A cute proof that makes e natural

a year ago

The article explains the natural properties of the exponential function, particularly focusing on the base 'e'.
Key properties include: the slope of the tangent line at any point on the curve y=e^x is equal to the function's value at that point, and the expression (1 + 1/n)^n approaches e as n grows.
Geometrically, all exponential functions with positive real bases are horizontal stretches of each other, similar to how ellipses are stretches of one another.
The base 'e' is defined as the unique positive real base where the tangent line at the y-intercept has a slope of 1.
An easy approximation method is provided to estimate e by considering the slope of the tangent line to the curve y=2^x at its y-intercept and stretching it horizontally by a factor of 1/ln(2).
The article demonstrates that the slope of the tangent line to y=e^x at any point (x, e^x) is e^x, which is why e^x is its own derivative.
The compound interest limit definition of e is reconciled with the geometric definition by showing that (1 + 1/n)^n approaches e as n grows, using visual proof involving inverse functions and reflections.
The proof concludes by showing that the slope of the tangent line to the natural logarithm function ln(x) at x=1 is 1, which is consistent with the definition of e.

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