Euler's formula — the equation that unifies everything

Three unrelated subjects. One equation.

You have learned about the unit circle from trigonometry — the circle of radius 1 where $\cos\theta$ and $\sin\theta$ live. You have also used exponential functions like $e^x$, which model growth and decay. These feel completely separate: one is about angles and waves; the other is about repeated multiplication.

Now for something strange. A Swiss mathematician named Leonhard Euler noticed a pattern in 1748: when you substitute an imaginary number into the exponential function, something impossible happens. The exponential — which normally grows without bound — wraps itself into a perfect circle. It stops growing. It starts rotating.

The formula is simple to state. It takes a little more work to understand why it's true. This lesson does both — the intuition first, the derivation second, the interactive third. By the end, $e^{i\theta}$ won't look like notation. It will look like a point moving around a circle.

Think of a wheel turning — not a value growing.

Ordinary exponentials grow along a line. $e^1 \approx 2.7$, $e^2 \approx 7.4$, $e^3 \approx 20$. Each step multiplies by the same factor. The result races off toward infinity in one direction.

Here's the key observation: ordinary real exponentials multiply along a line. When you replace the exponent with an imaginary number $i\theta$, you're essentially saying "instead of growing forward, turn left by $\theta$ radians." The imaginary unit $i$ acts like a 90° rotation instruction — and repeated application keeps you on a circle of radius 1.

The result is that $e^{i\theta}$ is always a complex number of magnitude 1, positioned at angle $\theta$ on the unit circle:

$$e^{i\theta} = \underbrace{\cos\theta}_{\text{real part}} + i\,\underbrace{\sin\theta}_{\text{imaginary part}}$$

The real part is $\cos\theta$ — the horizontal coordinate on the circle. The imaginary part is $\sin\theta$ — the vertical coordinate. The magnitude is always $\sqrt{\cos^2\theta + \sin^2\theta} = 1$ — the unit circle, guaranteed.

From Taylor series to $e^{i\theta} = \cos\theta + i\sin\theta$, every step shown.

The cleanest path to Euler's formula runs through Taylor series — a method for writing any smooth function as an infinite sum of simpler terms. You don't need to know Taylor series in advance; this section explains them from scratch.

Step 1: What is a Taylor series?

Every smooth function can be written as an infinite polynomial — a sum of powers of $x$ with specific coefficients. The idea is: a polynomial is determined entirely by its derivatives at a single point. If you know all the derivatives of $f(x)$ at $x = 0$, you can reconstruct $f$ everywhere.

The general formula is:

$$f(x) = f(0) + f'(0)\,x + \frac{f''(0)}{2!}\,x^2 + \frac{f'''(0)}{3!}\,x^3 + \cdots$$

where $n!$ means $n$ factorial: $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, and so on. Each term divides by the factorial of its power.

Step 2: Taylor series for $e^x$, $\cos\theta$, and $\sin\theta$

Let's write down the Taylor series for three functions we'll need. These are standard results — you can verify each by computing derivatives and applying the formula above.

Step 3: Substitute $x = i\theta$ into $e^x$

Now comes the key move. The exponential formula works for any $x$ — including complex numbers. Let's replace $x$ with $i\theta$ and expand term by term:

$$e^{i\theta} = 1 + (i\theta) + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \frac{(i\theta)^5}{5!} + \cdots$$

Now we simplify each power of $i\theta$. Recall from M04 that $i^2 = -1$, which gives us a cycling pattern:

The signs cycle with period 4: $+1,\, +i,\, -1,\, -i,\, +1,\, +i,\, \ldots$ This pattern is exactly what makes the next step possible.

Step 4: Substitute the simplified powers back

Replacing each $(i\theta)^n$ with its simplified form:

$$e^{i\theta} = 1 + i\theta + \frac{-\theta^2}{2!} + \frac{-i\theta^3}{3!} + \frac{+\theta^4}{4!} + \frac{+i\theta^5}{5!} + \cdots$$

Step 5: Group real and imaginary parts separately

Now collect all the real terms (those without $i$) and all the imaginary terms (those with $i$):

Step 6: Arrive at Euler's formula

Combining real part plus imaginary part:

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

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

That's Euler's formula. It follows directly from the Taylor series of three functions — no guesswork, no magic, just careful substitution and regrouping.

Step 7: The special case $\theta = \pi$

Let's plug in $\theta = \pi$ and see what happens:

This identity has been called "the most beautiful equation in mathematics" — not because of aesthetics, but because it means something precise: rotating by $\pi$ radians (one half-turn) on the complex plane takes $+1$ to $-1$. The multiplicative inverse of going all the way around. A 180° phase flip, which is exactly what the $Z$ gate does to a qubit on the Bloch sphere.

Drag $\theta$ and watch the unit circle come alive.

The formula $e^{i\theta} = \cos\theta + i\sin\theta$ places a point on the unit circle at angle $\theta$. As $\theta$ increases from $0$ to $2\pi$, the point completes one full revolution. The real part traces $\cos\theta$; the imaginary part traces $\sin\theta$. Both projections are shown live as you drag.

Micro practice — three quick checks