The naturalness of the number e can be understood through its rich mathematical properties and its relevance in various areas of mathematics and science. Let’s delve into the reasons why e is considered the natural number.

1. Compound Interest: One of the earliest applications of e can be found in compound interest. Imagine you invest a sum of money in a bank account that compounds interest continuously. The balance of your account after a certain period of time can be calculated using the formula A = P * e^(rt), where A is the final amount, P is the principal amount, r is the interest rate, and t is the time. The presence of e in this formula highlights its natural connection to growth and exponential functions.

2. Calculus and Differentiation: In calculus, the exponential function f(x) = e^x plays a fundamental role. It has a special property where its derivative is equal to itself, i.e., f'(x) = f(x). This property makes e the base for natural logarithms (ln), which are logarithms with base e. The logarithm with base e is intimately tied to the properties of differentiation and integration, making it a natural choice for many mathematical applications.

3. Euler’s Identity: Euler’s identity is often hailed as one of the most beautiful equations in mathematics. It combines five fundamental mathematical constants: e, π (pi), i (the imaginary unit), 1 (the identity element for multiplication), and 0 (the additive identity). The equation is e^(iπ) + 1 = 0. This stunning relationship between these key mathematical elements highlights the naturalness and interconnectedness of e within the mathematical realm.

4. Probability and Statistics: The natural logarithm base e is also prevalent in the field of probability and statistics. The exponential distribution, which describes the time between events in a Poisson process, is characterized by the constant e. Additionally, e is crucial in the study of continuous probability distributions and the calculation of probabilities using the natural logarithm.

5. Growth and Decay Processes: Many natural phenomena follow exponential growth or decay patterns. For instance, radioactive decay, population growth, bacterial growth, and the charging or discharging of capacitors in electrical circuits can all be modeled using exponential functions with base e. This natural occurrence of e in various natural processes further solidifies its significance as the natural number.

6. Limit Definition: The number e can also be defined as the limit of the expression (1 + 1/n)^n as n approaches infinity. This limit arises in calculus and is intimately linked to the concept of continuous compound interest. The fact that e is the limit of this expression suggests its inherent connection to continuous and smooth growth.

The naturalness of the number e stems from its prevalence in various mathematical and scientific contexts. Its appearance in compound interest, calculus, probability, growth processes, and its involvement in Euler’s identity all contribute to its status as the natural number. Through its unique properties and wide-ranging applications, e has solidified its place as a fundamental constant in mathematics and science.