Euler's Formula — Definition, Proof & Examples

Q: What is the difference between Euler's formula and Euler's identity?

Euler's formula is the general equation $e^{i\theta} = \cos\theta + i\sin\theta$, which holds for every real number θ. Euler's identity is the specific case when θ = π, giving $e^{i\pi} + 1 = 0$. The identity is famous because it links the five fundamental constants, but it is just one instance of the broader formula.

Q: Why is Euler's formula considered the most beautiful equation in math?

Euler's identity ($e^{i\pi} + 1 = 0$) unites five constants from different branches of mathematics — arithmetic (0 and 1), analysis ($e$), geometry ($\pi$), and algebra ($i$) — in one short equation. It also uses addition, multiplication, and exponentiation, three core operations. Many mathematicians view this unexpected connection as a sign of deep unity across mathematics.

Euler's Formula

e^iπ + 1 = 0. This remarkable equation combines e, i, π (pi), 1, and 0, which are arguably the five fundamental numbers of mathematics. It also includes addition, multiplication, exponentiation, and composition, four of the fundamental operations of mathematics.

Note: Euler is pronounced "Oiler".

See also

Euler's formula for polyhedra, formula

Key Formula

e^{i\theta} = \cos\theta + i\sin\theta

Where:

$e$ = Euler's number, approximately 2.71828, the base of the natural logarithm
$i$ = The imaginary unit, defined by i² = −1
$\theta$ = Any real number, representing an angle in radians
$\cos\theta$ = The cosine of the angle θ (the real part)
$\sin\theta$ = The sine of the angle θ (the imaginary part)

Worked Example

Problem: Use Euler's formula to verify that e^(iπ) + 1 = 0.

Step 1: Write Euler's general formula and substitute θ = π.

e^{i\pi} = \cos\pi + i\sin\pi

Step 2: Evaluate cos(π). On the unit circle, the cosine of π radians (180°) is −1.

\cos\pi = -1

Step 3: Evaluate sin(π). The sine of π radians (180°) is 0.

\sin\pi = 0

Step 4: Substitute these values back into the formula.

e^{i\pi} = -1 + i(0) = -1

Step 5: Add 1 to both sides to obtain Euler's identity.

e^{i\pi} + 1 = -1 + 1 = 0

Answer: This confirms that e^(iπ) + 1 = 0, which is Euler's identity.

Another Example

This example uses a different angle (π/2 instead of π) to show that Euler's formula works for any value of θ, not just the one that produces Euler's identity.

Problem: Use Euler's formula to express e^(iπ/2) in the form a + bi.

Step 1: Substitute θ = π/2 into Euler's formula.

e^{i\pi/2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2}

Step 2: Evaluate cos(π/2). Cosine of 90° is 0.

\cos\frac{\pi}{2} = 0

Step 3: Evaluate sin(π/2). Sine of 90° is 1.

\sin\frac{\pi}{2} = 1

Step 4: Combine the results.

e^{i\pi/2} = 0 + i(1) = i

Answer: e^(iπ/2) = i. Raising e to the power of iπ/2 gives the imaginary unit itself.

Frequently Asked Questions

What is the difference between Euler's formula and Euler's identity?

Euler's formula is the general equation

e^{i\theta} = \cos\theta + i\sin\theta

, which holds for every real number θ. Euler's identity is the specific case when θ = π, giving

e^{i\pi} + 1 = 0

. The identity is famous because it links the five fundamental constants, but it is just one instance of the broader formula.

Why is Euler's formula considered the most beautiful equation in math?

Euler's identity (

e^{i\pi} + 1 = 0

) unites five constants from different branches of mathematics — arithmetic (0 and 1), analysis (

e

), geometry (

\pi

), and algebra (

i

) — in one short equation. It also uses addition, multiplication, and exponentiation, three core operations. Many mathematicians view this unexpected connection as a sign of deep unity across mathematics.

How do you pronounce Euler?

Euler is pronounced "OY-ler," not "YOO-ler." Leonhard Euler was an 18th-century Swiss mathematician, and the name follows German pronunciation rules.

Euler's Formula (complex exponentials) vs. Euler's Formula (polyhedra)

	Euler's Formula (complex exponentials)	Euler's Formula (polyhedra)
Formula	$e^{i\theta} = \cos\theta + i\sin\theta$	$V - E + F = 2$
Field	Complex analysis / trigonometry	Geometry / topology
Variables	e, i, θ, cos, sin	V = vertices, E = edges, F = faces
When to use	Converting between exponential and trigonometric forms of complex numbers	Relating vertices, edges, and faces of convex polyhedra

Why It Matters

Euler's formula is central to any course that touches complex numbers, from precalculus through university-level math and engineering. In physics and electrical engineering, signals and waves are routinely written as

e^{i\omega t}

instead of separate sine and cosine terms, because the exponential form makes calculations far simpler. Understanding this formula also unlocks De Moivre's theorem, Fourier analysis, and the polar form of complex numbers.

Common Mistakes

Mistake: Thinking e^(iθ) equals cos θ + sin θ, forgetting the i in front of sin θ.

Correction: The correct formula is

e^{i\theta} = \cos\theta + i\sin\theta

. The sine term is the imaginary part, so it must be multiplied by

i

Mistake: Using degrees instead of radians for θ.

Correction: Euler's formula requires θ to be in radians. For example, use θ = π (not 180) to get

e^{i\pi} = -1

. If you substitute 180, the formula gives an incorrect result.

Related Terms

e — The base of the natural logarithm used in the formula
The Imaginary Number i — The imaginary unit that appears in the exponent
Pi — The angle (in radians) that yields Euler's identity
Exponentiation — The operation of raising e to a complex power
Equation — Euler's formula is an equation relating expressions
Euler's Formula (Polyhedra) — A different Euler formula for vertices, edges, and faces
Composition — One of the operations present in the identity
Formula — General term for a mathematical relationship