What is euler?

Euler’s number, written as e, is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and was popularized by the Swiss mathematician Leonhard Euler. In simple terms, e is the unique number that makes the function f(x)=eˣ grow at a rate equal to its own value.

Let's break it down

  • Definition: e can be defined as the limit (1 + 1/n)ⁿ as n approaches infinity.
  • Series: e equals the infinite sum 1 + 1/1! + 1/2! + 1/3! + … where ”!” denotes factorial.
  • Property: The derivative of eˣ is itself, meaning the slope of the curve at any point is the same as the function’s value there.
  • Irrational & transcendental: e cannot be expressed as a fraction and is not a root of any non‑zero polynomial with rational coefficients.

Why does it matter?

Because e describes continuous growth and decay, it appears everywhere in science and engineering. In calculus, it simplifies differentiation and integration of exponential functions. In computing, algorithms that model real‑world processes-like population growth, radioactive decay, or interest compounding-rely on e. Its unique properties also make it a natural choice for many mathematical models and optimizations.

Where is it used?

  • Finance: continuous compounding of interest.
  • Computer graphics: exponential easing functions for smooth animations.
  • Machine learning: activation functions such as the sigmoid σ(x)=1/(1+e⁻ˣ).
  • Signal processing: modeling decay of signals and filters.
  • Algorithms: Euler’s method for numerically solving differential equations.
  • Cryptography: certain random number generators and hash functions use exponentiation with e.

Good things about it

  • Natural base: aligns with real‑world continuous processes.
  • Simple calculus: derivative and integral of eˣ are straightforward.
  • Universal: appears in many unrelated fields, providing a common language.
  • Well‑studied: many efficient approximations and libraries exist for computing e to high precision.

Not-so-good things

  • Irrational: cannot be written exactly, so computers must approximate it, introducing tiny errors.
  • Non‑intuitive: many beginners find the concept of a number that grows at its own rate confusing.
  • Overuse: sometimes simpler linear models work just as well, but people default to exponential models because e is popular.
  • Precision limits: in very sensitive simulations, the approximation error of e can accumulate if not handled carefully.