iv

Page No.Part - 1 11. Mathematics in Vedas 32. Additions 63. Subtractions 94. Multiplications-1 125. Multiplications-2 156. Complements 167. Multiplications-3 178. Multiplications-4 239. Multiplications-5 3010. Multiplications-6 3711. Multiplications-7 4012. Multiplications-8 4213. Multiplications-9 47

Part - 2 5914. Vinculam Numbers 6115. Cube Values-1 7416. Cube Values-2 7817. Divisions-1 8418. Symmetry in the Quotient 8919. Divisions-2 9120. Divisions-3 9321. Divisions-4 9722. Multiplications-10 102

Part - 3 10723. Decimal number System in Vedas 10924. Measures of numbers 11725. Multiplications-11 12126. Multiplications-12 12727. Multiplications-13 13228. Multiplications-14 13429. Multiplications-15 13630. Multiplications-16 13831. Multiplications-17 14032. Multiplications -18 14433. Multiplications-19 147

Index

v

34. Multiplications-20 14935. Multiplications-21 15236. Akshahridayam 15337. Positional weightage system 15438. Infinity 15639. ‘AC’ and ‘HAL’ SYMBOLS 157

Part - 4 15940. Methodology for Expressing Numbers in Sanskrit 16141. Katapayadi System - 1 16242. Katapayadi System - 2 16443. Katapayadi System in Vedantasastra 16944. Katapayadi System in Sangita Sastra 17145. Magic Squares with Katapayadi Numbers 17246. Magic Squares with katapayadi Numbers 17547. Katapayadi System - 3 17848. Number of Planetary Revolutions with Katapayadi

System 18149. Logic Behind the Naming of Weekdaya in Sanskrit 18450. Vedic Numerical Codes / Algebraic Notation 18951. Matrhematics in Chandas Sastra (Prosody) 210

Part - 5 21352. Long term calender -1 21553. Long term calender -2 22054. Long term calender -3 23155. Multiplications - Lilavati Ganitam 23556. Method for validation of product of multiplication 24457. About Bhaskaracharya 24658. Divisions - 5 24859. Prime Numbers 25460. Reference to numbers like 19,29,39 and 49 in vedas 25561. Digits & Numbers -their Characteristics -1 25662. Mystic codes in our Philosophy 26163. Digits & Numbers -their Characteristics -2 26264. Divisibility 269

Part - 6 27365. Divisions - 5 27566. Divisions -6 27867. Divisions -7 28568. Divisions -8 286

vi

69. Ancient Mathematicians of India 28770. Some Contributions of Indian 289

Mathematicians71. Squares - 1 29272. Squares -2 29373. Squares -3 30174. Squares -4 30575. Squares -5 30976. Square Roots - 1 31277. Square Roots - 2 31478. Square Roots - 3 32579. Square Roots - 4 33180. Square Roots - 5 339

Part - 7 34781. Cubes - 3 34882. Cubes - 4 35783. Cubes - 5 36084. Cube Roots - 1 36385. Cube Roots - 2 36486. Cube Roots - 3 38387. Cube Roots - 4 38888. Cube Roots - 5 39589. Fourth Order Roots 40190. Fifth Order Roots - 1 40291. Fifth Order Roots - 2 403

3

1. Mathematics in Vedas

Introduction

Veda means “Knowledge”. It is the belief of Hindus that anykind of knowledge, either mundane or divine, can be tracedin the Vedas. It is also opined that the Vedas are eternal anddeal with the Reality. The Vedic Mantras, or the divinestatements, were discovered by the great Rishis during theSamadhi state of their penance.

Several Rishis were involved in the discovery of millions ofMantras, which were compiled by the Sage Veda Vyasa andwere segregated into four Vedas, viz., Rigveda, Yajurveda,Samaveda and Atharvaveda. The same were furthersubdivided into 1,131 branches. This information is found inSrimad Bhagavatham of Veda Vyasa, MahaBhashyam ofPatanjali and the commentary of Sri Vishnu Sahasra NamaStotra of Jagadguru Sankaracharya.

However, as on date, only 13 Vedic branches could belocated, after a search of past 150 years.Among these 13branches, only 7 branches are being studied. The rest of the6 branches are not having the teachers and students. Itmeans that more than 99% of Vedic literature hasdisappeared. Even this remaining part of Vedic literature istoo extensive to understand, as they deal with, in addition tothe main topics of Yajnas and Moksha, the physical sciences,biological sciences, Jyotisha, and Aeoronautics. They alsodeal with the subjects of Medicine, Structural Engineeringetc.

To understand the Vedas, it is a prerequisite to acquire theknowledge of Vedangas. Then only the meanings ofkeywords and other terms of Vedas can be followed. In thiscontext, it may be noted that the Vedas and Vedangas have

30

9. Multiplications-5

Sutra : Anurupyena

Meaning : Suitably / Proportionately

Details :

In all the examples taken so far, atleast one number is nearerto the base selected. Hence we were able to multiply veryeasily. But if the numbers are away from the base significantlythen how to go ahead?

When both the numbers involved in the multiplication areaway from the powers of 10, say 10, 100, 1000 etc., then themultiplications can be carried out with the help of two kindsof bases. One base is called Theoretical Base (TB) and thesecond one is called Working Base (WB).

The relation between WB and TB helps in finding the solution.

Example 1: 41˛41´?The given numbers are far away from their base 100. This100 can be taken as the TB. We can choose any roundednumber like 50 or 40 or 10 or any other number as the WB.

Method 1

1. Let us have the Theoretical Base as 100 and the WorkingBase as 50.

TB´100

WB ´ 502. Taking 50 as the reference, the given numbers can bewritten as follows.

61

14. Vinculam Numbers

Introduction :

The sign indicators of numbers involve the symbols of ({) or

(–). Examples are { 75, – 43 etc.

Here the positive sign ({) and the negative sign (–) areapplicable to the entire number containing a set of digits. Inthe above example of +75, the '+' symbol applies to both thedigits, viz., 7 in the 10's place and 5 in the units place. Similarlyin the example of -43, the '-' symbol applies to both the digits,viz., 4 in the 10's place and 3 in the units place.

The Vinculum method helps to attach the '+' or '-' symbols toany digit or a selected group of digits in the given number.

The word Vinculum in Latin language indicates the meaningof a chain or bond.

Normally the multiplication or division is convenient with digitsin the lower range, that is upto 5. The arithmetic operationswith numbers higher than 5 appear to be slightly inconvenient.The Vinculum method helps to convert the digits in the higherrange (>5) into lower range (<=5).

The numbers formed with the Vinculum method are calledVinculum numbers.

How to find out the Vinculum numbers

We can understand methodology of forming the Vinculum

numbers through illustrations.

Example 1 : To find out the Vinculum number for 6

6´10–4´ 14-

78

16. Cube Values-2

Sutra : Anurupyena

Meaning : Proportionately / as per the ratio

Details :

1. The Anurupyena sutra is useful in finding the cube valueof some numbers.

2. Let us take the given number as 'ab'

3. In the case, the digit a belongs to 10's place and b be-longs to units place.

4. To find (ab)3, we have to understand (a+b)3

5. (a+b)3 ´ a3+3a2b+3ab2+b3.

6. The above expression can be split into two rows.

First Row = a3 a2b ab2 b3

Second Row = 2a2b 2ab2

7. The sum of the above two rows will have the same

value as in step 5.

8. The terms in the first row are with the same ratio.

Ratio´ b3 ´ ab2 ´ a2b ´ b

ab2 a2b a3 a

9. The number of digits for each term will be decided

depending on the base.

If the Base ´ 10, each place will have one digit.

If the Base ´ 100, each place will have two digits.

109

23. DECIMAL NUMBER SYSTEM INVEDAS

Mathematics has earned the reputation as the queen of allsciences. Its significance was recognized by Indians long back.Lagadha, the great astronomer and mathematician of India in thehoary past (about 1500B.C.) states in his book, Vedanga Jyotisha,that:

Ü«∞ ä•tMÏ =∞Ü« ~å}ÏO <åQÍ<åO =∞}Ü≥∂ Ü«∞ ä• I« Œfi Õfi^•OQÆ âß„™êÎ}ÏO QÆ} «O =¸~°úx ã≤÷ «"£∞ II

Yathâ úikhâ mayûrânâm nâgânâm manayo yathâ

Tadvadvedâñga Úâstrânâm ganitam mûrdhani sthitam

“Like the crest of a peacock, like the gems on (the hoods of)snakes, so is mathematics at the top of all the Vedâñga Úâstras.”(Science and Technology in Ancient India, p45)

The contributions of India to mathematics are very remarkableand globally acknowledged. The concept of zero and place valuesystem of numeration are unique contributions which top the list.

The individual digits and the numbers of the decimal numbersystem, both in the lower range and upper range, are noticed atseveral locations in different Vedas as indicated in the followingsection:

ŒUKLA YAJURVEDA

1. A statement in Œukla Yajurveda reads as follows:

140

31. MULTIPLICATIONS-17

Topic: Multiplications with 11

Sutra :JO «ºÜ≥∂ˆ~=Antyayoreva

Meaning: Only at both the ends

Explanation:1. This is useful while multiplying the given numbers with 11.2. The first and the last digits of the given number will become

the first and last digits of the result also.3. The values obtained through addition of two digits at a time

of the given number from right to left will become the middledigits of the result in the same direction.

Example 1: 15*11=?

1. Given number=152. The right side (units place) digit of the result=The right side

digit of the given number=53. Present status:

Result= 54. The pair of digits in the given number from right to left are

5 and 1.5. Value obtained after addition=5+1=66. This value has to be written in the tens place of the result.7. Present status:

Result= 65

189

50. VEDIC NUMERICAL CODES/50. VEDIC NUMERICAL CODES/50. VEDIC NUMERICAL CODES/50. VEDIC NUMERICAL CODES/50. VEDIC NUMERICAL CODES/ALGEBRAIC NOTATIONALGEBRAIC NOTATIONALGEBRAIC NOTATIONALGEBRAIC NOTATIONALGEBRAIC NOTATION

The advantage and need of expressing knowledge in symbolicform was recognized in India several thousands of years back. Forthis purpose, two types of numerical codes had come into existence,viz., Bhûta Samkhyâ System, and Katapayâdi System. The detailsof both the systems are as follows:

BHÛTA SAMKHYÂ SYSTEM / SAMJÑÂ NIGHANTU

In this system, the numerals are expressed by objectstraditionally associated with some numbers. The stanzas describingthe numerals are as follows:

â◊j ™È=∞â◊≈âßOHõâ◊Û WO Œ∞â◊Û„#Ìó HõÖÏxkèó I~å*Ï q èŒ∞ã∞û è•Oâ◊√â◊Û Ü«∞=∞ UHõ[#ãÎ ä• II

JHΔ K«‰õΔΩó Hõ~À <Õ„ «O Ö’K«#O ÉÏÇïHõ~°‚HÍó IÑHõΔ Œ$+≤ì ŒfiÜ«∞O Ü«ÚQƇ=∞O|H“ #Ü«∞<ÕHõΔ}Ë I I

=Ç‘Ïfl ~å=∞t≈v Kåyfló áê=HÀ ŒÇÏ<å#Ö∫ Iâ◊OHõ~åHΔÑÙsÖ’HÍ„ã‘Î} HÍÅ„ãÎÜ≥∂QÆ∞}Ïó II

Jaú ™êQÆ~° K« åfii =#~åt~°∞ºQÀO|∞kèó IK« «∞~åfiiúQÆuâßÛÑ≤ [Åkèsfl~°kèãÎ ä• II

WO„kÜ«∞O ÑOK«=∞O *Ï#q∞+μ~åƒ}â◊Û =∂~°æ}ó I„= «O Éèí∂ «O â◊~°ó Ñ~åfi „áê}â◊Û q+Ü«∞ãÎ ä• II

âß„ãÎO +@Û ~°∞zâ‹ÂÛ= HÍÅâ◊Û |∞∞ «∞ãOlHõ"£∞ I~°ã„ Œ=ºO K« HÀâ◊â◊Û +_»Ì~°≈#+_®QÆ=∞ø II

215

52. LONG TERM CALENDAR -1

Topic : To find the weekday name, given the date

Explanation:

1. It is well known that the calendars display the relation betweenthe dates and weekday names. But the calendars are printedwith the details of one or two years generally. We do not getthe calendar of our required year when it is very remote past.Similarly we do not get the calendars of future years also,because they are neither printed nor accessible. With a viewto help in this kind of situations, few persons have attemptedand found out some methods. One among such attempts wasby Sri Vedagiri Subbarayudu of Kavali, Nellore District,Andhra Pradesh, in 1952.

2. In this method, three Tables were prepared linking centuries,years, months and dates with weekday names.

3. Two strings of letters are used for indices of the tables.

First string of 14 letters: A to N (viz., A B C D E F G H I JK L M N)( In Telugu, the string contains the letters equivalent toSRI YU TA VE DA GI RI RA MA CHAN DRA YYAGA RU )

The format of any given date can be “DD MM YYYY”.As shown here, the year is expressed with the help of fourdigits. The first two digits from left decide the century, andthe remaining digits indicate the year within that century.For example, the year 2008 has four digits. It belongs to

275

65. DIVISIONS -5

The European mathematicians uptp 15th century wereunder the impression that process of division is a complexoperation and that it requires special skills to accomplish it.But the division process was well known in India by that timeitself. In fact, Indians reached the stage of just giving a referenceonly, without mentioning the details.

They used to carry out their mathematical operations onsand slabs, as the facilities for writing were not extensivelyavailable. This method was known as ‘Paatii Ganitam’.

During their studies, the division was written as a fraction.Finding the factors for both the numerator and denominator andeliminating the common factors was called as ‘apavartana’. Thisis illustrated with an example.

Example : 2576”16´?2576”16´?2576”16´?2576”16´?2576”16´? Explanation Numbers to be divided Remainder Quotient Re-sult

Write ‘25’ of thegiven number2576 under‘Present’,and ‘76’under ‘Later’

Present status 25 76

Write 16under 25

Present status 25 76 16

Present Later

TOPIC: Paatii Ganitam

349

81. CUBES – 3

TOPIC:To find cube values of numbersã∂„ «O :1. ã=∞„uѶ∂ «â◊Û Ñ¶∞#: „Ñk+ì :

™ê÷áȺ Ѷ∞<ÀO «ºãº « ÀO «º=~°æ:Pk „uxѶ∞fl ãÎ « Pk=~°æ:„ «ºO`åºÇÏ`À ä•k Ѷ∞#â◊Û ã~fi II

2. ™ê÷<åO «~° Õfi# Ü«Ú`å Ѷ∞# ™êûº ü„ÑHõÅʺ « «öO_» Ü«ÚQÆO « ÀO «º"£∞ IU=O =ÚÇï~°fi~°æ Ѷ∞# „Ñã≤ •ú"å^•ºOHõ À "å qkè~+ HÍ~°º : II

1. Samatrighaatascha ghanah pradishtahSthaapyo ghanontyasya tatontyavargahAadistrinighnastata aadivargahTryantyaahatothaadi ghanascha sarve

2. Sthaanaantaratvena yutaa ghanassyaatPrakalpya tatkhandayugam tatontyamEvam muhurvargaghanaprasiddhaaVaadyaamkato vaa vidhiresha kaaryah

Summary:1. The cube of a number is obtained by multiplying three

numbers of equal value2. Let the given number be taken as ‘ab’.

Start analyzing the number from LHS.Identify the digit on the left hand side. Take it as ’a’. It isthe last one when we see from RHS. It is called as ‘antyanka’

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