The Complex Plane & Euler's Formula
“Euler's formula: an imaginary exponent is not growth but rotation.”
The formula
e^(iθ) = cos θ + i sin θ, e^(iπ) + 1 = 0How to read it: a rotation by angle θ is the complex exponential e^(iθ), which is the point on the unit circle given by cosine and sine
- e^(iθ)
- — a rotation by angle θ—a point on the unit circle
- θ
- — the angle of rotation (in radians)
- cos θ, sin θ
- — the horizontal and vertical coordinates of that point
- e^(iπ) + 1 = 0
- — Euler's identity—one line linking five constants
The hook
e, i, π, 1, 0—the five most famous constants in mathematics meet in a single line: e^(iπ)+1=0. It's called 'the most beautiful equation'.
In plain words
See a complex number a+bi as a point (or arrow) in the plane, and you get the complex plane. Euler's formula e^(iθ)=cos θ+i sin θ says that the complex exponential e^(iθ) is the point on the unit circle rotated by angle θ. In short: an imaginary exponent means rotation.
The intuition
Real numbers live on a line, but complex numbers add an axis and live in a plane. Multiplying by i rotates a point 90° (so multiplying by i twice turns 180° to −1). Euler's formula extends this—multiplying by e^(iθ) rotates by exactly θ. It is the moment the exponential (the language of growth) and circular motion (the language of rotation) become one on the imaginary axis.
How it's built
The unit circle and the angle θ are the parts. The horizontal component of e^(iθ) is cos θ, the vertical is sin θ, so its magnitude is always 1 (on the unit circle). Increase θ and the point travels around the circle. At θ=π it turns halfway to −1, and adding 1 gives 0—that is Euler's identity.
Example
Put θ=π: e^(iπ)=cos π+i sin π = −1 + i·0 = −1. So e^(iπ)+1 = −1+1 = 0. Put θ=π/2: e^(iπ/2)=cos(π/2)+i sin(π/2) = 0 + i·1 = i (a 90° rotation).
Common mistake
Don't read e^(iθ) as 'e multiplied by itself iθ times'. An imaginary exponent means rotation, not repeated multiplication (growth). That's why e^(iθ) never grows—it just circles with magnitude always 1.
Where it's used
AC circuit analysis, the Fourier transform in signal processing (decomposing waves into rotations), quantum-mechanical wavefunctions, rotations in computer graphics, control engineering—nearly all engineering and physics involving waves and rotation speaks the language of Euler's formula.
Where it came from
Euler published the formula in 1748. It revealed that the exponential and the trigonometric functions—two worlds that looked utterly different—were one through complex numbers, a decisive bridge in the history of mathematics.
Prerequisites
Quick check
By Euler's formula, what is the value of e^(iπ)?
- 1
- −1✓
- i
- 0
Practice
Use Euler's formula to find the value of e^(iπ).
Answer: -1
- e^(iπ) = cos π + i sin π
- = −1 + i·0 = −1
Takeaway: θ=π is a half turn → −1.
For every angle θ, what is |e^(iθ)| (the magnitude of the complex number e^(iθ))?
Answer: 1
- e^(iθ) is a point on the unit circle
- magnitude = √(cos²θ + sin²θ) = 1
Takeaway: e^(iθ) always has magnitude 1—it only rotates, never grows.
It holds that e^(iπ/2) = i. Find the imaginary part of this value.
Answer: 1
- e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i·1
- = i, so the imaginary part is 1
Takeaway: θ=π/2 is a 90° turn → i.
Explain in your notebook why multiplying a complex number by e^(iθ) rotates the point by θ.
Answer: undefined
- e^(iθ) has magnitude 1 and angle θ (on the unit circle)
- complex multiplication multiplies magnitudes and adds angles
- so the magnitude is unchanged and only the angle gains θ—a rotation
Takeaway: Complex multiplication multiplies magnitudes and adds angles; e^(iθ) is pure rotation.