Euler’s Nu The Magic of What Is Euler’s Number? Have you ever come across a number that seems to connect math, science, and the world around us? One of the most fascinating is Euler’s number, written as e, a special constant approximately equal to 2.718, central to countless natural and scientific phenomena. It’s the foundation of the natural logarithm, a tool that helps us understand how things grow or shrink over time. From bacteria multiplying in a lab to stars fading in the sky, e appears in countless natural processes [1]. Surprisingly, this number was first uncovered not in a science lab but in a puzzle about money. Let’s explore how e came to be and why it’s so extraordinary. By Jane Yang 楊靜悠 A Mathematical Gem Discovered in Finance The story of e begins in the 1600s with Jacob Bernoulli, a mathematician curious about how small changes add up [2, 3]. Imagine you have $1.00, and you’re offered an unrealistic 100% annual growth rate. If this growth is added once at the year’s end, your $1.00 doubles to $2.00. But what if the growth is calculated more often? Suppose it’s added twice a year. Every six months, you gain 50%, so your $1.00 grows to $1.00 × 1.5 × 1.5 = $2.25 by year’s end. If it’s calculated four times a year, each period adds 25%, turning your $1.00 into $1.00 × 1.25 × 1.25 × 1.25 × 1.25 = $2.44. Monthly calculations yield $1.00 × (1 + 1/12)^12, about $2.61. The pattern is clear: More frequent additions mean a larger result. Here’s the exciting part. What if the growth is calculated every day, every minute, or even every second? The formula becomes $1.00 × (1 + 1/n)^n, where n is the number of times the growth is added. As n grows larger — approaching infinitely frequent additions — the result doesn’t climb endlessly but settles around 2.718281828459045… . This number is e! Bernoulli discovered this constant, revealing a mathematical gem that would resonate far beyond his original question. The Power of e in Our World Why is e so important? Named after Leonhard Euler, who further explored its properties in the 1700s, this number became a universal key to understanding exponential change, appearing in fields like biology, physics, medicine, and engineering. Its unique properties make it ideal for describing processes that speed up or slow down, like a snowball growing larger
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