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Math · Complex Numbers and Parametric Motion

Complex Numbers on the Plane

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Unit Circle / Sine and Cosine from RotationKeep the same turn-on-a-plane picture and reuse it for trig coordinates.

Key takeaway

A complex number can be read as the point (a, b) and as the vector from the origin to that point.
Addition on the complex plane is the same head-to-tail story as vector addition.
Multiplication changes size and direction together because magnitudes multiply and arguments add.

Common misconception

Complex multiplication is not only a symbolic component rule. The geometry is the clearest way to read what the product is doing.

The algebraic rule and the geometric picture describe the same operation.

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Turn the same plane into sine and cosine projections from rotationComplex Numbers and Parametric Motion

Unit Circle / Sine and Cosine from Rotation

Keep one rotating point, its x and y projections, and the sine-cosine traces linked so the unit circle becomes the live source of both functions.

Reuse magnitude and angle as one live radius-angle point on the same planeComplex Numbers and Parametric Motion

Polar Coordinates / Radius and Angle

Keep one point visible in polar and Cartesian views at the same time so radius and angle turn directly into x and y on the plane.

Use the same plane and unit-circle point to justify the core trig identities geometricallyComplex Numbers and Parametric Motion

Trig Identities from Unit-Circle Geometry

Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.

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More practice optionsReview on the bench · Worked examples

Practice optionReview on the benchReplay the guided setup with the key controls and graphs in sight.
Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
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Reference and support

Use these calmer sections when you want a fuller explanation, examples, or accessibility support after the guided flow.

Return here when you want the slower reference pass.

Open reference and support

Equation snapshotComplex-coordinate form · Geometric multiplication ruleShow 2 equations

Keep the plane picture in mind: one formula tells you where the point is, and the other tells you how multiplication turns and scales it.

Complex-coordinate form
$z = a + bi$
Reads one complex number as the point (a, b), or equivalently as the vector from the origin to that point.
Geometric multiplication rule
$∣ z w ∣ = ∣ z ∣∣ w ∣, ar g (z w) = ar g (z) + ar g (w)$
Explains why multiplying by w changes both the size and the direction of z.

Why it behaves this way

Explanation

Complex numbers become easier to trust when the algebra and geometry stay on the same plane. This bench keeps z, w, and the current result visible together so a + bi can be read as both the point (a, b) and the arrow from the origin to that point.

Addition should look like vector addition: add real parts and imaginary parts, or place one arrow head-to-tail with the other. Multiplication should look different: it changes size by multiplying magnitudes and changes direction by adding arguments, so multiplying by w scales z and rotates it at the same time.

Key ideas

01A complex number z = a + bi is one object with two views: the point (a, b) and the vector from the origin to that point.

02Adding complex numbers adds their horizontal and vertical components, just like vector addition on the plane.

03Complex multiplication multiplies magnitudes and adds arguments, so a multiplier near the unit circle mainly rotates while a larger or smaller magnitude also stretches or shrinks.

Worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live plane instead of a detached worksheet. First read z and w as points or arrows, then read the result from the same geometry.

Supporter unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

View plans

Example 1 of 2

Frozen valuesUsing frozen parameters

For the current addition view, where does z + w land, and how do the components and the head-to-tail guide show the same result?

a

Real part of z

2.2

b

Imaginary part of z

1.6

c

Real part of w

1.1

d

Imaginary part of w

1.8

1. Read z and w on the plane

The current points are z = 2.2 + 1.6i and w = 1.1 + 1.8i.

2. Add the real and imaginary parts

The sum keeps the plane honest: (2.2 + 1.1, 1.6 + 1.8).

3. Match that sum to the endpoint on the plane

That gives z + w = 3.3 + 3.4i, so the sum lands at (3.3, 3.4).

Current sum on the plane

z + w = 3.3 + 3.4 i

The tip-to-tail move reinforces the two arrows enough that the sum lands farther from the origin.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a complex plane with z, w, and the current result arrow. Sliders or dragging change the real and imaginary parts of both points, and a toggle switches between addition view and multiplication view. Optional guides can show head-to-tail addition, a one-unit reference circle, and a rotation cue for multiplication.

Graph summary

One graph shows how the real and imaginary parts of z + w change as the real part of w changes. A second graph shows how the real and imaginary parts of z · w change under the same sweep, so addition and multiplication can be compared as two different geometric actions on the same plane.

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Current bench

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Featured setups

Add as vectors

Open a clean addition case with the head-to-tail guide visible.

Multiply by i

Open the multiplication case where the multiplier acts like a quarter-turn.

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Starter track

Step 1 of 6

Complex and Parametric Motion

Unit Circle / Sine and Cosine from Rotation is what this track opens next.

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