Euler’s Number “e”: A Mathematical Marvel
When we think of famous numbers in mathematics, we often recall π (pi) or the imaginary unit i. However, another number stands shoulder to shoulder with these in its significance and ubiquity: Euler’s number, denoted as e.
The number e is often linked to Leonhard Euler, but it was first found by Jacob Bernoulli in 1683. While Bernoulli wasn’t specifically searching for a new mathematical constant, he inadvertently discovered the first approximation of e during his studies on continuous compound interest. This makes the story of e particularly intriguing; it wasn’t the product of abstract mathematical thought, but rather a result of tangible financial studies.
e is approximately equal to 2.71828, though like π, it’s an irrational number, meaning it has an infinite number of decimal places without any repeating pattern.
e = 2.71828182845
Common definitions of e
The number e is a mathematical constant that is approximately equal to 2.71828. It can be defined in various ways, but here are three of the most common definitions:
The number e can be defined using a limit as:
Sum of an Infinite Series
e can also be expressed as the sum of an infinite series
Compound Interest Definition
As Jacob Bernoulli discovered, the number e arises in the study of continuously compounded interest. If you invest a sum of 1 unit of currency at an annual interest rate of 100%, and let the interest compound more and more frequently, the amount of money approaches e when the interest is compounded continuously:
where n is the number of times interest is compounded per year.
All these definitions converge to the same value for e, which is approximately 2.71828.
What makes e so special?
Exponential Growth and Decay: the natural exponential function e^x is central to understanding various phenomena that involve growth or decay, such as populations, radioactive decay, and, as Bernoulli discovered, compound interest.
Calculus: e plays a pivotal role in calculus. The function e^x is unique because it is its own derivative. This property is foundational to many concepts and calculations in differential equations and other advanced areas of mathematics.
Complex Numbers: in the realm of complex numbers, Euler’s formula e^ix = cos(x) + i sin(x) is a profound relationship between exponential functions and trigonometric functions. It’s through this formula that we get the beautiful Euler’s identity: e^iπ + 1 = 0, which astonishingly connects five of the most important numbers in mathematics: 1,e,i,π,1, and 0.
As with all things in mathematics, there’s always more to explore. Some intriguing questions and topics about e include:
- How do other mathematical constants, like π and the golden ratio, relate to e?
- Why does e appear in so many different areas of science, from biology to quantum physics?
- Are there undiscovered applications of e in modern technology or future innovations?
Remember, every answer in mathematics often leads to a dozen more questions. So, the next time you see e, give a nod to its mystery and wonder. Who knows what we might discover about it next?