Aryabhata I, the father of trigonometry and zero

Aryabhata I, the father of trigonometry and zero 
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One of the, if not the, greatest formulations in the human history is the Indo-Arabic system of numerals, that we currently use in most places. This elegant and simple system of numbering things, using just 10 symbols - 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 0 - it's an absolute beauty. And without it, had we been using the old western (Roman) system of numbering, with I, II, XI etc mathematics would certainly have taken a blow. 

This number system was first put in place by the great Indian mathematicians of the classical age of Indian history (during the Gupta dynasty). And arguably, much of the credit for formalizing this system can be given to Aryabhata I. 

Who was Aryabhata? 

Aryabhata, also called Aryabhata the first or Aryabhata the elder, to distinguish him from Aryabhata the second, is regarded as the greatest mathematician and astronomer of India'a classical age. Before him, there were great scholars like Baudhayana, Lagadha but it was Aryabhata who gave a formal touch to the subject of mathematics, founded trigonometry (especially the sine and versine functions), formally introduced 0 as a number and much more. 

Born in central India (although his birthplace is debated) in 476 AD, Aryabhata witnessed an age of tremendous scientific and mathematical progress in India, thanks to the patronage of the powerful Gupta emperors. He made use of this opportunity, joined the prestigious Nalanda University (an ancient seat of learning in eastern India) and authored some of the greatest mathematical works. 

Contributions to science and mathematics 

Without Aryabhata, many of the things we take for granted today, would not have been around. Whenever we dial a phone number, do some transactions or programming, use the sine function in calculus or just know that a new day has begun as soon as the clock struck 00:00 AM, we are unknowingly indebted to Aryabhata. 

As a mathematician, Aryabhata modified the already present Indian place-value system, and formalized it to make room for 0. This is open to debate, though, but it is usually accepted that Aryabhata was the first to develop the concept of Shunya in mathematics, which still serves as the Indian term for zero. 

Contributions in mathematics 

1. Zero and the place-value system: As already mentioned, Aryabhata modified and formalized the Indian number system. In the tabular arrangement of digits, he used the term Kha to denote zero. It is unclear though, whether Aryabhata crafted the symbol for 0, or it was done by some other Indian mathematician, but it is accepted that the symbol for zero is round because it is made to mimic the round depression in the sand left, after we take out a pebble from it (one of the ways the Indians loved to calculate). 

2. Discovery of the value of pi: Although pi, our beloved number, was discovered long before Aryabhata, and was used in cultures such as ancient Egypt, Aryabhata was the first to correctly estimate it to upto four decimal places, using his elegant and purely arithmetic approach. In his works, we find the value of π to be: 

 π = 62,832/20,000 = 3.1416 

3. Sine function and trigonometry: Although basics such as the Pythagorus Theorem existed before Aryabhata, it was he who virtually founded the discipline, giving rise to the very common functions sine, cosine, (Jya and Koti-jya). Using nothing other than Pythagorus theorem, Aryabhata wrote the first ever trigonometric tables, carefully calculating the values.

Using this great knowledge, later Indiana astronomers were able to estimate the relative distances of the sun and other planets, right from the earth itself.

4. Powered arithmetic series: Aryabhata formulated the equations for calculating the sum of any finite series of squares and cubes of natural numbers. The following equations appear in his work:

1^2 + 2^2 + 3^2 + ... + N^2 = N x ( N + 1 ) x ( 2N + 1 ) / 6,

1^3 + 2^3 + 3^3 + ... + N^3 = ( 1 + 2 + 3 + ... + N ) ^ 2 = { N x ( N + 1 ) / 2 } ^ 2

5. Aryabhata algotirhm and Diophantine equations: Aryabhata's work on the Diophantine equations (of the form ax + by = c) were referenced by later mathematicians like Bhaskara I, Mahavira and Brahmagupta. His method involved breaking down the first-order standard Diophantine equations into smaller and newer equations with smaller coefficients, thereby utilizing the method of continued fractions finally. 

Aryabhata referred to this method as Kuttaka (fragmenting into smaller pieces). This method is used in modern disciplines such as cryptology with great vigour. 

6. Ganita and the systematization of mathematics: Aryabhata used the term Ganita to refer to his mathematics, although Ganita was much more than that. It was essentially a philosophy of mathematics, which was subsequently carried by the Arabs of the Islamic empire to Europe. Many great early mathematicians were greatly influenced by this concept of Ganita, and Aryabhata's number system. Fibonacci, for example, had argued in the favour of the Indian number system as opposed to the Roman system, after virtually bringing this system to Europe. 

Contributions in astronomy 

Other than being a first-class mathematician, Aryabhata was also hugely interested in astronomy. Long before Newton would eomply the full might of mathematics to bind the heavens to our intuition, Aryabhata utilized simple observations and geometry to frame what would finally be the first truly Heliocentric theory of the solar system. 

1. Rotation of earth: Before Aryabhata, it was believed that the cosmic bodies like stars and planets themselves moved, while the earth stayed in the same place. But Aryabhata correctly explained the concept of terrestrial rotation, that the earth itself rotates on its axis daily, thereby causing the illusion of a rotating sky. In Aryabhatiya, he describes the development of this radical new theory, using the counterexample of asterisms. 

2. Explanations of eclipses: Aryabhata also gave a very detailed description of the eclipses, and established them as optical phenomena rather than events caused by pseudo-planetary nodes of the lunar orbit, called Ketu and Rahu. 

3. Geometric model of the solar system: Although Aryabhata's own model was geocentric, it contained the roots of Heliocentrism in it. In addition, Aryabhata developed an observationally accurate model of the solar system, which was also the first truly geometric model in the Indian subcontinent. This influenced Arabian thinkers of the middle ages, and helped them further improve the model. 

4. Times of rotational and revolution periods: Aryabhata estimated the time of earth's rotation to 23 hours 56 minutes 4.1 s, while the modern value is 23 hours 56 minutes 4.091 s! This would tell us how accurat and sophisticated his calculations were. 

He also estimated the time of earth's revolution to be 365.25858 days, as opposed to the modern value of 365.25636 days. These were the most accurate values for thousands of years. 

5. Midnight day: Although the prevalent Hindu model was to consider dawn to be the beginning of a new day, Aryabhata's genius was to first formally establish the notion of midnight days. 

Legacy 

Aryabhata's works Aryabhatiya and Arya-Siddhantam are widely famous in India, although the latter has not been recovered fully. It's only through commentaries on his work and references, that we can estimate what was in his second book. 

The Muslim invaders obviously ransacked the great Nalanda university, after defeating the Indian kings through the use of treachery (battle of Tarain, 1192). Thousands of scholars were beheaded and burnt alive by the invaders, while great books and texts were plundered and burned. This, according to many, destroyed much of the data on Aryabhata we could have recovered. 

However, Aryabhata, the immortal mathematical giant of India, continues to live today through his works, without which much of modern-day mathematics would have been obsolete. The Indian government named its first satellite Aryabhata in 1975, in the honour of this great personality. 

To translate a portion of his biography from a Bengali text: 

"It was the middle of 499 AD, and the great scholars of the Nalanda university had gathered together - to welcome a young 23-year-old, polite of speech, radiant as the midday sun, before whom, even the elderly thinkers seem to feel nervous. With the sound of conch shells, Aryabhata, pride of the _Aryaas, walked in front, lifted his hands to the midday sun, and began his great work Aryabhatiya, that would one day redefine how we view the universe itself...*

References 

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