e^(iθ) = cos θ + i sin θ is one arrow spinning on a circle. Complex multiplication rotates and scales, roots of unity turn the circle into a frequency ruler, and RoPE turns it into relative position inside a transformer.
Defined by i² = −1, i looks like a trick — until you draw it. Multiplying by i is a 90° turn, so i² is two quarter turns: a half-turn, which is multiplying by −1. The powers cycle 1, i, −1, −i and come home after four steps.
1 → i → −1 → −i → 1(a + bi)(c + di) looks like bookkeeping, but on the plane it is a single motion: rotate by the sum of the angles and stretch by the product of the lengths. Rectangular form is for adding; polar form makes multiplying trivial.
|z·w| = |z|·|w| · arg(zw) = arg z + arg we^(iθ) = cos θ + i sin θ traces the unit circle. Its N evenly spaced stops, the roots of unity, sum to zero — that is what makes the DFT invertible (next lesson). Rotating q by m·θ and k by n·θ makes attention depend only on m − n: RoPE.
e^(iθ) = cos θ + i sin θA complex number is a point in a plane written a + bi, and multiplying by i is a quarter turn: two quarter turns make a half turn, which is multiplying by −1, and that single fact is all “i² = −1” means.
By the end you will be able to do complex arithmetic by hand, convert between rectangular and polar form, explain why multiplication rotates and scales, derive Euler’s formula from power series, use roots of unity to see what the DFT measures, and say exactly why RoPE makes attention relative.
On the number line the answer does not exist: every square is non-negative. Step off the line into the plane and the question turns into a rotation.
A complex number has two parts: z = a + bi. The real part a sits on the familiar number line; the imaginary part b is measured along a second axis, and i is a new number defined by the single property i² = −1. That is the whole definition. The word “imaginary” is a 17th-century insult that stuck — negative numbers were called absurd once too.
The definition looks like a trick until you stop treating numbers as positions and treat them as moves. Multiplying by −1 flips an arrow to point the other way: a half turn. Multiplying by i is a quarter turn. Two quarter turns are a half turn, so i·i = −1, exactly as promised. Multiplication by i is not an amount, it is an instruction: turn 90° counter-clockwise.
Numeric check on a point that is not 1: take 2 + 0.5i and apply i once. (2 + 0.5i)·i = 2i + 0.5i² = −0.5 + 2i. The length is unchanged (√4.25 on both sides); only the direction turned by 90°. Multiplying by i never stretches, only rotates.
Each press sends z₀ through another 90° rotation. Four turns bring it home, which is why i⁴ = 1 and i² = −1.
i is not an amount, it is an instruction: rotate a quarter turn. Everything else about it follows.
Multiplying a complex number by i twice is the same as…
Plot the real part across, the imaginary part up, and the whole number system becomes a map. The map is called the complex plane, and it is where complex arithmetic turns into geometry.
The horizontal axis is the real axis; the vertical axis is the imaginary axis. The number z = a + bi is the point (a, b). So 3 + 2i is (3, 2), the real number −1 is (−1, 0), and the purely imaginary 4i is (0, 4). Nothing new is hidden: the plane is just a place to put the two parts.
The second reading is the one that matters. A point is also an arrow from the origin — a vector, in the language of Lesson 01. Two complex numbers can be added as arrows (head to tail) and multiplied as rotations and stretches. That double life — point and vector, position and transformation — is why complex numbers keep showing up wherever something spins.
Drag the blue point z (or use the sliders). It is multiplied by w = r·e^(iθ); the orange arrow is the product, turned by θ and scaled by r. The purple dashed arrow is the conjugate, reflected across the real axis.
Magnitudes multiply, angles add. Every multiplication in the plane is one rotation (by the angle) and one scale (by the length).
A point can be described by its coordinates (a, b) or by its distance from the origin r and its angle θ measured from the positive real axis. The conversion is the one from Lesson 01, with new names.
In words: the magnitude is Pythagoras, the phase is atan2 (which reads the quadrant from the signs), and the pair (r, θ) names the same point as (a, b). Polar form is just that pair, written r·(cos θ + i sin θ).
Addition walks two arrows head to tail. Multiplication looks like bookkeeping, but on the plane it does exactly two things: turn by the sum of the angles, stretch by the product of the lengths.
Addition is part-by-part: add the real parts, add the imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)i, so (3 + 2i) + (1 + 4i) = 4 + 6i. On the plane this is vector addition — walk one arrow, then the other.
Multiplication expands like any two brackets, with one extra rule: every i² becomes −1. (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The formula looks lopsided, but the geometry it produces is perfectly even: the lengths multiply, and the angles add.
Edit z = a + bi and w = c + di, pick an operation, and read every step of the arithmetic before the result.
FOIL the source example, then check the answer against the two rules that do not mention FOIL at all.
Plain-English reading: multiplication never moves a point to a random place. It picks the arrow up, turns it by the second number’s angle, and stretches it by the second number’s length.
The conjugate reflects a point across the real axis. Multiply a number by its own reflection and the imaginary parts cancel, leaving a single real number: the squared length.
The conjugate of z = a + bi is z̄ = a − bi: same point, mirrored across the real axis. That little sign flip is the workhorse of complex algebra, because the product z · z̄ = a² + b² is always real — the cross terms cancel. And a² + b² is Pythagoras, so z · z̄ = |z|², the squared distance from the origin.
The magnitude |z| = √(a² + b²) is the ruler: the length of the arrow. The conjugate is also how division works. Dividing by a complex number means multiplying top and bottom by its conjugate, which makes the denominator real — no different in spirit from clearing a square root from a denominator.
In plain English: conjugation turns magnitude into arithmetic. The mirror image is chosen precisely so that the two imaginary parts annihilate, and what survives is the squared length.
What is (3 + 4i)(3 − 4i)?
The most important formula in this lesson compresses the two projections of a rotating arrow into a single exponential: e^(iθ) = cos θ + i sin θ.
Polar form says z = r·(cos θ + i sin θ), where r is the length and θ the angle. Euler’s formula replaces the bracket with a complex exponential: z = r·e^(iθ). The two forms are the same point; the exponential is shorter, and — as the previous chapter showed — multiplication in it is trivial.
On the unit circle (r = 1) the formula reads e^(iθ) = cos θ + i sin θ. At θ = 0 the point is 1; at θ = π/2 it is i; at θ = π it is −1, which is the famous e^(iπ) + 1 = 0 linking e, i, π, 1 and 0. A full trip is θ = 2π and lands back at 1.
Slide θ around the circle. The arrow is e^(iθ); its shadow on the real axis is cos θ and its shadow on the imaginary axis is sin θ.
The point never leaves the circle: the magnitude is always 1. Only the angle changes, which is why e^(iθ) is a pure rotation.
The exponential, cosine and sine all have power series (Lesson 04). Substitute x = iθ into the exponential’s and use the cycling powers of i: i² = −1, i³ = −i, i⁴ = 1.
Plain-English reading: an exponential with an imaginary exponent is not a growth curve, it is a point going around a circle. The real part of that motion is a cosine wave, the imaginary part is a sine wave — the same wave, shifted a quarter period.
What is e^(iπ/2)?
Take the unit circle and mark N equally spaced points. Those points are the roots of unity — and their sum is zero for every N. That one fact is what makes the Fourier transform work.
The N-th roots of unity are w_k = e^(2πik/N) for k = 0, 1, …, N−1. Each is a point on the unit circle, and together they sit exactly 360°/N apart. For N = 4 they are the compass points: 1, i, −1, −i. For N = 8, add the diagonals.
Two properties make them the natural frequency vocabulary. First, every root has magnitude 1, so multiplying by a root never changes a signal’s size — it only rotates its phase. Second, the roots sum to zero: N arrows pulling equally in all directions cancel. That cancellation is what lets the DFT separate one frequency from another, and it is the forward link to Lesson 20.
N evenly spaced points on the unit circle. The grey arrow is their vector sum — it stays pinned at zero for every N.
Multiplying any root by w₁ jumps to the next one, so w₁ generates all of them — that is what “primitive” root means.
Let w = e^(2πi/N) be the first root. The others are w², w³, …, and w^N = e^(2πi) = 1 closes the loop. Multiply the sum by w and watch what happens.
The geometry and the algebra say the same thing: evenly spaced arrows have no preferred direction, so their vector sum is the only point with no direction at all — zero.
For N = 6, the primitive root w₁ = e^(2πi/6) is one sixth of the way around the circle. What is w₁ · w₁?
Attention compares a query vector with a key vector by a dot product. RoPE rotates each one by its token position before the comparison, so the score ends up depending on m − n — the relative distance.
A transformer has no built-in sense of order: shuffle the tokens and the attention scores are identical. Position must be injected. The original Transformer adds sinusoidal patterns to the embeddings; RoPE (Rotary Position Embedding) takes the rotation idea literally and multiplies the query and key vectors instead.
Concretely, RoPE splits a query into 2-D pairs and rotates each pair by an angle proportional to the token’s position: q at position m becomes q·e^(imθ), and k at position n becomes k·e^(inθ). When attention takes their dot product, the rotation angles meet as an angle difference: what is left is e^(i(m−n)θ). Absolute position vanishes; only the gap between the tokens remains.
Give a query position m and a key position n. RoPE rotates each by its own position; the score on the right depends only on m − n. Shift both tokens by five and nothing changes.
This lab rotates one 2-D pair. Real RoPE does this to every pair of coordinates inside a head, each pair at its own frequency — the same idea, repeated. Simplified teaching model.
Write the 2-D pair of the query as a complex number q and the key as k. RoPE rotates them by their positions, and the attention score is the real part of q′ times the conjugate of k′ — the complex spelling of the 2-D dot product.
Plain-English reading: positions are not added on as separate numbers, they are encoded as angles. Two tokens 10 apart look the same whether they are at positions (0, 10) or (90, 100), because only their angular difference is visible to the dot product.
Answer before you look. The Euler and roots-of-unity questions are the ones that separate “I can compute with i” from “I can see the rotation”.
0 / 5 answered · 0 correct
01What is the imaginary unit i defined by?
02What does Euler's formula e^(iθ) equal?
03What is (3 + 2i)(1 + 4i)?
04Why are complex numbers used in Rotary Position Embedding (RoPE) for transformers?
05The N-th roots of unity are N equally spaced points on the unit circle. What is their sum?
Click a card to swap the lazy description for what it actually means.
Four problems, all fully worked with numeric checks. Try first; the answer is one click away.
Multiplication: (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 8 + 10i + 3 = 11 + 10i. Check: |2 + 3i| = √13 ≈ 3.6056, |4 − i| = √17 ≈ 4.1231, so the product magnitude should be √221 ≈ 14.8661, and |11 + 10i| = √(121 + 100) = √221 ✓. Angles: atan2(3, 2) ≈ 56.31° and atan2(−1, 4) ≈ −14.04° sum to 42.27°, and atan2(10, 11) ≈ 42.27° ✓. The product is the first factor turned by the second factor's angle (−14.04°) and stretched by its length (√17 ≈ 4.1231). Division: (5 + 2i)/(1 − 3i) = (5 + 2i)(1 + 3i) / ((1)² + (−3)²) = (5 + 15i + 2i + 6i²)/10 = (−1 + 17i)/10 = −0.1 + 1.7i. Check by multiplying back: (−0.1 + 1.7i)(1 − 3i) = −0.1 + 0.3i + 1.7i − 5.1i² = −0.1 + 2.0i + 5.1 = 5.0 + 2.0i ✓. Geometrically the division rotated 5 + 2i by the negative of the denominator's angle (75.96° → −(75.96°)) and scaled it down by |1 − 3i| = √10 ≈ 3.1623.
Each multiplication adds 30°: after k steps the point is (cos(kπ/6), sin(kπ/6)). After k = 12 the angle is 12·30° = 360°, so the point is (cos 2π, sin 2π) = (1, 0) ✓. A sample of the walk: k = 0 (1, 0); k = 1 (0.8660, 0.5); k = 2 (0.5, 0.8660); k = 3 (0, 1); k = 6 (−1, 0); k = 9 (0, −1); k = 12 (1, 0). Each point is exactly 30° from its neighbours, and every point has magnitude 1 by construction, so the points are equally spaced on the unit circle — a regular 12-gon. In general, multiplying by e^(iθ) n times gives e^(inθ): n rotations through θ are one rotation through nθ.
A real sine at an integer bin k₀ concentrates all of its energy at bins k₀ and 32 − k₀: a full sine of amplitude A sums to |X[k₀]| = A·N/2 = 16A. For the amplitude-1 tone at bin 3 that gives |X[3]| = 16 (and |X[29]| = 16); for the amplitude-0.5 tone at bin 7 it gives |X[7]| = 8 (and |X[25]| = 8). Every other bin is zero up to round-off, because the roots of unity are orthogonal: Σₙ e^(2πi(k−k′)n/N) = 0 whenever k ≠ k′. So the spectrum has peaks at 3 and 7 with heights 16 and 8 — the second exactly half the first, matching the input amplitudes. That is the DFT read as a bank of N correlations, one per frequency.
The roots are w_k = e^(2πik/8) = cos(45k°) + i sin(45k°): k = 0 → 1; k = 1 → 0.7071 + 0.7071i; k = 2 → i; k = 3 → −0.7071 + 0.7071i; k = 4 → −1; k = 5 → −0.7071 − 0.7071i; k = 6 → −i; k = 7 → 0.7071 − 0.7071i. Sum: reals 1 + 0.7071 + 0 − 0.7071 − 1 − 0.7071 + 0 + 0.7071 = 0 and imaginaries 0 + 0.7071 + 1 + 0.7071 + 0 − 0.7071 − 1 − 0.7071 = 0 ✓. Stepping: w₁·w₁ = e^(iπ/4)·e^(iπ/4) = e^(iπ/2) = i = w₂; likewise w₁·w₇ = e^(2πi) = 1 = w₀. Matrix equivalence: rotating (2, 1) by 45° by hand gives (2·0.7071 − 1·0.7071, 2·0.7071 + 1·0.7071) = (0.7071, 2.1213); multiplying (2 + i)(0.7071 + 0.7071i) = 1.4142i + 0.7071i² + 1.4142 + 0.7071i = (1.4142 − 0.7071) + (1.4142 + 0.7071)i = 0.7071 + 2.1213i ✓. The two computations agree to floating-point precision (about 1e−16 for random inputs).
You do not need to master these here. Each one gets a proper treatment in its own lesson; the one-line meaning is enough to keep reading. Orange dotted underlines in the prose point back to this list.
This lesson is a port of an open course. Everything here traces back to it — and the next step is running the code yourself.
Lesson text adapted from AI Engineering from Scratch (Phase 01, Lesson 19) and the Math Foundations Notebook reference build. The Euler dial, complex plane playground, multiply-by-i lab, roots-of-unity lab, RoPE lab, complex arithmetic console, the second worked multiplication and division examples, the numeric series check inside the Euler derivation, the N = 4 and N = 3 root-sum checks, and the relative-position invariance demo are original to this page. Every number shown is computed in your browser. All labs run in your browser.