Three separate methods for squaring: A connective prospective on Lilavati, Vedic and Trachtenberg

In the twelfth-century, Lilavati of Bhascaracharya-II contributed significantly making its occurrence up to the splendid history of mathematics, which has been used as a text-book for mathematics over 800 years in South Asian countries. However, in the nineteenth century, two independently developed mathematical systems namely, Trachtenberg and Vedic system of mathematics were significant all over. This paper mainly accentuates on tracing the connection of Lilavati to Vedic and Trachtenberg system of mathematics. This research is based on a comparison among the three treaties by their mathematical systems on squaring the numbers. All three mathematical treaties have their separate methods for finding the square. The Lilavati appears to have more or less influence on Vedic and Trachtenberg systems. Moreover, there are many similarities between the Vedic and Trachtenberg mathematical system in squaring the numbers.

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

Conceptual Relationship MapClose ×

AI-generated conceptual illustration (Three distinct ancient and modern mathematical manuscripts). It explains the idea and does not show study data.

Open full-size image ↗

Introduction

Mathematics carries a great history and is almost in par with human evolution. The History of Mathematics is the man-made science of human interest and reflects the growth of any civilizations perfectly. The history of Mathematics is a powerful tool for every learner as it suggests the evolution of new concepts. History of Mathematics will not only remind us of what we have but will also guide us as a strong future reference. Many mathematicians have contributed their lifetime and work ethics fortifying this beautiful subject. Bhaskaracarya-II from the twelfth century is one of the important figures regarding the contributions for the glorious history of mathematics. Lilavati, one of his fine creation gathered some remarkable attention as it encompassed some of the unsurpassed, easy, and pertinent procedures for solving many a mathematical problem from different fields. Besides these, Lilavati of Bhaskaracarya-II was the famous leading Mathematics books which have been used as a text-book for the last eight hundred years in south Asian country (Patwardhan et al. 2015, xvxxv). The mathematical tools developed after the upsurge of Lilavati of Bhaskaracarya-II directly or indirectly were certain to be affected by the procedures provided by him. Amongst them, the independently developed concepts of Vedic mathematics and the Trachtenberg system in the nineteenth-century were more likely to get influenced by the methods of Bhaskaracarya-II. In the process of developing mathematics, mental mathematics was also developed by many mathematicians. Among them, two independently developed mathematical systems Trachtenberg and Vedic were remarkable in the history of Mathematics. And this paper mainly emphasizes answering whether these two above mentioned concepts of Vedic mathematics and the Trachtenberg system in the nineteenth-century are affected by Bhaskaracarya-II's conception or not for squaring the numbers.

A glance of three mathematicians and their Mathematics

Many mathematicians contributed a significant role in making the glorious history of Mathematics. Among them, three mathematicians Bhascaracharya-II, Bharati Krishna Tirthaji, and Jakew Trachtenberg and their mathematics are more relevant for this article.

Lilavati of Bhaskaracarya

Bhaskaracarya or Bhaskara-II (1114 -1193) was the famous leading Mathematician, Poet, Astronomer of ancient Indian during the twelfth century. Bhaskaracarya wrote his Siddhantasiromani when he was 36. It consists of four parts: Lilavati, Algebra, Planetary motion, and Astronomy (Patwardhan et al. 2015, xix) [14] . The first part Lilavati mainly deals with Arithmetic but also contains portions of Geometry, Trigonometry and Algebra. Bhaskaracarya's Lilavati has been used as a text-book for the last eight hundred years in many south Asian countries and is still in practice in many Indian provinces (Patwardhan et al. 2015, xv xxv) [14] .

Squaring in Lilavati

There are three methods in Lilavati for squaring a number. The first method is written in Sanskrit as (Patwardhan et al. 2015, 19) [14] below.

समद्व िघातः क ृ द्व तरूच्यते ऽथ स्थाप्योन्त्यवर्गो द्व िर्गु णान्त्यद्व िघ्ाः । स्वस्वोपरिष्ठाच्च तथाऽपिे ऽङ्कास्त्यक्त्त् वान्त्यमु त्सार्य पु िश्च िाद्व िम् ।।

(Samadhighaataha kritiruchyateatha sthapyontyawargo dhigunantyanighnaha, Swoswoparisthachcha tathaapareankastyakttwantyamutsarya punascha raashim.) Meaning: First write the square of the extreme left-hand digit on its top. Then multiply the next (i.e., second) digit by the double of the first digit and write the result on the top. Next, multiply the third digit by the double of the first digit and write the result on the top. In this way arrive at the unit's place. Next, cross the first digit and shift the number so formed one place to the right. Then repeat the same procedure. Finally, add all the products written at the top and the sum is the required square (Patwardhan et al. 2015, 19) [14] , which is a general method for finding the square of the numbers.

The first method uses the formula (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca.

Example: To find the square of 396 = (300+90+6) 2 . By using the above formula, (396) 2 = 156816.

In Lilavati, the second and third methods for squaring the numbers is mentioned in Sanskrit verses as (Patwardhan et al. 2015, 19) [14] follows.

खण्डिर्स्यद्व िद्व िद्व तद्र ् द्व वद्व िघ्ी तत्खण्डवर्गै क्यर्ु ता क ृ द्व तवाय । इष्टोिर्ु ग्राद्व िवधः क ृ द्व तः स्याद्व िष्टस्य वर्गे ण समन्व ितो वा ।।

(Khandadhyasyavhihitirdhinighnee tatkhandawargaikyayutta kritirwa, Ishtonyugurashiwadha kritiha sydishtasya wargena samanwito waa.) Meaning of the first line: Split the given number into two parts. To the sum of the squares of the two parts add twice the product of the two parts. The result is the square (Patwardhan et al. 2015, 20) [14] , which is known as the second method of squaring in Lilavati. By second method, the formula is used as (a + b) 2 = a 2 + 2ab + b 2 . For the square of 396 by the second method, it is written as (300 + 96) 2 and getting the result = 156816.

Meaning of the second line: Add and subtract a suitable of the two numbers thus obtained and add the square of the suitable number chosen above. The result is the required square (Patwardhan et al. 2015, 20) [14] , which is known as the third method of squaring in Lilavati. The Sloka 1 (in Sanskrit) command: The number is added and subtracted to the given number, multiply them and add the square of the Ista 2 . If x is the number to be squared and choose the Ista number in such a way that (xa) and (x + a) can be easily multiplied.

Then x 2 = (xa) (x + a) + a 2 .

Example: To find the square of 396, Here, 4 is the best Ista number, such that (396 -4)  (396 + 4) can be multiplied easily.

Then, (396) 2 = (396 -4)  (396 + 4) + 4 2 = 392  400 + 16 = 156800 + 16 = 156816.

Bharati Krishna Tirthaji

A very renowned cultured personality of India, Jagadguru Sankaracharya Sri Bharti Krishna Tirthaji (1884 -1960) was the glorious and divine person. He is well known for his formulation and reconstruction of Vedic Mathematics from certain Sanskrit text Veda. According to him, the Vedic system is based on Sixteen Sutras and an equivalent number of sub-sutras cover all branches of mathematics, pure and applied. Vedic Mathematics was formulated and reconstructed by Tirthaji in the time of 1911-1918, was only published in 1965 after his death in 1960 (Tirthaji 2015, xxiii xxxi). Vedic Mathematics also relates to mental calculations.

Squaring in Vedic Mathematics

The Vedic method consists of two methods for squaring the number, they are specific and general. There are four methods including specific and general in Vedic Mathematics to find the square of a number. Among these methods, the first two methods belong to specific parts and the second two methods belong to general parts. These methods can be mentioned as (Tirthaji 2015; Singhal 2014; Thakur 2013).

1. Ekadhikena Purvena (एकाद्व धक े ि पू वे ण) 3 2. Yavadunam Tavadani Kritya Varganea Yojayet (र्ाविू िम ताविू द्व िक ृ त्य वर्गे ण र्ोजर्े त् ) 4 3. Dwanda Yoga (िन्द् ि र्ोर्ग)foot_0 iv) Urdhya Tiryagbhyam (उध्र्वद्व तर्य र्गभ्याम् ) 6 1 Sloka(श्लोक) = A couplet of Sanskrit Verse. (iii) Above two methods are useful for special types of numbers only, but there is another method i.e. Dwanda Yoga (or Duplex) which is general method, so it is applicable for all types of numbers. For this, we need know about the duplex of different digits numbers, which is denoted by D. (iv) The method of squaring a number by Urdhava-Tiryak method is nothing new but it is the same as the multiplication of two numbers by the same Vedic method.

Jakow Trachtenberg

Russian Jewish Engineer Jakow Trachtenberg (1888 -1953) invented a mathematical system when he was sentenced to prison (Cutler et al. 2008, 7 -17). He was sentenced as he was against the philosophy of Adolf Hitler during the time of the second world war. He was mostly known for the development of mental calculation techniques which today is widely known as the Trachtenberg method. The Trachtenberg system was developed around 1944-1945. Trachtenberg introduced his methods and started to experiment on his students in Switzerland in 1950 (Cutler et al. 2008, 7 -17).

His system was easy and still followed by many of the people.

The efforts he applied is remarkable and is comparable to that of Bharati Krishna Tirthaji.

Squaring in Trachtenberg System

Trachtenberg's mathematical system consists of two types, special and general. The specific type of system is restricted by special conditions whereas general doesn't consist of any type of barrier or conditions, i.e. general is applicable for every type of number. For squaring the numbers, the Trachtenberg system includes two special cases of numbers (Cutler 2008, 188 -197). The first case related to the numbers in which the unit digit number is 5, like 45 or 65 or 85, etc. and the second case related to the numbers in which the tenth digit number is 5, like as: 56 or 58 or 59, etc.

For the first case: Taking an example of 65 for squaring the first case numbers. The result consists of two parts: LHP and RHP such that RHP is always 25 (i.e. square of unit place number) and LHP is the multiplication of 6 and (6+1) i.e. 42. Hence the square of 65 is 4225.

For the second case: Taking an example of 58 for squaring the second case numbers. This case also consists of LHP and RHP parts. RHP is the square of unit digit number i.e. 64 and RHP is obtained by 25 + unit digit number i.e. 25 + 8 = 33. Hence the square of 58 is 3364.

Note: Above cases can be extended for a further number of digits.

The specific are those which are fast and effective but can be applied only to a particular case of numbers. But, the general pattern of squaring by the Trachtenberg system is applicable in all the cases of all types of numbers. To clarify the idea for squaring the numbers in general cases, it should be better to take an example for illustration.

For two-digits number: For squaring the two-digits number ab = (ab) 2 = (10a+b) 2 = 100a 2 +20ab+b 2 . But this result can be obtained mentally by Trachtenberg mathematical system as described stepwise (Cutler et al. 2008, 190 -193) below. Let's forward the number 73 for an example with stepwise.

1. Ann Cutler and Rudolph Mcshane, The Trachtenberg speed system of basic mathematics (English edition), New Delhi, India: Asia publishing house, 34 th impression, 2008, ISBN 0-285-62916-6.
2. B Mohan, Ganitka Itihas, Lakhnou, Up, India: Hindi samiti suchana bibhag, 1965.
3. Bharati Krishna Tirthaji, Vedic mathematics, New Delhi, India: Motilal banarsidass publishers Pvt Ltd, 2015, ISBN 978-81-208-0164-6
4. Bhaskaracharya, Bhascaracharya pranitam bijganitam, Choukhamba Bhawan, India: Avyktaganita pransamsa, 1992.
5. D Sengupta, 'Approach towards designing fast arithmetic circuits using ancient Vedic mathematics sutras', Jadavpur university: A published Ph D thesis by shodhganga, a reservoir of Indian theses, 2016. http://sodhganga.inflibnet.ac.in
6. James Glover, Vedic mathematics for schools -1, 2 and 3, New Delhi, India: Motilal banarasidass publishers Pvt Ltd, 2002. http:// www.vedicmaths.org
7. Parajuli KK, 'A mental mathematics-trachtenberg system', Sodh research journal. Kathmandu, Nepal: Nepal professor association, Nepal Sanskrit University 2012, 218-224.
8. Parajuli KK, Jha K, Acharya SR, Maskey SM. 'A research on efficacy of teaching Vedic mathematics in school level', International journal of mathematical sciences and engineering applications, Ascent publication, Pune, India 2020;14(1):35-41. ISSN 09739424. http://www.ascent.journals.com/ijmsea
9. Parajuli KK, Jha K, Acharya SR, Maskey SM, 'Connection of paravartya sutra with vedic and non-vedic mathematics', proceeding of 4 th international Vedic mathematics conference, IAVM publishing, 6 Barnfield gardens, Kingstone, Upon-Thames, UK: 2020, 20-25, ISBN 978-1-9161247-5-2. www.instavm.org
10. Kenneth William, The natural calculator, Scotland, UK: inspiration books, 2009.
11. Kenneth Williams, Discover Vedic mathematics, Scotland, UK: inspiration books, 2019. http:// www.vedicmaths.org
12. Kenneth Williams, Vedic mathematics teacher manual - 1, 2, and 3, Scotland, UK: inspiration books, 2019. http:// www.vedicmaths.org
13. Krishnaji Shankara Patwardhan, Somashekhara Amrita Naimpally and Shyam Lal Singh, Lilavati of Bhaskaracarya, New Delhi, India: Motilal banarasidass publishers Pvt Ltd, fourth reprint, 2015, ISBN 978-81208-1777-7
14. Rajesh Kumar Thakur, Essential of Vedic mathematics, New Delhi, India: Rupa publications Pvt Ltd, sixth impression, 2016. ISBN 978-81-291-2374-9
15. Vaidya SA. 'The contribution of Vedic mathematics in advance calculus': Shri jagadishprasad jhabarmal tibarewala university, a published Ph D thesis by shodhganga, a reservoir of Indian theses, 2019. http://sodhganga.inflibnet.ac.in
16. Vandana Singhal, Vedic mathematics for all ages, New Delhi, India: Motilal banarasidass publishers Pvt Ltd, second edition, 2014, ISBN 978-81-208-3230-5.