Aryabhata I, II of mathematics of both.pptx

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1. Great Mathematicians: Aryabhata I & Aryabhata II Presented by: Kashif, Ayush, Ayushman, Tanishk

2. INTRODUCTION TO ARYABHATA I Aryabhata I, also known as Aryabhata the Elder, was an influential Indian mathematician and astronomer who lived in the 5th century AD . He was born in 476 AD in Kusumapura, Pataliputra which is now known as Patna in India. Aryabhata I work, particularly the "Aryabhatiya," is celebrated for its contributions to mathematics and astronomy. He is renowned for his pioneering use of the decimal system and the concept of zero, which had a profound impact on mathematics and science.

3. INTRODUCTION TO ARYABHATA II Aryabhata II , distinct from Aryabhata I , was an Indian mathematician who lived in the 10th century CE. He was born around 920 CE, and his specific birthplace is not as well-documented as Aryabhata I. Aryabhata II made significant contributions to the fields of mathematics, particularly in algebra and number theory. While less known than Aryabhata I , his work has left a lasting impact on the world of mathematics.

4. Aryabhata I – Early Life He lived during the Gupta dynasty, a period known for its significant contributions to Indian science, mathematics, and culture. Aryabhata I early life is not extensively documented, but he likely received his education and training in mathematics and astronomy in Kusumapura. He was influenced by the mathematical and astronomical knowledge of his time, which was characterized by a strong interest in the movements of celestial bodies and the measurement of time. Aryabhata I early environment and access to mathematical and astronomical resources played a crucial role in shaping his intellectual development. 3/11/2023 Great Mathematicians: Aryabhata I & Aryabhata II 4

5. Contributions of Aryabhata I Aryabhatiya: Aryabhata I most famous work, "Aryabhatiya," is a foundational text in Indian mathematics and astronomy. It consists of 121 verses that cover a wide range of mathematical and astronomical topics. Decimal System and Zero: Aryabhata I is credited with introducing the decimal system and the concept of zero (0) as a placeholder in mathematical calculations. These innovations were groundbreaking and had a profound impact on the development of mathematics worldwide. Trigonometry: Aryabhata I made significant contributions to trigonometry. He provided sine and versine tables and advanced the understanding of trigonometric concepts. His work laid the foundation for future developments in trigonometry.

6. Contributions of Aryabhata I Astronomical Calculations: Aryabhata I made precise calculations related to astronomy, such as determining the length of a year with remarkable accuracy. He also explained the causes of various celestial phenomena. Pi (π) Approximation: Aryabhata I "Aryabhatiya" contains an approximation for the value of pi (π), which he calculated to be approximately 3.1416, a reasonably accurate value for the time. Summation Formula: Aryabhata I invented formula for summation of series of squares & cubes.

7. Aryabhata I – Astronomical Contributions Aryabhatiya: Aryabhata I's seminal work, the "Aryabhatiya," made substantial contributions to the field of astronomy. This text is divided into three parts: Ganita (mathematics), Kala-Kriya (time calculations), and Gola (celestial sphere). Year Length Calculation: Aryabhata I is known for his remarkably accurate calculation of the length of a year. He determined that a year is 365 days, 6 hours, 12 minutes, and 30 seconds. This was a highly precise estimate considering the tools and knowledge available during his time. Position of Celestial Bodies: Aryabhata I's work included detailed explanations of the positions and motions of celestial bodies, including the sun, moon, and planets. He contributed to the understanding of how these bodies move through the heavens.

8. Aryabhata I – Astronomical Contributions Eclipses: Aryabhata I provided a theoretical framework for explaining solar and lunar eclipses. He understood the role of the shadow of the Earth in lunar eclipses and the moon's position in solar eclipses. Zero and Place Value System: Aryabhata I's use of zero as a placeholder and the development of the decimal place value system were significant in astronomical calculations. These innovations greatly improved precision in recording and calculating celestial positions. Influence on Indian Astronomy: Aryabhata I's astronomical contributions influenced Indian astronomers for centuries. His work laid the foundation for future developments in Indian astronomy and had a lasting impact on the understanding of the cosmos.

9. Birth and Time Period: Aryabhata II, often referred to as Aryabhata the Younger, lived in India during the 10th century CE. His specific birth date is not as well-documented, but it is estimated to be around 920 CE. Geographical Origin: Aryabhata II's exact place of birth is not well-documented, but he is believed to have lived in the Indian subcontinent, possibly in the region that is now part of India. Cultural and Historical Context: Aryabhata II lived during a period of significant historical and cultural development in India. By the 10th century, India had seen the rise and fall of several dynasties, and it was a time of rich intellectual activity. Educational Background: Little is known about Aryabhata II's early education, but it is likely that he received training in mathematics and astronomy. He would have been influenced by the mathematical and scientific developments of his time. Legacy and Recognition: Aryabhata II is considered a significant figure in the history of Indian mathematics, particularly in the areas of algebra and number theory. While he might not be as well-known as Aryabhata I, his contributions left a lasting impact on the field. Aryabhata II - Background

10. Algebraic Contributions: Aryabhata II is celebrated for his significant contributions to the field of algebra. He made advancements in solving polynomial equations and quadratic equations, which played a pivotal role in the development of algebra in India. Number Theory: Aryabhata II's work extended into number theory, particularly in the study of divisibility, prime numbers, and the properties of integers. His contributions helped lay the foundation for further exploration of these topics. Diophantine Equations: He delved into Diophantine equations, which involve finding integer solutions to polynomial equations. Aryabhata II's work on these equations contributed to the broader field of number theory and the study of integer solutions. Aryabhata II - Mathematical Achievements

11. Continued Fractions: Aryabhata II is known for his work on continued fractions. He developed methods for approximating square roots and other irrational numbers through the use of continued fractions, which is a valuable mathematical technique. Legacy in Mathematics: Although Aryabhata II is less known than some other Indian mathematicians, his work in algebra and number theory has had a lasting impact on the field of mathematics. His contributions were built upon by later mathematicians and have influenced the development of mathematical knowledge. Aryabhata II - Mathematical Achievements

12. Astronomical Calculations: Aryabhata II made notable contributions to the field of astronomy during his time in the 10th century CE. He provided calculations and explanations for various astronomical phenomena. Planetary Movements: Aryabhata II work included an analysis of the movements and positions of celestial bodies, including the planets. He contributed to the understanding of planetary orbits and their motion in the night sky. Star Positions: Aryabhata II is believed to have worked on cataloguing the positions of stars and understanding their behaviour. This would have been crucial for both astronomical and astrological purposes in his era. Eclipses and Lunar Motion: He may have provided explanations for solar and lunar eclipses and studied the motion of the moon. His work could have included predictions and calculations related to lunar phases and eclipses. Aryabhata II - Astronomy

13. Both Aryabhata I and Aryabhata II have left an indelible mark on the history of Indian mathematics and astronomy. Their contributions have enriched our understanding of these fields and continue to inspire scholars and mathematicians to this day. CONCLUSION

14. Thank You Presented by: Kashif, Ayush, Ayushman, Tanishk 2/2/20XX Great Mathematicians: Aryabhata I & Aryabhata II 14