Skip to content
Mathematics

Euler’s Identity

Growth, circles, and imaginary numbers meet in one equation. Understand what each constant means, explore complex exponentials as rotation, and see why Euler’s identity follows from a half-turn.
Source code
eiπ+1=0e^{i\pi}+1=0

The connection is Euler’s formula: raising ee to a purely imaginary exponent describes a point moving around a circle. At an angle of π\pi radians, that point reaches −1-1.

Meet the three constants

Euler’s number

e≈2.71828e\approx2.71828

The natural base for continuous growth. If interest at a total rate of one per period is compounded more and more frequently, the growth factor approaches ee.

e=lim⁡n→∞(1+1n)ne=\lim_{n\to\infty}\left(1+\frac1n\right)^n

The circle constant

π≈3.14159\pi\approx3.14159

A circle’s circumference divided by its diameter. Radians measure an angle as arc length divided by radius, so a half-turn is π\pi radians.

C=2πr,θ=srC=2\pi r,\qquad \theta=\frac{s}{r}

The imaginary unit

i2=−1i^2=-1

A complex number has a real part and an imaginary part. Plotting them on perpendicular axes gives the complex plane. Multiplying by ii makes a counterclockwise quarter-turn.

i(a+bi)=−b+aii(a+bi)=-b+ai

Euler’s formula connects them

eiθ=cos⁡θ+isin⁡θ(θ∈R)e^{i\theta}=\cos\theta+i\sin\theta\qquad(\theta\in\mathbb R)

Here θ\theta is a real angle measured in radians. The real part is cos⁡θ\cos\theta; the imaginary part is sin⁡θ\sin\theta. Together they locate a point on the unit circle, centered at the origin with radius 11.

∣eiθ∣=cos⁡2θ+sin⁡2θ=1\left|e^{i\theta}\right|=\sqrt{\cos^2\theta+\sin^2\theta}=1

Start at 11 on the positive real axis. As the angle increases, the point moves counterclockwise. A real exponent controls growth; a purely imaginary exponent controls rotation. More generally, a complex exponent can do both:

ea+ib=ea(cos⁡b+isin⁡b)(a,b∈R)e^{a+ib}=e^a(\cos b+i\sin b)\qquad(a,b\in\mathbb R)

The factor eae^a sets the distance from the origin, while bb sets the angle in radians.

Explore a complex exponential

Move the angle to rotate the point. The horizontal coordinate is the real part; the vertical coordinate is the imaginary part.

The point stays one unit from the origin. The highlighted arc traces the counterclockwise rotation from the positive real axis.
00π\pi2π2\pi

Angle in radians

θ=π\theta=\pi

Point on the complex plane

eiπ=−1e^{i\pi}=-1

A half-turn reaches the negative real axis: Euler’s identity.

The slider uses degrees for convenience. Euler’s formula uses radians, converted by:

θ=degrees180π\theta=\frac{\text{degrees}}{180}\pi

A half-turn gives Euler’s identity

Substitute θ=π\theta=\pi into Euler’s formula. On the unit circle, a half-turn lands on the negative real axis: its real coordinate is −1-1 and its imaginary coordinate is 00.

  1. Substitute the half-turn angle

    eiπ=cos⁡π+isin⁡πe^{i\pi}=\cos\pi+i\sin\pi

  2. Use the circle coordinates

    eiπ=−1+i⋅0=−1e^{i\pi}=-1+i\cdot0=-1

  3. Add one to both sides

    eiπ+1=0e^{i\pi}+1=0

Euler’s formula describes every real angle. Euler’s identity is its special half-turn case, bringing the five constants e,π,i,1,0e,\pi,i,1,0 into a single equation.

Why an exponential becomes a circle

The picture illustrates the formula, but the power series explain why it is true. The exponential extends to complex inputs using the same series as for real inputs. All three series below converge absolutely for every complex input, so we can group their terms.

Follow the power series derivation

Start with the exponential, cosine, and sine series.

ez=1+z+z22!+z33!+⋯e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots
cos⁡θ=1−θ22!+θ44!−⋯\cos\theta=1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots
sin⁡θ=θ−θ33!+θ55!−⋯\sin\theta=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots

Substitute z=iθz=i\theta. Powers of the imaginary unit repeat in a cycle, separating even powers from odd powers.

i0=1,i1=i,i2=−1,i3=−i,i4=1i^0=1,\quad i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1
eiθ=1+iθ−θ22!−iθ33!+θ44!+iθ55!−⋯e^{i\theta}=1+i\theta-\frac{\theta^2}{2!}-i\frac{\theta^3}{3!}+\frac{\theta^4}{4!}+i\frac{\theta^5}{5!}-\cdots

Group the real and imaginary terms. The real terms are exactly the cosine series, and the coefficient of the imaginary unit is exactly the sine series.

eiθ=(1−θ22!+θ44!−⋯ )+i(θ−θ33!+θ55!−⋯ )e^{i\theta}=\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right)+i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right)
eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta+i\sin\theta

What to remember

  • The exponent is the product iπi\pi. Complex exponentiation is defined through an extension such as the power series above.
  • Use radians in Euler’s formula. A half-turn is 180∘=π radians180^\circ=\pi\ \text{radians}.
  • The identity is exact. A calculator may show a tiny imaginary remainder because it uses finite numerical approximations.

Complex exponentials are useful for waves, oscillations, and signals. Their exponent law turns multiplication into addition of angles: rotating by one angle and then another adds the two rotations.

eiαeiβ=ei(α+β)e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}

Continue with logarithms to explore the inverse of exponential growth, or review number systems for the bigger picture of real and complex numbers.