Euler's Formula
$e^{ix} = \cos x + i\sin x$
One of the most beautiful and powerful equations in all of mathematics — connecting exponentials, trigonometry, and complex numbers in a single elegant statement.
Prerequisites
Before diving in, make sure you're comfortable with:
- Complex numbers: $z = a + bi$, where $i = \sqrt{-1}$
- Basic trigonometry: $\sin x$, $\cos x$, and the unit circle
- The exponential function $e^x$ and its properties
- Factorials and the idea of infinite series (helpful but not required)
1. What Is Euler's Formula?
Euler's formula establishes a fundamental relationship between trigonometric functions and complex exponential functions. It states that for any real number $x$:
$$e^{ix} = \cos x + i\sin x$$
Here:
- $e \approx 2.71828\ldots$ is Euler's number (the base of natural logarithms)
- $i = \sqrt{-1}$ is the imaginary unit
- $x$ is a real number (measured in radians)
- $\cos x$ gives the real part and $\sin x$ gives the imaginary part
Key Insight: Even though $e^{ix}$ looks like it should "blow up" like a normal exponential, it doesn't! For real $x$, $|e^{ix}| = 1$ always — it traces the unit circle, not an exponential growth curve.