Report length and direction instead of coordinates, and multiplication becomes rotation.
Up to now a complex number has worn its coordinate clothes: z = a + b i, the point (a, b). That dress is perfect for adding — join the arrows tip-to-tail — but clumsy for multiplying, where the cross terms always have to be chased. There is a second outfit that fits multiplication like a glove. Stop reporting where the arrow's tip lands and start reporting how long the arrow is and which way it points: its length r = |z| (the modulus) and its direction θ = arg z (the argument). In this trigonometric form every product becomes a single clean instruction — multiply the lengths, add the angles — so multiplying by a complex number is nothing more than a rotation paired with a scaling, and multiplying by i is exactly a 90° turn. We will lean on the unit circle (cos θ, sin θ) the whole way.
Drop a perpendicular from the tip of the arrow z to the real axis and you get a right triangle whose hypotenuse is the arrow. Its length is r = |z| = √(a² + b²); the angle it makes with the positive real axis is θ = arg z. Reading the two legs off that triangle — the same cos/sin reading we made on the unit circle — gives
a = r cos θ and b = r sin θ, so z = a + b i = r cos θ + (r sin θ) i = r(cos θ + i sin θ).
That last line is the trigonometric form (sometimes "polar form"). It carries exactly the same information as a + b i — just reorganized into size and direction.
z = r(cos θ + i sin θ), with r = |z| ≥ 0 the modulus and θ = arg z the argument. Going one way: a = r cos θ, b = r sin θ. Going back: r = √(a² + b²) and θ is the angle of (a, b) from the positive real axis.
From coordinates to trig form. Take z = 1 + i. Then r = √(1² + 1²) = √2, and the point (1, 1) sits on the 45° ray, so θ = 45°. Hence 1 + i = √2(cos 45° + i sin 45°). Check the legs: √2·cos 45° = √2·(√2/2) = 1 ✓ and √2·sin 45° = 1 ✓.
From trig form to coordinates. Take 2(cos 120° + i sin 120°). Since cos 120° = −½ and sin 120° = √3/2, the legs are a = 2·(−½) = −1 and b = 2·(√3/2) = √3, so the number is −1 + √3 i. The figure and the readout below will agree on this exactly.
The modulus is never negative: r = |z| ≥ 0. The argument is read from the positive real axis, counter-clockwise positive, and is only fixed up to whole turns (±360°); we report it in (−180°, 180°]. And 0 has a modulus (0) but no argument — a zero-length arrow points nowhere.
Here is the payoff. Write two numbers in trig form, z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), and multiply them out. Group the real and imaginary parts:
z₁z₂ = r₁r₂[(cos θ₁cos θ₂ − sin θ₁sin θ₂) + i(sin θ₁cos θ₂ + cos θ₁sin θ₂)].
The two brackets are precisely the angle-sum identities from trigonometry: cos(θ₁+θ₂) and sin(θ₁+θ₂). They collapse to
z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].
To multiply two complex numbers, multiply the moduli and add the arguments: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂. So multiplying by a complex number rotates (by its argument) and scales (by its modulus) — one geometric act, not two separate ones.
Worked example. Let z = 2 + i and w = 1 + 2i. The straight product is (2 + i)(1 + 2i) = 2 + 4i + i + 2i² = 5i. Now the trig story: |z| = √5 and |w| = √5, so the length should be √5·√5 = 5 — and indeed |5i| = 5 ✓. The arguments add to arg z + arg w ≈ 26.57° + 63.43° = 90°, and arg(5i) = 90° ✓. Same answer, told as a rotation-and-scale.
Division is the mirror image. If multiplying adds the angles and multiplies the lengths, then dividing must undo both: subtract the angles and divide the lengths.
z₁/z₂ = (r₁/r₂)[cos(θ₁ − θ₂) + i sin(θ₁ − θ₂)] (z₂ ≠ 0).
So |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg z₁ − arg z₂. This is the trig-form view of the conjugate trick from 31.3: rationalizing with w̄ was secretly subtracting w's angle and dividing by its length.
Worked example. Suppose z has length 6 at 80° and w has length 2 at 30°. Then z/w has length 6/2 = 3 and angle 80° − 30° = 50°: a shorter arrow, turned back by w's angle. The widget below dials this live.
What is i in trig form? Its modulus is |i| = 1 and its argument is arg i = 90°, so i = 1·(cos 90° + i sin 90°). By the rule of §31.5.2, multiplying any z by i keeps its length (×1) and adds 90° to its angle.
i · z rotates z by +90° counter-clockwise and leaves |z| unchanged — the cleanest rotation in mathematics. Do it twice and you have turned 180°: i²·z = −z, which is exactly the statement i² = −1 seen as a half-turn. Four times brings you home: z → iz → −z → −iz → z.
Why this is consistent. Take z = 3 + i. Then i·z = i(3 + i) = 3i + i² = −1 + 3i — a different point, but the same distance from O: |3 + i| = √10 and |−1 + 3i| = √10 ✓. And its angle climbed by 90°: arg(3 + i) ≈ 18.43°, while arg(−1 + 3i) ≈ 108.43° ✓. The arrow simply pivoted a quarter-turn.
When multiplying you add the arguments and multiply the moduli — never multiply the angles or add the lengths. And "rotate" is not a new operation bolted on: it is what multiplying by a complex number means.
(A taste of what's ahead: repeating "scale and turn" n times gives De Moivre's rule, zⁿ = rⁿ(cos nθ + i sin nθ) — powers scale the length and multiply the angle. It is the engine behind roots of unity, just over the horizon.)
One number, two outfits — and the second one is built for multiplication.
| Idea | The rule | In words |
|---|---|---|
| Trig form | z = r(cos θ + i sin θ) | r = |z| (length), θ = arg z (direction); a = r cos θ, b = r sin θ |
| Multiply | r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)] | multiply the lengths, add the angles → rotate & scale |
| Divide | (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)] | divide the lengths, subtract the angles |
| Multiply by i | i·z = z turned +90° | same length, a quarter-turn CCW; i² = −1 is a half-turn |
| Power (De Moivre) | zⁿ = rⁿ(cos nθ + i sin nθ) | scale the length to the nth power, multiply the angle by n |
Next, in 31.6, we cash all of this in: a negative discriminant is no longer a dead end, and every quadratic gets exactly two roots — arriving as a conjugate pair when Δ < 0.
r = √((−1)² + (−1)²) = √2. The point (−1, −1) is in quadrant III on the 225° ray, which we report as θ = −135° (in (−180°, 180°]). So z = √2(cos(−135°) + i sin(−135°)).
cos 30° = √3/2, sin 30° = ½, so a = 4·(√3/2) = 2√3 and b = 4·(½) = 2. Hence 2√3 + 2i (≈ 3.46 + 2i).
Multiply lengths: |zw| = 3·2 = 6. Add angles: arg(zw) = 50° + 70° = 120°. So zw = 6(cos 120° + i sin 120°) = −3 + 3√3 i.
Divide lengths: |z/w| = 8/4 = 2. Subtract angles: arg(z/w) = 110° − 35° = 75°. So z/w = 2(cos 75° + i sin 75°).
Each ×i is a +90° turn at fixed length √13. i·z = i(2 + 3i) = 2i + 3i² = −3 + 2i; i²·z = −2 − 3i; i³·z = 3 − 2i. The arrow has pivoted three quarter-turns (270° CCW), same distance from O the whole way.
Because |i·z| = |i|·|z| = 1·|z| = |z|: moduli multiply, and |i| = 1, so the length is untouched — only the direction turns (by +90°).
Six questions to lock it in. Tap the answer you think is right.
This lesson develops the trigonometric (polar) form z = r(cos θ + i sin θ) and the geometric meaning of multiplication and division, addressing CCSS HSN-CN.B.4 (represent complex numbers in rectangular and polar form and explain why they represent the same number) and HSN-CN.B.5 (multiplication and conjugation of complex numbers geometrically on the complex plane). The angle-sum identities from trigonometry (Stage 25) and the modulus from 31.4 are the only prerequisites. A good prompt at the dinner table: "If multiplying by i turns an arrow 90°, what does multiplying by i four times do — and why does that match i⁴ = 1?"