Explore the fascinating mathematical constant 'e' in this comprehensive lecture by Tanvi Jain. Delve into the history, properties, and applications of this fundamental number, starting with its origins in compound interest calculations. Learn about logarithms, their development by John Napier, and their connection to 'e'. Investigate the relationship between 'e' and hyperbolas, and discover how it appears in calculus through the study of areas under curves. Examine the series expansion of 'e' and its role in exponential functions, including its unique property as the base of the natural exponential function. Understand the proof of 'e' being irrational, its significance in complex numbers, and its various occurrences in mathematics and science. This in-depth exploration covers historical context, mathematical derivations, and practical applications, providing a comprehensive understanding of this important mathematical constant.