Ancient Indian Mathematical Treasure is a new series of articles by Salil Sawarkar, introducing the rich mathematical heritage of our country. This is the fifth article in the series. Read the first four articles here, here, here and here.
As one starts playing with numbers, various interesting patterns start showing themselves. For example, if one considers only the odd numbers and starts adding those (right from 1), then one starts getting square terms. For example, 1 = 12, 1 + 3 = 4 = 22, 1 + 3 + 5 = 9 = 32, and so on. As a consequence of this, if one goes through the table of squares, one observes that the addition of square of a number, twice the number itself and the number one (1) equals the square of the next number. For example, 102 = 100. Add 2×10 (= 20) and then 1 to it. We get 121, which is the square of 11. Similarly, 142 + (2×14) + 1 = 225 = 152. The difference of squares of two numbers equals the product of addition of the numbers and the difference of the numbers. For example, 122 – 92 = 63, which equals, 21×3 = (12 + 9)(12 – 9). The cubes also show some nice patterns. The addition of cubes of numbers (starting from 1) equals square of the addition of the numbers. For example, 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2. The story of such fascinating results doesn’t end here. It goes on and on! Of course, any matriculate reader must have realized that they have seen such results in high school or college level mathematics texts. Indeed, such results are stuffed in their book, called Algebra!
Whence has this word appeared in English? A few have shown the connection of this word with the name of some Arabic text, ‘Al Jebar’ or something similar. But what’s the origin of this text or its name? Recall that in the last article, we talked about one of the greatest mathematical gems, Aryabhata. Note that, the य (y) in Sanskrit gets corrupted and becomes ज (j) in regional languages, for example, योगि-जोगी (yogi-jogi). The र (r) gets corrupted and becomes ल (l) in Sanskrit itself (‘रलयोरभेद:’, says Panini). A study of such corruptions gives us a hint that it could be the Aryabhata or the Aryabhateeya, which became Algebra, in the due course of time. And if you still doubt, let us cast a glance at the contribution of Indians in the field of Algebra. At least it will help you get some ides of the immense knowledge of Algebra, which we had.
The examples of addition of numbers or the addition of cubes of numbers fall under the category of ‘Progression’ or ‘Sequence’ of numbers. Indians had a vast knowledge of obtaining any required term, situated at any place in a given progression. The old treatises also tell us that obtaining the sum of any number of terms in a given progression was also known to Indians. Here Aryabhata says,
इष्टं व्येकं दलितं सपूर्वमुत्तरगुणं समुखं मध्यम्।
इष्टगुणितमिष्टधनं त्वथवा आद्यन्तं पदार्थहतम्॥
(ishtam vyekam dalitam sapoorvamuttaragunam samukham madhyam|
ishtagunitamishtadhanam tvathavaa aadyantam padaardhahatam||)
Here Aryabhata is talking about what is now known as Arithmetic Progression. In such a progression, the difference between the consecutive terms is constant. (For example, a sequence like 5, 8, 11, 14, 17,… is an arithmetic progression, for the difference between consecutive terms is constant; here it is 3.) Mathematically, a sequence is always infinite. However, one may consider only a finitely many terms of such a sequence and then may think about the sum of these terms. According to the above verse, the average of such terms is a + d(n-1)/2, where a is the first term, n is the number of terms and d is the common (constant) difference. In the same verse, he also talks about the sum of the terms, but let us verify the first formula. Consider the numbers: 5, 8, 11, 14, 17. Here, a = 5, n = 5 and d = 3. So, a + d(n-1)/2 = 5 + 3(5-1)/2, which is 11. Now 5 + 8 + 11 + 14 + 17 = 55, and then dividing 55 by 5 yields 11, which is the same answer that we got by Aryabhata’s method.
Aryabhata also says that the sum of the terms in a given progression is “this average multiplied by the number of terms” (which is obviously true) or the sum equals “the addition of the first and the last terms, multiplied by half the number of terms”. If we add 5 and 17 and then multiply the sum by 5/2, then we do get the answer 55, which is the sum of the numbers in the given progression.
There is a story of the great mathematician Gauss that he had found out the sum of first hundred integers (1 + 2 + 3 + … + 100) in just five minutes, when he was in primary school. Gauss had observed that 1 + 100 = 2 + 99 = … = 50 + 51. So he multiplied 101 by 50 (that is, 100/2) and got the answer 5050. What Gauss had thought matches exactly with what Aryabhata had said thousands of years before Gauss (“the addition of the first and the last terms, multiplied by half the number of terms”)! This author has no intention of defaming Gauss, but it’s surprising and saddening that while many people know the story of Gauss, they have hardly any idea that the various formulae of progressions were given in Aryabhateeya.
To sharpen the intellect, it’s always advisable to try different twisted questions. This practice is being followed here since ages. For example, suppose the first term, the common difference and the sum of all the terms is given, can we find out the number of terms? Aryabhata considers this very problem and provides the required formula in the next verse. According to him,
गच्छ: अष्टोत्तरगुणिताद् द्विगुणाद्युत्तरविशेषवर्गयुतात्।
मूलं द्विगुणाद्यूनं स्वोत्तरभाजितं सरूपार्धम्॥
(gachchha: ashtottaragunitaad dvigunaadyuttaravisheshavargayutaat|
moolam dvigunaadyoonam svottarabhaajitam saroopaardham||)
Instead of translating this into English, it is better to write down the meaning as a formula. Note that from the previous verse, we have obtained s = (n/2)(a+l), where l is the last term. Noting that l=a+(n-1)d, one may write s = n(a + (n-1)d/2). Now from here, if one traces back carefully, one obtains n=1/2(1 + ((8ds + (2a-d)2) -2a1/2/d)) , which is the actual meaning of this verse. While ‘tracing back’, one encounters the quadratic equation n2d + n(2a-d) -2s = 0. Reader might recall the famous formula for obtaining the roots of a quadratic equation. Aryabhata has used the same.
Aryabhata doesn’t stop here. He further gives the formulae to obtain the sum of the squares of numbers (12 + 22 + 32 + … + n2) and also the sum of cubes of the numbers (13 + 23 + 33 + … + n3). Note that none of these are simple arithmetic progressions and so obtaining the sum is not an easy task. And then it’s not only Aryabhata, who came up with these rules.
In Bhaskaracharya’s Leelavati, we do find the formulae for obtaining the sum of integers from 1 up to any number n and sum of squares as well as that of cubes of the integers from 1 up to n. Bhaskaracharya also talks about भूमितिश्रेढि, now known as Geometric Progression. In such a sequence, the ratio between the consecutive terms is constant. For example, a sequence like 3, 6, 12, 24, 48,… is a Geometric Progression, for the ratio between consecutive terms is constant; it’s 2 here. Bhaskaracharya provides a formula for finding the sum of any number of terms from a Geometric Progression. According to him,
विषमे गच्छे व्येके गुणक: स्थाप्य: समेऽर्धिते वर्ग:।
गच्छक्षयान्तमन्त्याद् व्यस्तं गुणवर्गजं फलं यत्तत्॥
व्येकं व्येकगुणोद्धृतमादिगुणं स्याद् गुणोत्तरे गणितम्॥
(vishame gachchhe vyeke gunaka: sthaapya: samerdhite varga:|
gachchhakshayaantamantyaat vyastam gunavargajam phalam yattat||
vyekam vyekagunoddhrutamaadigunam syaat gunottare ganitam||)
As before, we will directly write down the formula. If a is the first term, r is the constant ratio and if n is the number of terms, then the sum of such terms is a(rn – 1)/(r – 1). It’s the same formula that is taught in the high school or college level mathematics.
The difference between squares of two numbers equals the product of addition and difference of the two numbers. The sum of squares of two numbers equals the square of their difference plus twice their product. What we have written is simply, a2 – b2 = (a + b)(a – b) and a2 + b2 = (a – b)2 + 2ab. These rules are seen in Bhaskaracharya’s Leelavati, as follows:
राश्योरन्तरवर्गेण द्विघ्ने घाते युते तयो:।
वर्गयोगो भवेदेवं तयोर्योगान्तरा हति:॥
वर्गान्तरं भवेदेवं ज्ञेयं सर्वत्र धीमता॥
raashyorantaravargena dvighne ghaate yute tayo:|
vargayogi bhavedevam tayoryogaantaraa hati:||
vargaantaram bhavedevam dneyam sarvatra dheemataa||)
Bhaskaracharya is famous not only for his mathematical ability but also for an interesting style of writing mathematics. It is clearly seen that he had a profound knowledge of poetry and was capable of creating nice verses. Many of his problems are based on lovely ideas and they show his fondness for the beauty around. While many of the today’s mathematicians freely proclaim the prosaic nature of their own subject, this fellow strove hard to make his subject poetic. After the formulae about progressions are described, he asks about the number of samavrutta-s, ardhasamavrutta-s, vishamavrutta-s of a particular Chhanda (meter). Is it not interesting that he brought poetry in mathematics? And this style of Bhaskaracharya is seen everywhere in Leelavati. As we go through the text, we meet peacocks, beetles, elephants, swans, lotuses and even couples involved in love making!!! In a nutshell, Bhaskaracharya exhibited his ability, with ease, as a hard core mathematician as well as a tender hearted poet.
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