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Ramanujan discovered remarkable formulas for calculating the value of pi with incredible precision. His work forms the foundation for many algorithms used today in computer-based calculations of pi.
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Known as the "Ramanujan number," 1729 is famous because it can be expressed as the sum of two cubes in two different ways: \( 1729 = 1^3 + 12^3 = 9^3 + 10^3 \). This discovery came from a casual conversation with his mentor, G.H. Hardy.
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Ramanujan introduced the concept of mock theta functions, which are special mathematical objects that extend ideas in modular forms. This groundbreaking work continues to influence mathematical physics and number theory today.
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Ramanujan developed the "circle method," which helped approximate the number of ways an integer could be expressed as the sum of smaller integers. This was pivotal in understanding number partitions and solving complex problems like Waring’s conjecture.
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Ramanujan expanded the theory of theta functions, which are key tools in mathematics and theoretical physics. His contributions are still used in advanced areas like string theory and quantum mechanics.
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Ramanujan’s work in this field provided the groundwork for future studies in algebra and geometry, particularly in relation to modular forms, which have applications in cryptography and theoretical physics.
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Ramanujan made significant advancements in the study of divergent series and hypergeometric series, providing insights that were both surprising and profoundly elegant. These contributions opened new avenues in mathematical analysis.
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