**Vedic Mathematics Newsletter No. 139**

A warm welcome to our new subscribers.

This issue’s article is titled **Developing Tirthaji’s Osculation Notation** and describes how the notation can describe numbers more than one unit from a base number. “This makes it possible to test numbers for divisibility by several primes in one go – by mere observation.”

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**NEWS**

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**IAVM WEBINAR - CONSTRUCTIVE GEOMETRY IN ANCIENT INDIA – 15 ^{th} June**

Constructive Geometry in Ancient India dates back to the Vedic period, around 800 BCE, with the sulbasutras. These ancient manuals, part of the Kalpasutras and one of the six Vedan gas, detail geometrical procedures for constructing ritual altars. The Sulbasutras showcase advanced geometry, from simple perpendicular bisectors to complex altar designs like falcons and tortoises. In this talk, the speaker will explore how shapes were converted while maintaining equal areas, the approximation of the square root of 2, and how cord geometry laid the foundation for later developments in Indian mathematics.

Speaker : DR. VANISHRI BHAT

Asst. Professor, Chanakya University, Bangalore

**ONLINE ANNUAL CONFERENCE – CALL FOR PAPERS AND PRESENTATIONS**

This is organized by the IAVM and is scheduled to take place on 13^{th} and 14^{th} July 2024. Please contact the IAVM for details and to offer papers and presentations.

**NEW ONLINE JOURNAL ARTICLE**

The paper “Generality & Applicability of Vedic Polynomial Factorization: Quadratics (Simple & Homogenous), Cubics and GCF” by Riddhiman Jain (Class 9) and Shreyansh Jain (Class 7) of Anand Niketan Shilaj School, Ahmedabad is the latest addition to the Online Journal.

**4TH LILAVATI MATHS POETRY CONTEST - 24TH JUNE 2024 DEADLINE**

**DR KAPOOR DOCUMENTS**

Dr Kapoor continues to write articles on Vedic geometry. Phase 5 is concluded and Phase 6, on the Atharav Ved, is started. **See here**.

**ROTATION OF LINES**

We can rotate lines, to get the equation of the rotated line, by using triple addition as follows.

If the line is ax + by + c = 0

and the triple is p,q,r

then (ap - bq)x + (bp + aq)y + cr = 0 is the equation of the rotated line. That is, the line is rotated anticlockwise around the origin by the angle in the p,q,r triple.

This is the triple addition pattern:

a b c

__ p q r __ +

__ap-bq bp+aq cr__

For example, to rotate the line 2x – y + 1 = 0 anticlockwise around the origin by the angle in the triple 4,3 5 we have:

2x – y + 1 = 0

__ 4 3 5 __

__11x + 2y + 5 = 0 __

That is, 4×2 – -1×3 = **11**

-1×4 + 2×3 = **2**

1×5 = **5**

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**ARTICLE FOR NEWSLETTER 139**

**DEVELOPING TIRTHAJI’S OSCULATION NOTATION Kenneth Williams**

Special numbers are numbers which are close to a base, where a base is a multiple of a power of 10. So, 19, 301, 4999, 303 are examples of special numbers.

Because of this special feature such numbers are easy to work with. And where the special numbers have factors they can be very useful. For example, 301 has factors 7 and 43, but 401 is prime.

Tirthaji, introduces special numbers in Chapter 29 of his book and develops their notation in the following chapter. This notation allows us to describe numbers that are 1 below, or 1 above, a base number.

The numbers initially described with osculator notation are numbers like 19, 301, 4999.

**Tirthaji’s Notation**

The ‘Positive Osculators’ are for numbers 1 **below** a multiple of a power of 10, and we say:

*P*_{1} = 5 means the number 1 below 50, i.e. 49,

*P*_{2} = 5 means the number 1 below 500, i.e. 499

and, in general, *P _{n}* =

*N*indicates the number

*N*×10

*– 1.*

^{n}Similarly, the osculator *Q _{n}* =

*N*indicates the number

*N*×10

*+ 1.*

^{n}For example,

*Q*

_{1}= 5 means the number 51, and

*Q*

_{2}= 5 means the number 501.

This, then, is Tirthaji’s notation. Some more examples:

*Q*_{3} = 6 means the number 6001,

*P*_{3} = 6 means the number 5999.

We can translate these little osculator equations to numbers and vice versa.

Given 301 we can translate to *Q*_{2} = 3. And given *Q*_{2} = 3 we can write this as 301.

The notation is very useful as it means we can easily represent large numbers 1 above or 1 below a base. *P*_{7} = 5 for example is more compact than 49999999.

Numbers ending in 3 or 7 can also be represented with this notation since we can take multiples of such numbers so that they end in 9 or 1.

For 13, for example, we can say *P*_{1} = 4 since 13 ×3 = 39.

Or, also for 13, we can say *Q*_{1} = 9 since 7 × 13 = 91.

Though not discussed here in detail, these osculator equations can be manipulated and combined. The present article is to show an extension of the above notation that is generally suitable for numbers **more than 1** below or above a multiple of a base.

**Numbers More Than 1 Below or 1 Above the Base**

The osculator equations, as described above, are limited to numbers 1 unit from a base. And so they do not apply immediately to numbers like, say, 203 or 3998.

201 is represented as the osculator equation *Q*_{2} = 2.

203 is represented as the osculator equation 3*Q*_{2} = 2. The coefficient, 3, indicates the deviation of the number from the base.

So, 2*Q*_{1} = 3 gives the number 32;

4*Q*_{2} = 5 gives the number 504;

3*P*_{2} = 4 gives the number 397.

In general, *kP _{n}* =

*N*gives the number

*N*×10

*–*

^{n}*k*.

And *kQ _{n}* =

*N*gives the number

*N*×10

*+*

^{n}*k*.

When *k* = 1 we have the formulas quoted earlier, so this is an extension of that.

With this extended notation we can represent numbers like 203 and 3997 and so on; and ultimately we can express any number in osculator form.

We may also note the case where both sides of an osculator equation are multiplied by the same value, which has its own uses.

**An Application**

We may express the number 696 as 4*P*_{2} = 7. And to illustrate an application of this unique notation we may seek common factors of **696** and **203** by combining their osculator equations.

**4 P_{2} = 7**

3Q_{2} = 2

Cross-multiplying the coefficients and adding we find 7×3 + 4×2 = **29**, which is the common factor of 203 and 696.

This method can also be used to factorise numbers by using suitable special numbers with known factors. This makes it possible to test numbers for divisibility by several primes in one go – by mere observation.

End of article.

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Editor: Kenneth Williams

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10^{th} June 2024