Skip to content
Open Model LabInteractive concept
HomeConcept libraryMathComplex Numbers and Parametric MotionComplex and Parametric Motion
MathComplex Numbers and Parametric MotionIntroStarter track

Concept module

Complex Numbers on the Plane

Read complex numbers as points and vectors on one plane, then keep addition and multiplication geometric instead of symbolic-only.

Interactive lab

Loading the live simulation bench.

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 1 compact task ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

Featured setups

Vector sum view

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

Rotate by i

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

Saved setups

Premium keeps named exact-state study setups in your account while stable concept links stay public below.

Checking saved setup access.

This concept can keep using stable links while the saved-setups capability resolves for this browser.

Copy current setup

Stable concept and section links stay public below while exact-state setup sharing stays behind premium.

Stable links

Starter track

Step 1 of 60 / 6 complete

Complex and Parametric Motion

Next after this: Unit Circle / Sine and Cosine from Rotation.

1. Complex Numbers on the Plane2. Unit Circle / Sine and Cosine from Rotation3. Polar Coordinates / Radius and Angle4. Trig Identities from Unit-Circle Geometry+2 more steps

This concept is the track start.

See next concept

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 reads as both an ordered pair and a directed arrow.

Addition should feel like vector addition, and multiplication should feel like one point being rotated and scaled by another complex number.

Key ideas

01A complex number z = a + bi can be read as the point (a, b) or as the vector from the origin to that point.
02The magnitude |z| is the distance from the origin, and the argument tells the direction of the same point.
03Complex multiplication combines scaling and rotation on the same plane.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough
Use the current plane state rather than a detached worksheet.

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

View plans
Frozen valuesUsing frozen parameters

For the current addition view, where does z + w land on the plane?

Real part of z

2.2

Imaginary part of z

1.6

Real part of w

1.1

Imaginary part of w

1.8

1. Write the current points

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

2. Add the real and imaginary parts separately

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

3. Read the new endpoint

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

Current sum

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

Common misconception

Complex multiplication is just a symbolic component rule, so the geometric picture is optional.

The component rule is real, but the geometry is what makes the result readable.

On this plane, multiplying by w changes both the size and the direction of z.

Mini challenge

Build a multiplication case where the product lands in the second quadrant without making the point much longer than the original z.

Make a prediction before you reveal the next step.

Decide whether the multiplier needs to act mostly like a rotation, mostly like a stretch, or both before you try it.

Check your reasoning against the live bench.

You need a multiplier whose argument turns z into the second quadrant while its magnitude stays near one.
The product direction comes from the added argument, while the size comes from the multiplier magnitude.

Quick test

Reasoning

Question 1 of 2

Answer from the live plane picture, not from a detached rule.

What is the most honest way to read z = a + bi on this bench?

Use the live bench to test the result before moving on.

Accessibility

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.

Graph summary

One graph shows the real and imaginary parts of z + w as the real part of w changes. A second graph shows the real and imaginary parts of z · w under the same sweep.