The Magic of Euler’s Number e
By Jane Yang 楊靜悠
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.
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 or a whisper fading away. It also simplifies complex problems, making it a go-to tool for scientists and engineers.
In biology, e models population growth. Picture a colony of bacteria doubling every hour. The smooth, accelerating curve of their growth uses e to predict how many bacteria there will be after a day or a week [4]. In ecology, e helps track how animal populations expand or how resources, like fish stocks, shrink when overharvested. For example, conservationists use e to estimate how quickly a threatened species might recover if protected [4]. In physics and chemistry, e describes decay, such as how radioactive elements like uranium lose energy over time. Scientists rely on e to calculate a substance’s half-life, the time it takes for half of it to break down, which is crucial for safe handling in medical treatments or power plants.
In daily life, e shines in engineering and technology that makes our life easier. It models how a capacitor stores charge in a circuit, essential for designing devices like your phone or computer. In medicine, e helps track how drugs are absorbed or cleared from the body, helping doctors determine safe dosages. In technology, e underpins algorithms for signal processing, ensuring clear audio in your earbuds or smooth video streaming.
Enormous Impact of the Seemingly Small Number
What makes e truly special is that it connects these diverse phenomena. Its value, about 2.718, may seem small, but its impact is enormous. Next time you hear about a virus spreading, a species recovering, or a cup of tea cooling, think of e — a quiet number with a massive role in unlocking the secrets of our world.
A Practical Archeological Question: Carbon-14 Dating
Carbon-14 dating is a method used to determine the age of organic archeological specimens from the age of 500 to 50,000 years [5]. Carbon-14 is an unstable radioisotope, which undergoes decay into nitrogen-14. Living plants incorporate naturally occurring atmospheric carbon-14 into their tissues through carbon fixation, and pass it on to animals through the food chain. The ratio of carbon-14 in living tissues is relatively stable because living organisms constantly take in air and food despite the constant decay of carbon-14, but once the organism dies, there will be a net reduction in carbon-14 content.
The decay process can be expressed by the exponential decay function: N = N0e-kt, where N is the number of undecayed nuclei, N0 is the initial number of undecayed nuclei, k is the decay constant, and t is the time lapsed [6, 7]. The half-life t1/2 of carbon-14, or the time needed for half of the radioisotope to decay, is roughly 5,730 years. If we have a piece of ancient wood whose carbon-14 content has four tenths of that in living trees, find:
- the decay constant k (to four decimal places).
- the age of that piece of wood (to the nearest year).
References
[1] 3Blue1Brown. (2017, May 2). What's so special about Euler's number e? | Chapter 5, Essence of calculus [Video]. YouTube. https://www.youtube.com/watch?v=m2MIpDrF7Es
[2] Kenton, W. (2025, May 10). Euler’s Number (e) Explained, and How It Is Used in Finance. Investopedia. https://www.investopedia.com/terms/e/eulers-constant.asp
[3] Reichert, S. (2019). e is everywhere. Nature Physics, 15(9), 982. https://doi.org/10.1038/s41567-019-0655-9
[4] Vandermeer, J. (2010). How Populations Grow: The Exponential and Logistic Equations. Nature Education Knowledge, 3(10), 15. https://www.nature.com/scitable/knowledge/library/how-populations-grow-the-exponential-and-logistic-13240157/
[5] Augustyn, A. (2025, May 28). Carbon-14 dating. Encyclopaedia Britannica. https://www.britannica.com/science/carbon-14-dating
[6] Friedrich, K. (2023, March 16). Euler’s Number Is Seriously Everywhere. Here’s What Makes It So Special. Popular Mechanics. https://www.popularmechanics.com/science/math/a43341607/what-is-eulers-number/
[7] Boelkins, M. (2022). Mathematics of carbon dating. EBSCO. https://www.ebsco.com/research-starters/mathematics/mathematics-carbon-dating