# Evaluating the Ancient Indian Method of Mathematics

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The mathematics of ancient India are some of the oldest and most complex in the world. Developed over centuries, they contain many sophisticated concepts that are still used today. Some of the most important aspects of ancient Indian mathematics include trigonometry, calculus, geometry, and probability. Many of these concepts were previously unknown to European mathematicians, and their discovery helped to shape the development of modern mathematics.

Mathematics as a discipline was studied from ancient times. Its roots in India can be traced back to the Indus Valley Civilization. The ancient mathematics of India prospered under notable mathematicians of the past. The contributions made by these exemplaries are indelible. It was pure passion and a deep interest in mathematics that led to important discoveries and inventions in ancient India. We get evidences of its early practices through manuscripts and texts. The research and views on mathematics conducted in India travelled far and wide until they developed into the current discipline of mathematics.

This article brings to you the rich ancient history of mathematics in India, sourced from various trusted scores. Right from its origin, important texts of mathematics to the earliest extant evidence, you can find all the information you need. This brings us to the beginning of a reading journey that is going to be quite interesting and engaging.

## Origin and The Earliest Evidence of Mathematical Application in Daily Life

• Archaeological excavations in the Indus valley Civilization brought forth the early instances of the use of mathematics in daily life. The earliest evidences of mathematics in India can be extracted from the wheel of a bullock cart which were depicted in artistic works. The bullock carts of the Harappan civilization had a metallic band wrapped around the rim of a wheel. It clearly hints at the fact that people were aware of the ratio of the circumference of the circle to the diameter and the pi system. All these facts help us conclude that Harappan people had the basic knowledge of geometry.
• A standardized system of weights (O’Connor Robertson) was used for measurement by the inhabitants of the civilization. These weights corresponded to the ratios of 1/20, 1/10, 1/5, ½, 1,2,5,10,20,50, 100, 200 and 500. Their enhancement in weights and measures led to the development of trade and commerce. Weights were produced in mass numbers and were of different geometric shapes like Hexahedra, barrels, cones, cylinders etc.
• The Mohenjo-Daro ruler stands testimony to the accurate use of the standardized measure of length. It was divided into 10 equal parts of 3.4 cm (1.32 inches) each. The dimensions of the bricks used in Mohenjo-Daro often were integral multiples of this unit of length (3.4 cm or 1.32 inches). Moreover, the bricks used for construction were made with the dimensions of 4:2:1 ratio. The dimensions were conventionally followed to ensure the stability of the bricks.

## Vedas and Mathematics

• The religious texts during the Vedic period contain the use of large numbers extending up to numerical like 1012. Vedas are the earliest literary source that includes mathematics and astronomy. The term “ganita” which means “the science of calculation” is found in Vedic works. Such Vedic texts include the mention of geometry, pythagorus, pi estimations etc.
• The Shulbasutras contains instructions for constructing fire altars. According to the Sutra, if any ritual sacrifice is successful, then the fire altar is to be built with precise measurements. The fire altar needs to be constructed with 5 layers of burnt bricks with each layer having 200 bricks. However, another important rule was that no two adjacent layers must coincide or superimpose one another.
• The Vedas consists of the 16 mathematical formulae as claimed by the Indian monk Bharati Krishna Tirtha in his book “Vedic Mathematics”. According to the author and monk, these formulae are present in the Atharva Veda. However, these claims have been refuted by scholars as they were not substantially evident.

## Notable Ancient Mathematicians and Their Contributions to Mathematics

Indian mathematics has rice sources that plays a central role in the development of history. Let us look into the details of the renowned mathematicians and their significant contributions to the field of mathematics

### Aryabhatta

Aryabhatta was a notable Indian mathematician and astronomer in the Golden Age (5th century). He made several developments in mathematics which he preserved in his books Aryabhatiya and Arya Siddhanta. The former is still extant while the latter is not traceable. His books discuss different topics on mathematics like square and cube roots, mensuration, value of pi, trigonometry, geometry etc. The concept of Zero is implicit in his works as per the French mathematician Georges Ifrah.

### Brahmagupta

• Brahmagupta was an Indian mathematician born in the 6th century. He was the first person to lay down rules stating the computation of zero.
• The primary book written by Brahmagupta is Brāhmasphuṭasiddhānta (c. 628 CE). It is a theoretical treatise on mathematical astronomy with 26 chapters. The contents of the text are present in the form of elliptic Sanskrit verses as were popular during those days. The other text written by Brahmagupta on mathematics and astronomy was Khaṇḍakhādyaka (c. 665 CE).
• Brāhmasphuṭasiddhānta contains a set of rules for working with negative and positive numbers, calculating square roots, solving linear and quadratic equations, summing series, the author’s identity and theorem. Moreover, the quadratic equations were clearly defined through the verses.

• Bhaskara I was an Indian mathematician and astronomer born in the 7th century. He is named Bhaskara I to differentiate him from the Indian mathematician (12th century) of the same name. Apart from Brahmagupta, he was the most renowned mathematician who made significant developments to the study of fractions. His most applauded contribution to the sphere of mathematics was the representation of numbers in a positional system.
• He set an example by writing numbers in the Hindu decimal system placing a circle for a zero. Moreover, he provided the commentary in his work Āryabhaṭīyabhāṣya, on the unique rational approximation of the sine function.
• Bhaskara also annotated Aryabhatta’s renowned work, Aryabhatiya. His other books on mathematical astronomy include the Mahābhāskarīya and the Laghubhāskarīya. The satellite Bhaskara was launched by ISRO in 1979 to honor the mathematician.

### Hemchandra

• Acharya Hemachandra, born in the 12th century in Gujarat, was a prodigy in multiple fields, including mathematics. He was named Candradeva at the time of his birth. The name “Hemachandra” was given to him later in life when he became a part of the Shvetambara sect of Jainism. A scholar of the excellent kind, he gained the title “kalikālasarvajña” which roughly translates to “omniscient one of the degenerate age”.
• He made contributions to the Fibonacci numbers half a century before Fibonacci introduced them in his book “Liber Abaci”.

## Important Discoveries in Ancient Indian Mathematics

• India is the first country to discover and use zero in mathematics. Calculus was already worked upon many years before it was actually used by its counterparts in other countries. Bhaskara, the noted mathematician of the 7th century already came up with various algebraic equations and methods. Scholars studied his techniques for a long period.
• Indian scholars in ancient times were deeply involved with the study of negative numbers. Brahmaputra in his notable text Brahmasphuta Siddhanta tried to demystify the concept of negative numbers. His elaboration on the subject dates many years before other mathematicians recognized its significance. His works have been translated into several languages and used as a main source for reference. Brahmasphuta presents the rules of addition and subtraction in case of negative numbers. Those rules acted as a fulcrum for development by later scholars.
• The concept of zero had its roots in Indian history. The concept emerged many years before it gained prominence in other parts of the world. The Bakshali manuscript which is the oldest manuscript in the world stands testimony to this fact. The use of zero made some major developments both in conceptual and applied mathematics.
• Although, Trigonometry as a branch of mathematics was born in Greece, the modern version was discovered by Aryabhatta. Thus, he laid the foundations of the modern trigonometry both in astronomy and general science. The fundamentals of Aryabhatta’s work travelled Europe all the way through Arab. The fractional multiples and trigonometric functions which we find today developed from his treatises.

## Oldest and Existing Manuscript of Mathematics

The oldest Indian manuscript that exists today is the Bakshali manuscript. It is not simply Indian’s oldest manuscript but is also the world’s oldest hand-written record on mathematics. This fact surely intrigues each one of us and makes us curious about what Bakshali manuscript has to offer. In brief, this old record consists of mathematical problems, their prosaic solutions, the concept of 0 and a lot more. So, let us look into the details of the Bakshali manuscript.

### Bakshali Manuscript

• The Bakhali manuscript was discovered in 1881 in Bakshali (in present-day Pakistan). It is an unfinished manuscript compiled in the 70 leaves of a birch bark. The script closely resembles Sharada script and the language is Sanskrit.
• It may date back to 224–383 AD or 885–993 AD as per carbon-dating studies. Also, scholars differ in their opinions regarding the period in which the manuscript was composed. Takao Hayashi claims the 7th century to be the age of the manuscript, whereas L. V. Gurjar proposes that it was written between the 2nd century BCE and 2nd century CE. Similarly, the author G.R. Kaye informs that the manuscript was written back in the 12th century.
• The mathematics in the manuscript is broadly divided into two portions. The first section consists of a poem that presents a problem economically. These problems were described in short due to the lack of writing equipment. The second section contains a prose commentary or a detailed solution of the problem. To be precise, an example is given followed by a statement stating the examples’ numerical in tabular form. In the next step, a detailed solution is provided which is followed by the justification of the solution.
• The Bakshali manuscript contains zero as a dot in the place-value calculation. The dot symbol came to be known as a shunya-bindu which literally means the dot of the empty place or dot of nothingness. The big dot was also alternatively used to denote the unknown. Fractions were not separated by a line as is the usage in recent times. The only similarity was that they were written one over the other. The concepts of equalization and square roots are equally present in the manuscript.

## Wrap Up

Ancient India was a breeding ground for scholars in mathematics. They created inventions that had the potential to revolutionise the world. Only their rich texts and old resources that provide information about them can help us track down these geniuses. This article offers a well-researched overview of ancient Indian mathematics. In every way possible, the scholars expanded and developed mathematics. Their contributions will undoubtedly inspire and be used for future generations.

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