Euler’s Formula

One of the most powerful identities in all of mathematics connects the complex exponential to trigonometry. For any real angle $\theta$ (in radians),

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

This is Euler's formula. It means that the complex exponential $e^{i\theta}$ is a point on the unit circle in the complex plane, at angle $\theta$ from the positive real axis. The real part is $\cos\theta$ and the imaginary part is $\sin\theta$, so its modulus is always 1:

$|e^{i\theta}|^2 = \cos^2\theta + \sin^2\theta = 1.$

The last equality follows from the Pythagorean identity, confirming that $e^{i\theta}$ stays exactly on the unit circle for every $\theta$.

Why it matters for quantum mechanics

A quantum state amplitude is a complex number, and rotating that amplitude by an angle $\theta$ multiplies it by $e^{i\theta}$. Because $|e^{i\theta}| = 1$, this rotation — called a phase — leaves the measurement probability $|z|^2$ unchanged. Euler's formula makes this fact transparent: phases are pure rotations on the unit circle. This is why quantum gates like the T gate (rotation by $\pi/4$) and S gate (rotation by $\pi/2$) are expressed in terms of $e^{i\theta}$.

Deriving the formula

A concise route uses Taylor series. Expanding $e^{x}$, $\cos x$, and $\sin x$ around $x = 0$:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

Substitute $x = i\theta$ and use $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, cycling every four powers:

$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!} + \frac{\theta^4}{4!} + \cdots$

Separating real and imaginary parts:

$e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right) = \cos\theta + i\sin\theta.$

The two bracket expressions are exactly the Taylor series for $\cos\theta$ and $\sin\theta$, completing the derivation.

Special values

Several angles give exact results worth memorizing:

| $\theta$ | $e^{i\theta}$ | |---|---| | $0$ | $1$ | | $\pi/6$ | $\dfrac{\sqrt{3}}{2} + \dfrac{i}{2}$ | | $\pi/4$ | $\dfrac{1}{\sqrt{2}} + \dfrac{i}{\sqrt{2}}$ | | $\pi/3$ | $\dfrac{1}{2} + \dfrac{i\sqrt{3}}{2}$ | | $\pi/2$ | $i$ | | $\pi$ | $-1$ | | $2\pi$ | $1$ |

The case $\theta = \pi$ gives $e^{i\pi} = -1$, or equivalently $e^{i\pi} + 1 = 0$ — often called Euler's identity.

Try it

This is a numerical exercise — your code should return a number. Using Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, find the real part of $e^{i\pi/3}$.

From the table above (and the 30-60-90 triangle), $\cos(\pi/3) = 1/2 = 0.5$ exactly.

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