Working note · mathematics · part V
Euler's formula: how rotation became algebra
One line from 1748, linking exponential growth with circular rotation — and also presenting an equation that Feynman called "the most remarkable formula in mathematics."
In 1748, Leonhard Euler, expanding excos x and sin x into infinite series and substituting an imaginary number for x, discovered that all three functions are actually different projections of the same object: rotation. The formula eiθ = cosθ + i sinθ translates the language of "growth" into the language of "turn" — and turns the analysis of oscillations and waves into simple power algebra. A special case at θ = π is an identity that unites five of mathematics' most important constants in one line.
Three views of one formula
| View | What is seen |
|---|---|
| Algebraic | eiθ — a complex number equal to cosθ + i sinθ |
| Geometric | a point on the unit circle of the complex plane at an angle θ from the axis |
| Analytic | coincidence of the infinite series ex, cos x, sin x after substituting x = iθ (§3) |
None of the three views is "more important" — the power of Euler's formula lies exactly in the fact that it is simultaneously an algebraic identity, a geometric fact, and an analytic coincidence of series. Let's start with the most visual one — geometry.
Geometry: a point on a circle
Statement. The complex number eiθ is a point on the unit circle (radius 1) of the complex plane, placed at an angle θ from the positive real axis:
As the point travels around the circle (θ increases from 0 to 2π), its horizontal coordinate faithfully traces the cosine, and the vertical one — the sine — the very same functions you've seen on trigonometry graphs, only now as projections of UNIFORM rotation, rather than as separate wavy curves. This image is precisely the content of the formula: "e to an imaginary power" is not some abstract number, but the coordinate of a point moving uniformly around a circle.
cos²θ + sin²θ = 1 — the fundamental trigonometric identity, known from school. For eiθ this means: the distance from the origin to the point is always exactly 1 for any θ — the point cannot physically leave the unit circle, only run along it. Hence the role of θ — it is literally the traversed angle, the only thing that changes.
Where it comes from: Taylor series
The formula is not a postulate, but a consequence of the fact that ex, cos x and sin x can be written as infinite sums (Taylor series):
Let's substitute into the series for ex the imaginary number iθ instead of x and recall that i² = −1, i³ = −i, i⁴ = 1 (powers of i cycle with period 4):
Let's group separately the terms without i (even powers of θ) and the terms with i (odd powers) — the first group will exactly coincide with the series for cosθ, the second (without the common factor i) — with the series for sinθ:
The coincidence is not "forced" — it is inevitable, once we define ex for complex numbers via the same series as for real numbers. This explains why the formula works: three seemingly completely different objects (exponential, cosine, sine) mathematically belong to one family of functions, differing only in what argument — real or purely imaginary — is substituted into them.
Euler's identity
Let's substitute the specific angle θ = π (half a turn, the point on the circle ends up exactly to the left of the center, at point −1) into Euler's formula:
In surveys of mathematicians (for example, the poll by the journal Mathematical Intelligencer 1990), Euler's identity consistently takes first place. The reason is not mysticism, but extreme economy: five constants from completely different areas of mathematics (0 and 1 — arithmetic, e — analysis/growth, i — algebra/complex numbers, π — geometry/circles) are linked by three simplest operations (addition, multiplication, exponentiation) without a single extra symbol. Richard Feynman called it "our jewel" and "the most remarkable formula in mathematics."
Running example: alternating current
Electrical engineers rarely write cos and sin explicitly — instead, they represent AC voltage as a rotating vector (phasor) on the complex plane, using Euler's formula:
Why this is more convenient than working with cos/sin directly: adding two AC voltages of the same frequency (for example, in a grid) reduces to adding complex numbers (vectors) — simple parallelogram geometry, instead of cumbersome trigonometric angle addition formulas. Euler's formula is a dictionary/translator between "inconvenient" trigonometry and "convenient" algebra of complex exponentials.
The Fourier transform (which decomposes any signal — sound, image — into a sum of simple oscillations of different frequencies) in its practical form is almost always written using eiθ, rather than cos/sin separately — for exactly the same reason: adding and multiplying complex exponentials is computationally simpler. This is precisely what underlies audio (MP3) and image (JPEG) compression.
Where it leads: quantum phase
Euler's formula did not remain a mathematical curiosity — in the 1920s, Erwin Schrödinger discovered that the wave function of a quantum particle is naturally written via the same exponential with an imaginary exponent:
Here the role of "angle θ" is played by the combination of the particle's momentum p, energy E, and time t. The modulus of such a wave function (the distance of the point from the center, §2) is always 1 — this mathematically guarantees unitarity of quantum evolution: the total probability of finding the particle somewhere always remains equal to 100%, no matter how the phase changes. Without Euler's formula, one would have to separately track the cosine and sine parts of the wave function — with it, everything reduces to one rotating exponential.
An interesting quantum nuance: the angle θ of an individual wave function in isolation is not physically measurable (one can multiply the entire formula by any eiα and the predictions will not change). But if a particle can travel two different paths simultaneously (as in the double-slit experiment), the PHASE DIFFERENCE between the paths becomes measurable — it creates the interference pattern. The reality of the rotation itself on the complex plane remains a subject of philosophical debate, but its mathematical consequences are not.
On the website, this is a separate law, if you want to go deeper:
Schrödinger equation · de Broglie formula · Planck-Einstein equation
Experiment at home
Attach a string to a nail or door handle, tie a small weight (an apple, keys) to the other end, and spin it so that it rotates uniformly in a horizontal plane (like a poi toy). Shine a flashlight on it from the side, strictly horizontally, so that the shadow falls on the wall.
What to notice the shadow on the wall will move not in a circle, but strictly back and forth in a straight line, sinusoidally slowing down at the edges and speeding up in the center — this is the projection of uniform rotation onto one axis (§2, cosθ or sinθ, depending on where you shine the light). Exactly the same thing Euler's formula does mathematically: it takes circular motion and honestly computes its projection.
If you have a spirograph at home (a set of gears with holes for drawing patterns) or you can find an online simulator — draw a few patterns, changing the ratio of the gear radii.
What to notice the beautiful "flower" patterns of the spirograph are literally sums of several rotations with different frequencies (mathematically — the sum of several eiθ with different coefficients on θ), the same principle that underlies the Fourier decomposition of any complex signal into simple oscillations (§5).
Practice problems
First think for yourself, then open the solution. The solution always follows the scheme: Given → Law → Solution → Answer.
1. What is e^(iπ/2) in algebraic form (a + bi)?
Given θ = π/2 (a quarter turn).
Law Euler's formula (§2): eiθ = cosθ + i·sinθ.
Solution cos(π/2) = 0, sin(π/2) = 1 ⇒ eiπ/2 = 0 + i·1.
Answer i — the point exactly at the top of the unit circle.
2. Without explicit calculation, explain why |e^(iθ)| (the modulus of the number) equals 1 for ANY value of θ.
Law geometric meaning of Euler's formula (§2) + the fundamental trigonometric identity.
Solution |eiθ|e^(iθ)| = √(cos²θ + sin²θ) = √1 = 1 — the identity cos²+sin²=1 holds for any angle, so the distance of the point from the center is always exactly one, regardless of θ.
Answer always 1 — the point is obliged to stay on the unit circle.
3. What is e^(i·2π) (a full turn)? What does this say about the periodicity of e^(iθ)?
Given θ = 2π (a full turn).
Law Euler's formula (§2).
Solution cos(2π) = 1, sin(2π) = 0 ⇒ ei·2π = 1 + i·0 = 1.
Answer ei·2π = 1 — after a full turn the point returns to the start; eiθ is periodic with period 2π, as befits a point on a circle.
4. Two AC voltages of the same frequency are represented by phasors U₁ = 3 (along the real axis) and U₂ = 4i (along the imaginary axis). What is the magnitude of the total voltage?
Given U1 = 3, U2 = 4i (perpendicular phasors, §5).
Law addition of complex numbers (geometrically — parallelogram) + modulus of a complex number |a+bi| = √(a²+b²).
Solution U1+U2 = 3+4i; |3+4i| = √(9+16) = √25 = 5.
Answer 5 — the classic 3-4-5 triple, the total voltage is greater than each term individually, but less than their arithmetic sum (7), because the phasors are not aligned in direction.
5. Express cos θ in terms of e^(iθ) and e^(-iθ), using the fact that e^(-iθ) = cos θ − i·sin θ.
Given eiθ = cosθ+i·sinθ, e−iθ = cosθ−i·sinθ.
Law addition of two forms of Euler's formula — the imaginary parts cancel out.
Solution eiθ+e−iθ = 2cosθ ⇒ cosθ = (eiθ+e−iθ)/2.
Answer cosθ = (eiθ+e−iθ)/2 — a standard formula that physicists and engineers use almost more often than the very definition of cosine.
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