What Is the Mathematical Constant 'e'?
Euler's number, denoted by the letter 'e', is a fundamental mathematical constant that serves as the base of the natural logarithm. It is an irrational number, meaning its decimal representation never ends or repeats, with an approximate value of 2.71828. The constant 'e' naturally arises in processes involving continuous growth or decay.
Section 2: The Origin of 'e'
The value of 'e' can be formally defined as the limit of the expression (1 + 1/n)^n as n approaches infinity. This concept originated from studying compound interest. As the frequency of compounding interest increases towards infinity (i.e., becomes continuous), the final amount approaches a value defined by 'e'.
Section 3: A Practical Example
Consider investing $1 at a 100% annual interest rate. If compounded just once a year, you would have $2. If compounded monthly, you'd have about $2.61. If compounded daily, it's approximately $2.71. If the interest were compounded continuously (every possible instant), the amount after one year would be exactly 'e' dollars, or $2.71828...
Section 4: Importance in Science and Math
Euler's number is critical across many fields. In calculus, the exponential function e^x is unique because it is its own derivative, which greatly simplifies calculations. It's used in formulas for radioactive decay, population growth models, financial calculations, and even in probability theory to describe the distribution of certain events.