1. The problem is to understand the mathematical constant $e$ and its significance. 2. The number $e$ is an irrational constant approximately equal to 2.71828. 3. It is the base of the natural logarithm and is used extensively in calculus, especially in exponential growth and decay problems. 4. The number $e$ can be defined by the limit: $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ 5. Another important representation is the infinite series: $$e = \sum_{k=0}^\infty \frac{1}{k!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots$$ 6. This constant appears naturally when calculating continuous compound interest, solving differential equations, and in many areas of mathematics and physics. 7. Understanding $e$ helps in solving problems involving exponential functions and logarithms. Final answer: $e \approx 2.71828$ is the base of natural logarithms and a fundamental mathematical constant.