Euler’s number (or Napier’s constant), e ≈ 2.718281828459045, is one of mathematics’ most important constants. Like π, it appears in diverse fields, from finance to physics. This blog post dives into its mathematical definitions, fascinating properties, and real-world relevance.
What is Euler’s Number e
The number e is the base of natural logarithms and a cornerstone of calculus. Discovered in the 17th century, it governs exponential growth and decay. Let’s explore its formal definitions.
Mathematical Definitions of e
1. e as the Limit of (1 + 1 / x)^x When x Approaches Infinity
Formula:
e = lim(x→∞)(1 + 1 / x)^x
This limit models continuous compound interest. As x grows, the expression approaches e. For example, if x = 1000, the value is ≈ 2.7169. Taking the natural logarithm and applying L’Hospital’s Rule confirms the limit equals e.
2. e as the Limit of (1 + x)^(1/x) When x Approaches 0
Formula:
e = lim(x→0)(1 + x)^(1/x)
Substitute x = 1 / n (where n→∞), transforming this into the previous definition. This dual limit underscores e’s role in calculus, especially derivatives of exponential functions.
3. e Expressed via Natural Logarithms: x^(1/ln(x))
Formula:
e = x^(1/ln(x)), (x > 0, x ≠ 1)
Taking the natural logarithm of both sides:
ln(e) = ln(x) / ln(x) = 1
This identity holds since e is the base of natural logarithms.
4. Euler’s Formula and the Complex Exponential
Formula:
e = cos(i) – i sin(i)
Derived from Euler’s formula e^iθ = cos(θ) + i sin(θ) (which means e^iπ + 1 = 0), substituting θ = -i gives:
e^1 = cos(i) – i sin(i)
This is because cos(-θ) = cos(θ) and sin(-θ) = -sin(θ). The function cos(θ) is the only even trigonometric function and as such, it is symmetric about the y axis. On the other hand, sin(θ) as an odd function is symmetric with respect to the coordinate origin. Also, Since i² = -1, it means –ii = 1.
This links e to hyperbolic functions and complex analysis.
Other Fascinating Facts About e
Irrationality and Transcendence
- e is irrational (cannot be written as a fraction) and transcendental (not a root of any polynomial with rational coefficients)
Unique Calculus Properties
- Power series expansion: e^x = ∑[n = 0 to ∞] x^n / n!
- The derivative of eˣ is itself, making it central to differential equations:
Proof:
Let it be:
y = e^x
After logarithmization we have:
ln(y) = ln(e^x)
that is (knowing one of the properties of logarithms):
ln(y) = x ln(e)
After differentiating the left and right sides, and since y is a dependent variable, we get:
y’ / y = ln(e)
Given that ln(e) = 1, then y’ = y or (e^x)’ = e^x.
Applications in Probability and Statistics
- e appears in the Poisson distribution, modeling rare events, and the normal distribution’s bell curve
Real-World Applications of Euler’s Number
Finance: Calculating continuously compounded interest using:
A = P e^rt
where are:
- A = The total amount of money after a certain time period
- P = The principal amount or the initial amount of money
- e = Euler’s number
- r = The interest rate (always represented as a decimal)
- t = The time duration in years
Physics: Describing radioactive decay and heat transfer
Biology: Modeling population growth under ideal conditions
Conclusion
Euler’s number e is a mathematical marvel with endless applications. From its limit definitions to its role in complex analysis, e bridges abstract math and the real world. Understanding e enriches your grasp of calculus, physics, and beyond.
Awesome post, Slavisa! You made e sound way cooler than just some mysterious math constant. I loved how you explained all those definitions without turning it into a brain twister—seriously, props for that! It’s crazy how e shows up in so many places, from finance to physics to probability. That part about e being its own derivative? Totally blew my mind. Quick question though—how does e tie into real-world situations where growth isn’t ideal or constant, like in unpredictable markets or chaotic systems? Would love to hear your take! Thanks for making math feel fun and surprisingly relatable. Can’t wait to read more!
Thank you so much! I’m really glad the post made e feel more approachable. It’s one of those constants that’s everywhere once you start looking for it!
That’s a great question about how e ties into less-than-ideal or chaotic systems. While the classic exponential model assumes smooth, continuous growth (like in idealized finance or biology), e still shows up in more complex or erratic systems, but often in a slightly different way.
For example, in unpredictable markets, models like stochastic calculus come into play (think Black-Scholes for option pricing), and even there, e is baked into the math through differential equations that account for randomness. In chaotic systems, e appears in formulas that describe sensitivity to initial conditions. One famous case being the Lyapunov exponent, which measures how quickly nearby states diverge. If that exponent is positive (and it often involves e), it signals chaos.
So even when growth isn’t smooth or predictable, e still lurks in the math under the hood, just in more advanced or nuanced ways.
I am not a mathematician, and after reading this article I gained a real respect for those like the author of this article, who can thoroughly understand the world of mathematics and physics and apply it so that those of us who can barely spell “e” can benefit from their knowledge and applications to our world.
Hey Slavisa!
This exploration of e is so well done – it’s both educational and super engaging! I especially enjoyed learning about its properties and how it shows up in things like compound interest and natural processes. You’ve made a potentially intimidating topic feel so approachable. Thank you for this great piece! For those of us who aren’t math experts, what’s one fun or intuitive way to visualize or understand the significance of e? Thanks for the fantastic content!
Sincerely,
Steve
Hi Steve,
Thank you so much for your kind words! I’m really glad you found the post engaging and approachable. That means a lot to me.
As for a fun way to visualize or intuitively understand the significance of e, here’s one I really like:
Imagine you have $1 in a bank account that earns 100% interest over one year. If the interest is compounded just once at the end, you’d double your money to $2. But if the interest is compounded more often, say, every month, day, or even every second, your balance grows a bit faster. If you compounded continuously, meaning infinitely many times, the final amount would approach e dollars, about $2.718.
In a way, e captures the “ultimate” growth you can achieve when compounding is happening constantly. It’s like the mathematical limit of growth at its purest form!
Thanks again for reading and for the great question!
Best regards,
Slavisa
This post definitely has i a target audience in mind. Many would be lost right after the “What is Eulers Number” question, Math is everywhere, most things can be explained using math, as an engineer it was part of my toolbelt.
But I never had to go this deep to get the job done.
Good article, I may have to keep coming back to cross reference and study on understand it though.
e is different!
Wow, this post is incredible. I do not have near enough mathematical knowledge to completely understand these calculations but I have always been interested in them. I had never heard of Euler’s e until now. Thank you for introducing me to a whole new concept to explore.
You explained everything in a very clear cut way that I able to understand which often not the case when I read about mathematical concepts and formulas. With your post, I can actually feel confident that I learned something.
Hello, this post is excellent. It was clear enough for someone like me who is not a math expert to walk away feeling confident that I have learned something. I have always loved numbers and I am especially intrigued with PI since I was born on 14th of March.
Thank you for introducing me to Eulers’ e in a way that is clear and concise. I will google more on it as I google PI occasionally. MAC.
This was such a fascinating read! My husband actually stumbled upon Euler’s number e while working on a project related to exponential growth, and we both got curious about it. Reading your post really helped deepen our understanding.
I love how you explained not just the definition but also the properties and historical significance—it made it so much more relatable! I’m curious though: are there real-world applications of e beyond finance and population growth that you find especially interesting? And do you think e is as “fundamental” to mathematics as π is to geometry? Would love to hear your thoughts!
Thank you so much for your kind words! I’m really glad the post helped make Euler’s number feel more relatable and interesting. To your questions, absolutely, there are many more real-world applications of e! Beyond finance and population growth, it plays a major role in areas like electrical engineering (for example, in the analysis of RC and RL circuits where voltages and currents decay exponentially), thermodynamics (in describing cooling rates), and even in machine learning algorithms where e is essential in functions like the softmax function and logistic regression.
As for whether e is as “fundamental” to mathematics as π is to geometry, I would definitely say yes, but in a different realm. While π governs the geometry of circles and spatial relationships, e governs the behavior of growth, change, and systems in motion. It’s like e is fundamental to processes, just as π is fundamental to shapes. Both constants are deeply woven into the fabric of mathematics, just on different fronts!
You’ll be pleased to know that your effective explanation ran rallys around what I remember my high school Calc teacher lectured us with. Perhaps I was not prepared for such conditioning in my youth. However, I applaud your resurfacing of this almost forgotten lesson. I especially enjoy applying it to interest amounts over time. I might find myself playing with the number e the next time I open up my banking folders. Thank you.