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.
Overview
Syllabus
The number ' e'
What is e? Is this just a number
The first hint of
How will you solve this? We will need to solve for the equation:
A question on compound interest
If the amount is compounded twice, each time at the interest rate of 50%.
Observations:
Let's see...
Logarithm and John Napier
So the problem of multiplication and division can be reduced to that of addition and subtraction.
Squaring a hyperbola
Coming to the early seventeenth century
Area of the region bounded by y = x", x = and x = a.
Similarly, the area of the region bounded by the curves y = x-",
A look at the hyperbola
The appearance of logarithm
The series for and the work of Newton
which implies ao = 0. Comparing the coefficient of y.
Derivative of an exponential function
Derivative of bx
The "natural "exponent Derivative of any exponential function is a constant multiple of the function itself.
The notation for the natural exponent
Is an integer?
rational or irrational
Since is not an integer, q 2 2. Hence q + 1 2 3.
Euler's proof of the irrationality of
The exponential function on complex numbers
Occurrences of
e'" Paradox
Taught by
International Centre for Theoretical Sciences
