Euler's constant

Why is e everywhere? It’s the natural rate of growth?

Think about this formula: if you add a tiny bit, many times.

$$\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n=e$$

The constant ( e ), approximately equal to 2.71828, is indeed ubiquitous in mathematics, and its prevalence is rooted in its fundamental properties, especially in relation to growth processes and calculus. Here are some key reasons why ( e ) appears in so many different contexts:

1. Natural Rate of Continuous Growth:

   • Compound Interest: The concept of ( e ) originally emerged in the context of compound interest. If you compound interest continuously, the limit of this process as the compounding frequency becomes infinitely large is ( e ).
   • Exponential Growth: In any process that grows continuously at a constant rate (like populations, investments, or radioactive decay), the base of the exponential function is naturally ( e ).

2. Calculus and Functions:

   • Derivatives: The exponential function ( e^x ) is unique because it is its own derivative and integral. This property makes ( e^x ) naturally arise in solutions to differential equations, which are fundamental to many physical and engineering problems.
   • Euler’s Formula: ( e ) is central in Euler’s formula ( e^{ix} = \cos(x) + i\sin(x) ), which is a cornerstone in complex analysis and has deep implications in fields like electrical engineering and quantum physics.

3. Statistics and Probability:

   • Standard Normal Distribution: In statistics, ( e ) appears in the formula for the standard normal distribution, a fundamental concept in probability theory and statistics.
   • Limiting Processes: As seen in bootstrap sampling, ( e ) naturally arises in the limit of sequences and series, especially those involving probabilities and growth rates.

4. Number Theory and Discrete Mathematics:

   • Euler’s Number and Pi: ( e ) is related to other fundamental constants like pi (( \pi )), especially in expressions involving complex numbers and trigonometry.

5. Natural Logarithms:

   • Logarithmic Functions: ( e ) is the base of the natural logarithm, ln. The natural logarithm has properties that make it more convenient and natural in many mathematical settings, particularly in calculus.

In summary, ( e ) is not just a number but a fundamental constant that naturally arises in many areas of mathematics, especially those involving growth, change, and complex patterns. Its properties make it an indispensable tool in various fields, from finance and economics to physics and engineering.