Euler's Formula in its Full Elegance

They tell you that the most beautiful equation in mathematics is

e^iπ + 1 = 0

Five fundamental constants — 0, 1, e, i, π — bound together in a single line, together with the fundamental signs: power, multiplication, addition, and equal. It is on T-shirts, tattoos, dorm-room posters. It is the cliché of mathematical beauty.

It is not the most beautiful equation, because it only tells half of the story — literally. The version that tells the full story, without rearrangement, without padding, without any patching, is

exp(iτ) = 1

Throughout this article, τ (tau) stands for one full rotation. It is the ratio of a circle's circumference to its radius: wrap a thread around any circle and the thread is τ ≈ 6.283 radii long. One full turn is τ radians.

This article derives Euler's formula from a single observation, finds the unit circle hiding inside it, walks through where the formula gets used in the world, and finally shows why the often-quoted identity is simply a half-frame of a movie that was never paused there for any reason but a historical accident.

Suppose we have a chart of some quantity v plotted against time t. We want to find the total area underneath that curve — how much has accumulated from the start to any given point.

The strategy: divide the area into thin vertical strips. Each strip is narrow enough to treat as a flat-topped rectangle. Stack them side by side and you have an estimate of the whole area; the thinner the strips, the better the estimate.

We name the strip width dt — the letter d is short for delta, meaning a tiny piece of the quantity that follows it. The height of the strip at that moment is v, so the strip's area is v · dt. We call that tiny area dx — a small addition to the running total x:

dx  =  v · dt

The right chart starts at zero and adds each dx one by one — it is the running total x of accumulated area. Select a function on the left to see how the strips build it up:

Each right-hand chart is the accumulated area of the function on its left. Starting from v = 1, a flat constant, the area grows linearly into x = t. Take v = t next, and the area curves into x = t²/2. One more step: v = t²/2 accumulates into x = t³/6. Each accumulation adds one power of t.

Part 1 — When both charts show the same curve

The charts above always show different functions — the left and right curves never coincide. But here is a different question: can we find a function where both charts show the same curve — where v and x are one and the same?

Such a function would be self-amplifying: its rate of growth at every moment equals its current value. The taller the right chart at any point, the steeper its slope — the more it has grown, the faster it keeps growing. A savings account compounds this way: a larger balance earns more interest, which grows the balance further, which earns still more. Population growth, radioactive decay, and the charging of a capacitor all work the same way — all governed by the same curve.

How do we find it? We use the fold-and-add-the-difference method. Start with any trial v — it does not have to be right. Fold it: build the change-accumulation chart to get x. The two charts will not match at first. But x tells you exactly what v is missing: anything that is in x but not yet in v is the gap. Add that gap back to v to produce a better trial, then fold again. Each round the gap gets smaller. Keep going, and the two charts are forced closer and closer together until they are one and the same.

The simplest trial is v = 1 — a flat constant. Its integral is x = t. Not the same, but the right chart immediately hands us the missing piece: +t. Add it: v = 1 + t. Integrate again: x = t + t²/2. Another missing piece. Add it. Watch each repair generate the next:

As the terms accumulate, both charts converge toward the same shape. At the limit they trace the same curve — separated by exactly 1, because v starts at 1 while x starts at 0.

That gap closes the moment we let x start at 1 instead of 0. Both charts then carry the same equation, and their blocks and risers match at every step:

The rate of change of this function equals the function itself — everywhere, always. It is its own derivative. This is a remarkable constraint, and it pins down a unique function. We name it exp:

exp(t)  =  1 + t + t²2! + t³3! + t⁴4! + ···

The symbol n! is called a factorial: the product of every whole number from 1 up to n. So 2! = 1×2 = 2, 3! = 1×2×3 = 6, 4! = 1×2×3×4 = 24. These are exactly the denominators we saw from accumulation in Part 1: t²/2 and t³/6. The 2 in 2! is the triangle factor from 2D; the 3 in 3! builds on top of it with the pyramid factor from 3D; each new dimension multiplies in one more number. A later series will explain exactly why.

What other functions share this property — rate of change equal to value? The answer is exactly the family x = c · exp(t), where c is any constant. Each choice of c gives a different member: the same shape, just scaled. The constant is simply the starting value, since x(0) = c · exp(0) = c. There are no other solutions. The case x = 0 is not an exception — it is the same family with c = 0.

Part 2 — One key property, and one question

Two facts first. The easier one: plug t = 0 into the series. Every term after the first contains t as a factor, so all of them vanish:

exp(0)  =  1 + 0 + 0²2! + ···  =  1

The second fact: x = c · exp(t) is the only family where rate of change equals value — every starting value c gives one member, and there are no other solutions. The proof of this is skipped here.

Now for the key identity. Write out both series and multiply every term of the first by every term of the second:

Every cell on the same diagonal has the same total power of a and b. Add each diagonal:

Each diagonal sum equals (a+b)ⁿ/n! by the binomial theorem. Adding all diagonals together:

exp(a) · exp(b)  =  1 + (a+b) + (a+b)²2! + (a+b)³3! + ···  =  exp(a + b)

This article now asks: what happens when we feed exp a number that rotates rather than scales? To answer it, we need two more functions: cos and sin.

In algebra, the difference of squares has a standard factorization:

x² − y²  =  (x + y)(x − y)

The sum of squares, x² + y², refuses the same treatment. But suppose we allow a number i with one defining property: i² = −1. Call it an invisible number — it has no place on the ordinary number line, yet it obeys all the usual rules of algebra. With it, the sum of squares becomes a difference in disguise:

x² + y²  =  x² − (iy)²  =  (x + iy)(x − iy)

A number combining an ordinary part and an invisible part — like x + iy — is called a complex number.

Part 1 — The unit circle

Place a point on the circle of radius 1. It starts at (1, 0) and walks counterclockwise. After covering arc length θ, its horizontal and vertical positions are two numbers we name cos(θ) and sin(θ). That is their definition — they are the coordinates of the point.

Since the point always sits on the unit circle, the Pythagorean theorem gives:

cos²(θ) + sin²(θ) = 1

Walking the arc the other way — angle −θ clockwise — lands the point at the same horizontal reach but a flipped vertical height:

The Pythagorean identity is a sum of squares. Using i² = −1, a sum of squares factors just like a difference:

(cos(θ) + i sin(θ)) · (cos(θ) − i sin(θ)) = 1

The chart showed exactly this: cos(−θ) = cos(θ) and sin(−θ) = −sin(θ). Substituting into the second factor:

cos(θ) − i sin(θ)  =  cos(−θ) + i sin(−θ)

That is the same pattern as the first factor — but at angle −θ. Name the pattern and write both factors at once:

Their product is the Pythagorean identity:

p(θ) · p(−θ) = 1

A familiar property

This product-equals-one pattern appeared in §1. From the exp property:

exp(t) · exp(−t)  =  exp(t − t)  =  exp(0)  =  1

Both p and exp satisfy the same equation. Is p secretly the same function as exp, seen from a different angle? Let us expand p(θ) as a power series and find out.

Part 2 — Constructing cos

I will state some facts (to be proved in a later series): on a circle, each fold of cos produces −sin, and each fold of sin produces cos. So folding cos twice gives back −cos: the function returns but with its sign reversed.

To construct cos, we use the same "fold-and-add-the-difference" method as in §1, but with a crucial difference. In §1, exp equalled its own fold: v = x. Whenever the fold result x showed something v was missing, we added it back. Here, cos equals the negative of its double-fold. So when the double-fold gives us a difference, instead of adding it back (which would make things worse), we subtract it. Wherever x shows a gap, we add a negative version of that gap back to v, in the hope of bringing v closer to its negative self after a double-fold. We start the trial again with v = 1:

As the repairs accumulate, v converges to cos. The charts below show the converged state. The left chart shows v ≈ cos as blocks. The right shows the double-fold result x ≈ 1 − cos as risers. Each fold of cos produces −sin; folding again produces −cos. With the initial condition folded in, the accumulated double-fold becomes 1 − cos. The two charts are complementary: at every θ, left plus right equals 1. That is the defining equation: v = 1 − x.

Part 3 — Constructing sin

Sin has the same fighting property: fold sin twice and you land on −sin. So the same logic applies: the double-fold result x is the opposite of what v should be, so we subtract it. The starting value this time is θ (because sin(0) = 0 and the slope at 0 is 1), so the rule is: new v = θ − x. Subtract the double-fold, and the trial is driven step by step onto sin.

As the repairs accumulate, v converges to sin. The left chart shows v ≈ sin as blocks. The right shows the double-fold result x ≈ θ − sin as risers. The two charts sum to θ at every point — that is the defining equation: v = θ − x.

As the terms accumulate, both approximations lock onto their targets.

The three functions each have a power series. Placed side by side, the connection becomes visible:

Reading down the columns: cos collects every even-degree term of exp. The i-powers in those columns (i⁰ = 1, i² = −1, i⁴ = 1, …) are real and alternating — exactly the signs of the cos series. sin collects the odd-degree terms. The i-powers there (i¹ = i, i³ = −i, i⁵ = i, …) are imaginary and alternating — the signs of the sin series, each carrying a factor of i.

Add the two rows together. Each term is i^kθ^k/k! = (iθ)^k/k! — the same pattern as the exp series, but with argument iθ instead of t. This is Euler's formula:

exp(iθ)  =  cos(θ) + i sin(θ)

The exponential that grew without bound in §1, when given an invisible input, is tamed onto the unit circle and rotates forever. At θ = τ — one full circle — the point returns to where it started:

exp(iτ)  =  1

N is the number of terms taken from the power series. At N = 1 only the constant term appears; each additional term refines the approximation. The chart titles update to show which terms are included. Press ▶ to watch the curves converge, or drag θ to see the current value on every chart at once.

Part 1 — The equation at work

The spinning-arrow picture appears across all of modern science and engineering. The same idea, four times:

AC circuits. A sinusoidal voltage is just the shadow (real part) of a spinning arrow:

V₀ cos(ωt) = Re[V₀ · exp(iωt)]

Each component — resistor, capacitor, inductor — just multiplies the arrow by one complex number. Differential equations become arithmetic.

Fourier analysis. Any signal is a sum of spinning arrows, one per frequency:

f(t) = Σ c_n · exp(inωt)

Arrows at different frequencies are deaf to each other — they cancel when averaged over a full cycle. This lets phone calls, Wi-Fi, and radio share the same air without colliding.

Quantum mechanics. A particle with energy E carries a spinning arrow:

ψ(t) = exp(−iEt/ℏ) · ψ₀

Probabilities (length squared) are unchanged by spinning. But when two paths reach the same point, their arrows either add or cancel. The double-slit experiment is two arrows doing arithmetic.

Electromagnetic waves. Light, radio, X-rays — all are oscillating fields described by a spinning arrow moving through space and time:

E = exp(i(kx − ωt))

The physical field is the real part — the horizontal shadow of the arrow as it spins in space and time. When two waves meet, their arrows add: aligned arrows give a bright fringe, opposed arrows cancel to darkness. That is interference — the geometry behind diffraction gratings, soap-bubble colours, radio antenna arrays, and medical imaging.

Part 2 — Full circle vs half circle

Part 3 — Why π dominated for 2000 years

Nobody measures circles by diameter anymore. Every modern formula that involves a circle uses the radius: areas, oscillation periods, Fourier kernels, phase angles. The natural constant for all of these is the ratio of circumference to radius, which is τ. Instead we inherited π (circumference to diameter) from ancient geometers who measured by laying a string across, not around. That historical accident costs a factor of 2 everywhere:

Every 2π is a paper cut from measuring the wrong thing 2000 years ago. And the symbol itself carries its own accident: in Euler's time Greek letters were used the way we use x and y today — no fixed meanings, just convenient variables. Euler's most influential work, Introductio in analysin infinitorum (1748), happened to use π for the circumference-to-diameter ratio. Once Euler used it, the entire mathematical world followed — not because the half-circle ratio is more natural, but because one famous book crystallised the convention. τ arrived later and found the field already occupied. In terms of naturalness, τ almost always appears more simply or more intuitively: it is the full circle, not half of one. (See also: Hartl, The Tau Manifesto.)

Part 4 — The aesthetic question

The defence of e^iπ + 1 = 0 is that it contains five fundamental constants: 0, 1, e, i, π. But 0 and 1 are grammatical — any equation A = B can be rewritten as A − B = 0 or A · 1 = B. The τ version gives the same count:

The substantive difference: e^iπ + 1 = 0 is three statements glued together (rotate halfway, reach −1, add 1, reach 0). e^iτ = 1 is one statement: rotate fully, return home. Side by side:

One statement is more elegant than three. And it generalises cleanly: add any integer k to the exponent and it still holds —

exp(i · k · τ) = 1   for any integer k

because each extra τ is one more full rotation, always landing back at 1. The same trick fails for the π version: e^ikπ + 1 = 0 only holds for odd values of k — for even k the point lands at +1, not −1, and the equation breaks. The τ identity works for every integer without exception. The π identity works for half of them.

If e^iπ + 1 = 0 deserves a place on your T-shirt, it is on the back — a half turn. The front belongs to e^iτ = 1.

Give exp a full turn — iτ — and it traces a perfect circle and returns home. Every π in the formulas was τ folded in half, a compass quietly hidden in the algebra.

Law 1: wherever you see π, τ fits more naturally — π is always half a circle. Law 2: whenever τ appears in an equation, a circle will always emerge somewhere. The next article asks: what is ∫ exp(−x²/2) dx? The answer hides √π — which turns out to be √τ once you find the circle inside. Continue to: Natural Derivation of the Gaussian Integral →