\(e\), defined by the limit \(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\) and approximating 2.71828, is characterized by its irrational and transcendental nature, indicating it cannot be depicted as a quotient of two integers nor as a solution to any non-trivial polynomial equation with rational coefficients.

John Napier, a Scottish mathematician and theologian, significantly contributed to the field of mathematics through his invention of logarithms, detailed in his seminal work *“Mirifici Logarithmorum Canonis Descriptio,”* (1614). This development facilitated the transformation of cumbersome multiplicative calculations into simpler additive operations. Napier’s logarithms laid the groundwork for the natural logarithm concept, where \(e\) emerged as the fundamental base that equates the value of the logarithm to unity.

One hundred years later, Jacob Bernoulli, Swiss, explored the concept of compound interest, leading to the discovery of \(e\)’s significance in financial mathematics. His investigations, as detailed in his posthumously published research “*Ars Conjectandi*,” (1713), demonstrated that continuous compounding yields a growth factor of \(e\), showing \(e\)’s utility in not only financial models but also in natural processes like population dynamics and radioactive decay.

References

- Napier, John (1614), Mirifici Logarithmorum Canonis Descriptio.
- Bernoulli, Jakob (1713), Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis.