if a certain number is divisible by 23, Tirthaji’s osculation technique allows us to work 2 figures at a time, while dividing by 3, if we know that 299 is a multiple of 23.
Or, we can work 6 digits at a time, dividing by 4, if we know that 3999999 is a multiple of 23.
How can we find these convenient multiples?
In fact, for whole numbers, D, that end in 1, 3, 7 or 9 we can always get a multiple that ends in one 9, two 9s, three 9s, or indeed any number of 9s.
There will even be some which consists entirely of a series of 9s.
In Tirthaji’s osculation technique we begin by finding a multiple of D that ends in 9. So in the case of D = 23 we multiply 23 by 3 to get 69. The osculator is then 7: one more than the number before the 9. We can say that if D = 23 then P = 7, where P is called the osculator.
For the number 23, the multiples that end in 1, 2, 3 etc. 9s are:
69, 299. 20999, 89999, 1699999, 3999999 and so on.
The osculators for these – one more than the number before the 9s are:
7, 3, 21, 9, 17, 4 etc.
Now, these can be obtained by repeatedly osculating by the 1st osculator (7 in this example).
Osculation consists of multiplying the last digit of a number by the osculator,
then adding on any other digits and
subtracting D if necessary.
To make this clear let us generate the osculators for D = 23. Those osculators are 7, 3, 21, 9, 17, 4 etc.
This is how it is done…
7×7 = 49 and 49 – 2×23 = 3; next we osculate this 3 with 7
3×7 = 21; next we osculate this 21 with 7
1×7 + 2 = 9;
9×7 = 63 and 63 – 2×23 = 17;
7×7 + 1 = 50 and 50 – 2×23 = 4.
And so on.
This means we can easily generate the osculators from the first one by repeated osculation. We just osculate until we get a suitable one that we can use.
Let us take another example: D = 49.
The first osculator is 5 (one more than 4).
Now osculate repeatedly with 5, casting out any 49s as needed: we get 5, 25, 27, 37, 38 etc. That is:
5×5 = 25;
5×5 + 2= 27;
7×5 + 2 = 37;
7×5 + 3 = 38;
8×5 + 3 = 43.
And so on. (No casting out of 49s is necessary up to here.)
This tells us that 49, 2499, 26999, 369999 etc. are multiples of 49.
The number of 49s contained in each of these multiples is also easily available by an application of the Sutra The Remainders By the Last Digit.
For D = 49 there are 42 osculators before they start to repeat (though we can generate multiples with any number of 9s at the end!), and we can generate them from right to left if we wish.
We can also jump forwards or backwards through the osculators, getting every 2nd, 3rd, 4th etc. one. And we can get specific osculators as required, or find the number of osculators in the cycle (which is also the number of digits in the recurring decimal for n/D).
There are many other beautiful and astounding properties within these osculation cycles.
End of article.
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Editor: Kenneth Williams
The Vedic Mathematics web site is at: https://www.vedicmaths.org
22 – 02 – 2022