The Aryabhata Number System

This article was originally posted on the Varnam Substack.

In Computer Science, there are computations using binary or hexadecimal systems, but for most people, the common system is the decimal system. Indian mathematicians did not restrict themselves to one system for computation. During the time of Aryabhata, there were at least three methods of writing numbers. The most popular way of writing was using the Samskritam number system. Mathematicians like Varahamihira and Bhaskaracharya used a different system called the bhooth sankhya. Aryabhata, though, invented his own system which was a new contribution.

In the Aryabhata number system, the Samskritam letters from क to म carry values from 1 to 25. Letters from य to ह carry values 30, 40, 50… 80. Whenever an इ-kaara is used, the value is multiplied by 100. When an उ-kaara is used, the multiplier is 10,000, and ऋ-kaara multiplies it by 1,000,000. To illustrate with examples:

च = 6
चि = 600
चु = 60,000
च्र = 6,000,000
कुचि = कु + चि = 10,000 + 600 = 10,600

Reference: Aryabhateeya by Aryabhata (by Prof K S Sukla & Prof. K V Sarma. Commentary by Dr. N. Gopalakrishnan), Published by Indian Institute of Scientific Heritage, Thiruvananthapuram.

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Aryabhatiya of Aryabhata

The Aryabhatiya, written by the ancient Indian mathematician and astronomer Aryabhata, is one of the seminal works in the history of mathematics and astronomy. Composed in 499 CE, this treatise is a comprehensive collection of mathematical and astronomical knowledge that was prevalent during Aryabhata’s time.

Structure of Aryabhatiya

The Aryabhatiya is divided into four sections:

Gitikapada (13 verses):
This section includes a large amount of astronomical constants and tables. It sets the stage for the later sections by introducing fundamental concepts.
Ganitapada (33 verses):
The Ganitapada is the mathematical section of Aryabhatiya. It deals with arithmetic, algebra, plane trigonometry, and spherical trigonometry. Aryabhata discusses topics such as quadratic equations, proportions, and the properties of triangles. This section also includes instructions on using the decimal number system and zero.
Kalakriyapada (25 verses):
This section covers the reckoning of time, including various units of time measurement, and calendars. Aryabhata introduces his method of calculating the length of the year, the month, and the day, and discusses the concepts of yugas (epochs) and kalpas (aeons).
Golapada (50 verses):
The Golapada deals with the geometry of spheres and the motion of celestial bodies. Aryabhata presents his astronomical theories, including the rotation of the Earth on its axis, which was revolutionary at the time. He also describes the positions and motions of planets, eclipses, and the phases of the moon.

Notable Contributions

Decimal System and Place Value:
Aryabhata’s use of the decimal system, including the concept of zero, was a significant advancement in mathematics. His methods for representing large numbers were innovative and laid the foundation for future developments in arithmetic.
Trigonometry:
Aryabhata’s work on trigonometry includes the introduction of sine (jya), cosine (kojya), and versine (utkrama-jya) functions. His trigonometric tables were accurate and useful for astronomical calculations.
Astronomy:
Aryabhata’s heliocentric model suggested that the Earth rotates on its axis, a groundbreaking idea that contradicted the prevailing geocentric models. He also accurately calculated the length of the solar year, the duration of a day, and the positions of planets.
Eclipses:
Aryabhata provided a detailed explanation of solar and lunar eclipses, attributing them to the shadows cast by and on the Earth, respectively. This explanation was a departure from mythological interpretations and represented a scientific approach to understanding celestial phenomena.

Legacy

The Aryabhatiya influenced a wide range of mathematical and astronomical work in both India and the Islamic world. It was translated into Arabic and subsequently had an impact on European astronomy and mathematics during the Renaissance. Aryabhata’s innovative approaches and profound insights established him as one of the most important figures in the history of science.

The Aryabhatiya remains a testament to the advanced state of mathematics and astronomy in ancient India and continues to be studied and respected by scholars worldwide.