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Complex Numbers Multiplication


Complex Multiplication

Multiply magnitudes, add angles
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₁θ₂θ₁+θ₂z₁z₂z₁z₂
Try these
z₁ — First factor
2 + i|z₁| = 2.24, θ₁ = 26.57°
z₂ — Second factor
-1 + 2i|z₂| = 2.24, θ₂ = 116.57°
z₁ · z₂ — Product
-4 + 3i
|z₁z₂| = 5, θ = 143.13°
Algebraic Method (FOIL)
(2 + i)(-1 + 2i)
= (2)(-1) + (2)(2i) + (1i)(-1) + (1i)(2i)
= -2 + 4i + -1i + 2i²
= -2 + 3i + (2)(−1)
= -2 − 2 + 3i
= -4 + 3i
Geometric Method (Polar)
|z₁| × |z₂| = 2.24 × 2.24 = 5
θ₁ + θ₂ = 26.57° + 116.57° = 143.13°
Multiply the lengths, add the angles. The product vector is 5 units long at 143.13° from the real axis.
Key Ideas
Algebraically, complex multiplication uses FOIL (distributive property) and the rule i² = −1. The real part of the product is ac − bd, the imaginary part is ad + bc.
Geometrically, multiplying z₁ × z₂ scales z₁ by |z₂| and rotates it by θ₂ (or equivalently, scales z₂ by |z₁| and rotates by θ₁). The result: |z₁z₂| = |z₁|·|z₂| and arg(z₁z₂) = θ₁ + θ₂.
Multiplying by i rotates by 90°. Try the “i × i” preset — i has modulus 1 and angle 90°, so i² has angle 180° = −1. This is why i² = −1 makes geometric sense.
Multiplying by a real number just scales (stretches or compresses) without rotating. Try “2(−3+4i)” — the angle stays the same, only the length doubles.
This opening pair hides a 3-4-5 triangle: √5 × √5 = 5, at 26.6° + 116.6° = 143.1°. Learn more about the angle arcs · Getting started

Getting Started — Drag and Multiply
Reading the Three Angle Arcs
Multiplication by a Pure Real Number
Multiplication by Pure Imaginary Numbers
Conjugate Pair Multiplication
Off-Screen Products and the Zoom Inset
The Algebraic (FOIL) Method
The Geometric (Polar) Method
Complex Multiplication as Rotation and Scaling
Why i² = −1 Makes Geometric Sense
Related Concepts and Tools




Getting Started — Drag and Multiply

Two draggable points represent the factors z1z_1 (navy) and z2z_2 (orange) on the complex plane. Grab either point and move it to see the green product vector z1⋅z2z_1 \cdot z_2 update in real time, along with the three angle arcs and both step-by-step solution panels.

You can also type values directly into the input fields on the right — each factor has separate real and imaginary inputs accepting values from −10-10 to 1010. Five preset configurations are available below the plane: (2+i)(−1+2i)(2+i)(-1+2i), (1+i)(1−i)(1+i)(1-i), 3×2i3 \times 2i, i×ii \times i, and 2(−3+4i)2(-3+4i). Click Random to generate two arbitrary factors. Each preset has a dedicated section below with the tool frozen on it: the general case, real scaling, pure imaginary factors, the conjugate pair, and i × i.

Each input panel shows the complex number in rectangular form alongside its modulus and argument, so you can track both the algebraic and geometric perspectives simultaneously.

Reading the Three Angle Arcs

Three concentric arcs near the origin display the arguments of both factors and their product. The inner navy arc shows θ1\theta_1 (the argument of z1z_1), the middle orange arc shows θ2\theta_2 (the argument of z2z_2), and the outer green arc shows θ1+θ2\theta_1 + \theta_2 (the argument of the product).

This is the geometric rule in action: when you multiply complex numbers, angles add. The green arc visually spans the combined rotation from both factors. Try clicking the preset (2+i)(−1+2i)(2+i)(-1+2i) — the navy arc covers about 26.6°26.6°, the orange arc covers about 116.6°116.6°, and the green product arc covers their sum at 143.1°143.1°.

Click i×ii \times i to see the cleanest case: each factor has an angle of 90°90°, so the product arc spans 180°180°, placing the result at −1-1 on the negative real axis. This is the geometric proof that i2=−1i^2 = -1 — two quarter-turn rotations compose into a half-turn.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₁θ₂θ₁+θ₂z₁z₂z₁z₂
(2+i)(−1+2i), frozen

All three arcs in play: navy 26.6° plus orange 116.6° gives the teal 143.1°, and the product −4 + 3i lands at modulus 5 — the lengths √5 × √5 multiplied.

The frozen frame is the opening preset, and it rewards close reading. The product (2+i)(−1+2i)=−4+3i(2+i)(-1+2i) = -4+3i has modulus exactly 55 — a hidden 33–44–55 triangle — and the polar check agrees: 5×5=5\sqrt{5} \times \sqrt{5} = 5. The geometric method panel runs this same computation live.

The three arcs stack outward in the order of the calculation: inner factor angle, middle factor angle, outer sum. Whenever the outer arc crosses the 90°90° line while both inner arcs are in the first quadrant, you are watching the product leave the quadrant its factors live in — angle addition making something visibly new.

Multiplication by a Pure Real Number

Click the preset 2(−3+4i)2(-3+4i) to see what happens when one factor is purely real. The navy vector for z1=2z_1 = 2 lies flat on the positive real axis with θ1=0°\theta_1 = 0°. Since its angle is zero, it contributes no rotation — only scaling.

The product −6+8i-6 + 8i points in the exact same direction as z2=−3+4iz_2 = -3 + 4i but is twice as long. The green product vector is a stretched copy of the orange vector. The product's angle equals θ2\theta_2 unchanged because θ1+θ2=0°+θ2=θ2\theta_1 + \theta_2 = 0° + \theta_2 = \theta_2.

Now try changing z1z_1 to −2-2. A negative real number has θ1=180°\theta_1 = 180°, so the product rotates z2z_2 by a half-turn (flipping it through the origin) and scales by 22. The product is 6−8i6 - 8i — same magnitude, opposite direction.

This shows that multiplying by a positive real stretches without rotating, multiplying by a negative real stretches and flips, and multiplying by 11 or −1-1 produces no scaling at all.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₂θ₁+θ₂z₁z₂z₁z₂
2(−3+4i), frozen

Pure stretch: no navy arc exists because 2 has angle zero. The product −6 + 8i points exactly where −3 + 4i pointed, twice as far — modulus 10, grazing the window edge.

In the frozen frame the doubling is exactly at the limit of the window: ∣−3+4i∣=5|{-3+4i}| = 5, so the product −6+8i-6+8i has modulus 1010 and its point grazes the edge of the ±10\pm 10 grid. Push z1z_1 any higher and the green vector turns into the dashed ray described in off-screen products.

Note also which arcs are missing: z1z_1 sits on the positive real axis, so there is no navy arc at all — and the orange factor arc and teal product arc coincide at 126.9°126.9°. No rotation contributed, none observed. That is the pure-scaling half of the rotation-and-scaling decomposition.

Multiplication by Pure Imaginary Numbers

Click 3×2i3 \times 2i to see a real number multiplied by a pure imaginary. Here z1=3z_1 = 3 (angle 0°0°) and z2=2iz_2 = 2i (angle 90°90°). The product is 6i6i — the result sits on the positive imaginary axis at angle 90°90°. Multiplication by ii rotates any vector 90°90° counterclockwise.

Now click i×ii \times i. Both factors are the imaginary unit with modulus 11 and angle 90°90°. The product has modulus 1×1=11 \times 1 = 1 and angle 90°+90°=180°90° + 90° = 180°, landing at −1-1. This is the defining property i2=−1i^2 = -1 demonstrated geometrically — two 90°90° rotations compose into a 180°180° rotation.

Try setting z1=0+2iz_1 = 0 + 2i and z2=0+3iz_2 = 0 + 3i. Both are pure imaginary, so both angles are 90°90°. The product angle is 180°180° and the modulus is 2×3=62 \times 3 = 6, giving −6-6. Two pure imaginary numbers always produce a negative real product because 90°+90°=180°90° + 90° = 180°.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₂θ₁+θ₂z₁z₂z₁z₂
3 × 2i, frozen

A real number turned imaginary: the factor 2i doubles the length and turns 3 by a quarter circle, standing the product 6i upright on the imaginary axis.

The frozen frame captures the quarter-turn in its purest useful form: a real number, 33, lifted off the axis by the factor 2i2i. Length doubles (3×2=63 \times 2 = 6), direction turns by exactly 90°90°, and the product 6i6i stands vertically. One factor supplies all the length change, the other all the rotation.

This is the halfway case between the two extremes on this page: real factors rotate nothing, and the i × i preset rotates twice. The general rule that unifies them is spelled out in the polar method: moduli multiply, arguments add — whatever axes the factors start on.

Conjugate Pair Multiplication

Click (1+i)(1−i)(1+i)(1-i) to see a conjugate pair. These two numbers have the same modulus (2\sqrt{2}) but opposite angles (45°45° and −45°-45°). Their angles cancel: 45°+(−45°)=0°45° + (-45°) = 0°, so the product lies on the positive real axis.

The product is (1+i)(1−i)=1−i2=1+1=2(1+i)(1-i) = 1 - i^2 = 1 + 1 = 2. Geometrically, moduli multiply: 2×2=2\sqrt{2} \times \sqrt{2} = 2. The result is always a positive real number — this is a general rule: z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2.

In the visualizer, the green product vector sits flat on the real axis. The navy arc points upward, the orange arc points downward by the same amount, and the green product arc vanishes because the total angle is zero.

Try setting z1=3+4iz_1 = 3 + 4i and z2=3−4iz_2 = 3 - 4i. The product is 9+16=259 + 16 = 25 — again a real number equal to ∣z1∣2=52=25|z_1|^2 = 5^2 = 25. This pattern is used extensively in complex division and rationalizing denominators.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₁θ₂z₁z₂z₁z₂
(1+i)(1−i), frozen

Mirror-image factors: the navy arc rises 45°, the orange arc dips −45°, and no product arc is drawn — the angles cancel and the product 2 sits flat on the real axis.

The frozen frame shows the angle cancellation as symmetry: the navy arc rises to 45°45°, the orange arc dips to −45°-45°, and no teal arc is drawn at all — the product's argument is exactly zero. The two factor vectors are mirror images across the real axis, and their product 22 sits between them on it.

Work the same case through the FOIL panel and the cancellation reappears algebraically: the two cross terms −i-i and +i+i annihilate, leaving 1−i2=21 - i^2 = 2. Two different mechanisms — angle arithmetic and term cancellation — forced to agree.

Off-Screen Products and the Zoom Inset

When both factors have large moduli, the product can exceed the visible ±10\pm 10 range. In this case the green product vector is replaced by a dashed ray pointing toward the edge of the plane, with an arrow and the label "z1z2→z_1 z_2 \to". The right panel still shows the exact numerical result even though the point is off-screen.

Try setting z1=8+6iz_1 = 8 + 6i and z2=5+5iz_2 = 5 + 5i. The product modulus is 10×52≈70.710 \times 5\sqrt{2} \approx 70.7 — far beyond the grid. The dashed ray indicates the direction, and the result panel gives the exact value.

Conversely, when the product modulus is very small (below 11), a zoom inset appears in the upper-right corner of the plane. This magnified view shows a small grid around the origin with the product point plotted at readable scale. Try z1=0.3z_1 = 0.3 and z2=0.2+0.1iz_2 = 0.2 + 0.1i — the product is tiny, but the inset makes it clearly visible with its own labeled axes and coordinates.

These two features ensure every product is readable regardless of magnitude.

The Algebraic (FOIL) Method

The Algebraic Method panel shows the full FOIL expansion step by step. For two complex numbers (a+bi)(c+di)(a + bi)(c + di), the four terms are:

ac+adi+bci+bdi2ac + adi + bci + bdi^2


Since i2=−1i^2 = -1, the last term becomes −bd-bd. Collecting real and imaginary parts gives:

(ac−bd)+(ad+bc)i(ac - bd) + (ad + bc)i


The panel substitutes your current values and walks through each stage: the four FOIL products, combining the ii terms, replacing i2i^2 with −1-1, and arriving at the final result.

Try (2+i)(−1+2i)(2 + i)(-1 + 2i) to see a worked example: the four terms are −2+4i+(−i)+2i2=−2+3i−2=−4+3i-2 + 4i + (-i) + 2i^2 = -2 + 3i - 2 = -4 + 3i. Each intermediate value updates in real time as you adjust the inputs, making this a dynamic FOIL calculator for any pair of complex numbers.

The Geometric (Polar) Method

The Geometric Method panel shows the polar interpretation of multiplication in two lines:

∣z1∣×∣z2∣=∣z1z2∣|z_1| \times |z_2| = |z_1 z_2|

θ1+θ2=θproduct\theta_1 + \theta_2 = \theta_{\text{product}}


Moduli multiply and arguments add. This is why polar form makes multiplication simple — instead of four FOIL terms and an i2i^2 substitution, you just do one multiplication and one addition.

Using Euler's formula, z1z2=r1eiθ1⋅r2eiθ2=r1r2⋅ei(θ1+θ2)z_1 z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 \cdot e^{i(\theta_1 + \theta_2)}. The exponential form turns multiplication of complex numbers into multiplication of real magnitudes and addition of exponents.

The panel also provides a plain-English summary: "The product vector is [modulus] units long at [angle] from the real axis." This sentence combines both polar components into one geometric description. Compare this to the multi-line algebraic expansion above — for multiplication, the polar approach is dramatically simpler.

Complex Multiplication as Rotation and Scaling

Every complex multiplication can be decomposed into two geometric operations: scaling (changing length) and rotation (changing angle).

Multiplying z1z_1 by z2z_2 scales z1z_1 by the factor ∣z2∣|z_2| and rotates it counterclockwise by θ2\theta_2. Equivalently, it scales z2z_2 by ∣z1∣|z_1| and rotates by θ1\theta_1 — the result is the same either way because multiplication is commutative.

When ∣z2∣=1|z_2| = 1 (the factor lies on the unit circle), multiplication is a pure rotation with no scaling. Set z2z_2 to any point on the unit circle — for example z2=cos⁡(60°)+isin⁡(60°)≈0.5+0.87iz_2 = \cos(60°) + i\sin(60°) \approx 0.5 + 0.87i — and watch z1z_1 rotate by 60°60° without changing length.

When θ2=0\theta_2 = 0 (the factor is a positive real number), multiplication is pure scaling with no rotation. The product points in the same direction as z1z_1 but is stretched or compressed.

This rotation-scaling interpretation is the reason complex multiplication appears throughout physics, signal processing, and computer graphics — it naturally encodes combined scale-and-rotate transformations in a single operation.

Why i² = −1 Makes Geometric Sense

The preset i×ii \times i gives the most fundamental demonstration. The imaginary unit ii has modulus 11 and argument 90°90°. Multiplying by ii means "rotate 90°90° counterclockwise and scale by 11" — a pure quarter-turn.

Applying this twice: i×i=i2i \times i = i^2. Two quarter-turns make a half-turn (90°+90°=180°90° + 90° = 180°), which places the result at −1-1 on the negative real axis. The length stays 1×1=11 \times 1 = 1, so the result is exactly −1-1. This is the geometric reason behind the algebraic definition i2=−1i^2 = -1.

Continuing: i3=i2⋅i=−1⋅i=−ii^3 = i^2 \cdot i = -1 \cdot i = -i (a 270°270° rotation), and i4=i3⋅i=−i⋅i=1i^4 = i^3 \cdot i = -i \cdot i = 1 (a full 360°360° rotation returning to 11). This connects directly to the powers of i cycle: 1→i→−1→−i→11 \to i \to -1 \to -i \to 1.

You can verify each step in the visualizer by setting one factor to ii and the other to successive powers.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθ₁θ₂θ₁+θ₂z₁z₂z₁z₂
i × i, frozen

Both factor points coincide at i. Two 90° arcs stack into the teal half-turn, and the product lands on −1: the geometric fact behind i² = −1.

The frozen frame is the smallest possible multiplication picture: both factors are the same unit point, so the navy and orange markers coincide at ii, their arcs overlap at 90°90°, and the teal arc sweeps the full half-turn to land the product on −1-1. Nothing has any length to spare — every modulus involved is 11 — so the entire event is rotation.

Read against the three-arc anatomy, this is the case where the diagram's whole vocabulary collapses to one statement: a quarter turn done twice is a half turn. The algebra i2=−1i^2 = -1 is that sentence written in symbols.

Related Concepts and Tools

Multiplication is where complex numbers stop feeling like algebra and start feeling like geometry. The "multiply lengths, add angles" rule connects everything else in the toolkit.

The immediate next step from here is Division — it's the same geometric rule run backwards. Where multiplication adds angles and multiplies moduli, division subtracts angles and divides moduli. Running both tools with the same two points side by side makes the inverse relationship clear in a way that algebra alone doesn't.

The geometric method here — angles and moduli — only makes sense in polar form, so the Polar & Rectangular Converter is the natural companion. Every pair of values in the polar panel here (∣z1∣|z_1|, θ1\theta_1, ∣z2∣|z_2|, θ2\theta_2) corresponds directly to what that converter shows for a single number.

The "multiplying by ii rotates by 90°" idea in the i×ii \times i preset is the seed of something bigger. The Powers of i Calculator follows that single observation through its full cycle — each multiplication by ii is another 90° rotation, producing the i,−1,−i,1i, -1, -i, 1 pattern directly from the geometry you see here.

And if you want to see what happens when you apply this operation repeatedly to the same number, De Moivre's Theorem is exactly that — multiplying zz by itself nn times, with the spiral trail showing each intermediate step.