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**Mathematics in India**

**Mathematics in India****Chapter: 6**

EXERCISE |

**1. How many fundamental operations were known to the ancient mathematicians? What are they?**

Ans: “There are five fundamental operations in mathematics,” the German mathematician Martin Eichler supposedly said.

“Addition, subtraction, multiplication, division and modular forms.”

**2. Name the Ancient Indian Mathematicians and their period, who worked in Geometry and Trigonometry. Do you find any similarity between the ancient mathematical concepts and the present day mathematical concepts of Algebra, Geometry, and Trigonometry that you study? (You may also refer to the literature given in the references).**

Ans: Ancient Indian mathematics boasts notable figures who made significant contributions to geometry and trigonometry. Aryabhata (476–550 CE) is renowned for his work “Aryabhatiya,” where he introduced concepts such as sine. Following him, Brahmagupta (598–668 CE) expanded on these ideas in his “Brahmasphutasiddhanta,” offering rules for operations involving zero and negative numbers, as well as advancements in geometry. Bhaskara I (c. 600 CE) systematically used the sine function, while Bhaskara II (1114–1185 CE), in works like “Lilavati” and “Bijaganita,” explored various mathematical concepts, including geometry and trigonometric applications. Madhava of Sangamagrama (c1340–1425 CE) is noted for his contributions to trigonometric series. The similarities between ancient and present-day mathematical concepts are striking; foundational ideas in trigonometry, such as sine and cosine, remain integral to modern mathematics. Additionally, the systematic approaches to problem-solving and the emphasis on rigorous proofs seen in ancient texts echo through contemporary algebra, geometry, and trigonometry. While notation and specific methodologies have evolved, the core principles established by these ancient mathematicians continue to inform and influence current mathematical thought and practice.

**3. (a) Do you think there is any difference in the process of performing the basic operations on numbers in the earlier period and the present system which you studied?**

Ans: Yes, there is a notable difference in the process of performing basic operations on numbers between earlier periods and the present system. In the past, calculations were primarily done manually using tools like abacuses or counting boards, relying heavily on rote memorization and manual techniques. Today, advancements in technology have transformed this process; we now use calculators, computers, and smartphones that can perform complex calculations almost instantly. Furthermore, educational approaches have evolved, incorporating interactive learning and visual aids, making mathematical concepts more accessible. The methods of calculation have also improved, with modern algorithms streamlining operations and enhancing precision. Overall, while the fundamental operations remain unchanged, the tools, methods, and educational practices surrounding them have undergone significant transformation, reflecting the advancements in technology and learning.

**(b) Which process do you feel is easier? Why? Discuss with your friends.**

Ans: I find that the modern process of performing basic operations on numbers is generally easier than the earlier methods. The use of technology, such as calculators and computers, allows for quick and accurate calculations, reducing the likelihood of errors that can occur with manual methods. Additionally, the availability of educational resources online makes it easier to understand concepts and access help when needed.

Discussing this with friends, we might agree that while manual calculations can enhance understanding of mathematical concepts, the efficiency and convenience of modern tools are invaluable, especially in our fast-paced world. Technology not only saves time but also enables us to focus on more complex problem-solving instead of getting bogged down by basic arithmetic. Overall, the modern system streamlines the learning process and makes mathematics more approachable for everyone.

**4. Write at least three terms used by ancient mathematicians and give their meanings:**

**(a) Addition.**

Ans: Aryabhata II defines addition as “The making into one of several numbers is addition”. The ancient name for addition is saṁkalita (made together). Other equivalent terms commonly used are saṁkalana (making together), miśraṇa (mixing), sammelana (mingling together), prakṣepaṇa (throwing together), saṁyojana (joining together), eki – karaṇa (making into one), yukti, yoga (addition) and abhyāsa, etc. The word saṁkalita has been used by some writers in the general sense of the sum of a series. In all mathematical and astronomical works, knowledge of the process of addition is taken for granted. Very brief mention of it is made in some later works of elementary character. Thus Bhāskara II says in the Li – lāvati – “Add the figures in the same places in the direct or the inverse order.” In the direct process of addition referred to above, the numbers to be added are written down, one below the other, and a line is drawn at the bottom, below which the sum is written. At first the sum of the numbers standing in the units place is written down, thus giving the first figure of the sum. The numbers in the tens place are then added together and their sum is added to the figure in the tens place of the partial sum standing below the line and the result is substituted in its place. Thus the figure in the tens place of the sum is obtained, and so on. In the inverse process of addition, the numbers standing in the last place (extreme left) are added together and the result is placed below this last place. The numbers in the next place are then added and the process continues. The numbers of the partial sum are corrected, if necessary, when the figures in the next vertical line are added.

**(b) Subtraction. **

Ans: **Aryabhata II (A.D. 950) defines subtraction as:** “The taking out (of some number) from the sarvadhana (total) is subtraction; what remains is called Śeṣa (remainder).” The terms utkalita (made apart), vyutkalana (making apart), śodhana (clearing), patana (causing to fall), viyoga (separation), etc., have been used for subtraction. The terms śeṣa (residue) and antara (difference) have been used for the remainder. The minuend has been called sarvadhana or viyojya and the subtrahend viyojaka. Bhaskara II gives the method of subtraction thus: “Subtract the numbers according to their places in the direct or inverse order.” The direct process is explained with the help of an example say, 1000 – 360. Six cannot be subtracted from the zero standing in the tens place, so taking ten and subtracting six from it, the remainder (four) is placed below (six), and this ten is to be subtracted from the next place. For, as the places of unit, etc., are multiples of ten, so the figure of the subtrahend that cannot be subtracted from the corresponding figure of the minuend is subtracted from ten, the remainder is taken and this ten is deducted from the next place. In this way this ten is taken to the last place until it is exhausted with the last figure. In other words, numbers up to nine occupy one place, the differentiation of places begins from ten, so it is known ‘how many tens are here in a given number’, and therefore, the number that cannot be subtracted from its own place is subtracted from the next ten, and the remainder taken.”

**(c) Multiplication.**

Ans: The common Indian name for multiplication is guṇana. This term appears to be the oldest as it occurs in Vedic literature. The terms hanana, vadha, kṣaya, etc., which mean ‘killing’ or ‘destroying’, have also been used for multiplication. These terms came into use after the invention of the new method of multiplication with the decimal place-value numerals; for in the new method, the figures of the multiplicand were successively rubbed out (destroyed) and in their places the figures of the product were written. Synonyms of hanana (killing) have been used by Aryabhata I (A.D. 499), Brahmagupta (A.D. 628), Śri – dhara (A.D. 750), and later writers. These terms also appear in the Bakhshali manuscript. The ancient terminology proves that the definition of multiplication was ‘a process of addition resting on repetition of the multiplicand as many times as is the number of the multiplicator.’ This definition occurs in the commentary of the Āryabhaṭi – ya by Bhāskara I. The multiplicator was termed guṇya and the multiplier as guṇaka or guṇakara. The product was called guṇana-phala (result of multiplication) or pratyutpanna (‘reproduced’, hence in arithmetic ‘reproduced by multiplication’). Brahmagupta mentions four methods: (1) gomutrikā, (2) khanḍa, (3) bheda, and (4) iṣṭa. Āryabhata II (A.D. 950) did not name the method and stated: “Place the first figure of the multiplier over the last figure of the multiplicand, and then multiply successively all the figures of the multiplier by each figure of the multiplicand.”

**(d) Division.**

Ans: Division seems to have been regarded as the inverse of multiplication. The common Indian names for the operation are bhāgahara, bhājana, haraṇa, chedana, etc. All these terms literally mean ‘to break into parts,’ i.e., ‘to divide’, except haraṇa, which means ‘to take away’. This term shows the relation of division to subtraction. The dividend is termed as bhājya, hārya, etc., the divisor is termed as bhājaka, bhāgahara or simply hara, and the quotient is called as labdhi, which means ‘what is obtained’ or labdha.

**5. Find from the literature the concepts in mathematics other than those discussed in this chapter developed by the Indian mathematicians.**

Ans: Indian mathematicians have made significant contributions to various mathematical concepts beyond those typically discussed in basic arithmetic. One of the most notable is the development of the decimal system and the concept of zero, which revolutionized numerical representation and calculations. Additionally, ancient Indian scholars like Aryabhata and Brahmagupta contributed to algebra, particularly in solving equations and understanding number theory. The concepts of trigonometry were also advanced by Indian mathematicians, with works detailing sine, cosine, and tangent functions long before their adoption in Europe. Moreover, the principles of combinatorics and the early forms of calculus can be traced back to texts such as the “Siddhanta Shiromani” by Bhaskara. These contributions not only laid the groundwork for future mathematical developments but also significantly influenced the global mathematical landscape.