HomeConcept libraryMathComplex Numbers and Parametric MotionComplex and Parametric Motion

Math Complex 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

Keep the stage, graph, and immediate control feedback in one working view.

Complex numbers on the plane

Drag z and w directly on the plane so real part, imaginary part, magnitude, argument, addition, and multiplication still read as one geometric story.

Addition view

Graphs

Switch graph views without breaking the live stage and time link.

Sum components vs Re(w)

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.

Re(w): -4.5 to 4.5result component: -8 to 8

Re(z + w)Im(z + w)

Hover or scrub to link the graph back to the stage.Re(w) / result component

Controls

Adjust the live parameters and watch the bench respond.

Real part of z

a

2.2

Imaginary part of z

b

1.6

Real part of w

c

1.1

Imaginary part of w

d

1.8

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Multiplication view

\times

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

In addition view, the result arrow can be read both as component addition and as a head-to-tail sum on the same plane.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

a

Real part of z

2.2

Moves z left or right on the real axis.

Graph: Sum components vs Re(w)Graph: Product components vs Re(w)Overlay: Head-to-tail guideOverlay: Unit circle cue

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep the plane, the result arrow, and the active graph in view together.

ObservationPrompt 1 of 2

Graph: Sum components vs Re(w)

In addition view, the result arrow can be read both as component addition and as a head-to-tail sum on the same plane.

Control: Real part of zControl: Imaginary part of zControl: Real part of wControl: Imaginary part of wGraph: Sum components vs Re(w)Overlay: Head-to-tail guideEquation

a

Equation

b

Equation

c

Equation

d

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Head-to-tail guide

Show the translated second vector used in the sum.

What to notice

The sum still uses the same two component pairs.

Why it matters

It keeps complex addition tied to vector addition.

Control: Real part of zControl: Imaginary part of zControl: Real part of wControl: Imaginary part of wGraph: Sum components vs Re(w)Equation

a

Equation

b

Equation

c

Equation

d

Challenge mode

Use the plane honestly: build the right scale-and-turn case instead of guessing from formulas alone.

0/1 solved

ConditionCore

1 of 6 checks

Rotate onto the positive imaginary axis

Build a multiplication case where z · w lands almost on the positive imaginary axis while the multiplier magnitude stays close to one.

Graph-linkedGuided start

Pending

Open the Product components vs Re(w) graph.

Sum components vs Re(w)

Matched

Keep the Unit circle cue visible.

Pending

Keep the Rotation cue visible.

Off

Pending

Keep scale factor between 0.95 and 1.05.

2.11

Pending

Keep product real between -0.25 and 0.25.

-0.46

Pending

Keep product imaginary between 2.2 and 5.

5.72

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

z = 2.2 + 1.6i sits 2.72 units from the origin at about 36.03 degrees. Adding w = 1.1 + 1.8i lands z + w at 3.3 + 3.4i.

Equation detailsDeeper interpretation, notes, and worked variable context.

Complex-coordinate form

Reads one complex number as a point on the plane with real part a and imaginary part b.

z = a + bi

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

Magnitude and argument

Turns the same point into a distance from the origin and a direction on the plane.

∣ z ∣ = a^{2} + b^{2}, ar g (z) = tan^{- 1} (\frac{b}{a})

a

Real part of z 2.2

b

Imaginary part of z 1.6

Geometric multiplication rule

Explains why multiplying by w scales and rotates z.

∣ z w ∣ = ∣ z ∣∣ w ∣, ar g (z w) = ar g (z) + ar g (w)

c

Real part of w 1.1

d

Imaginary part of w 1.8

\times

Multiplication view Off

Progress

Not startedMastery: NewLocal-first

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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

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Checking saved setup access.

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Copy current setup

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Stable links

Starter track

Step 1 of 20 / 2 complete

Complex and Parametric Motion

Next after this: Parametric Curves / Motion from Equations.

1. Complex Numbers on the Plane2. Parametric Curves / Motion from Equations

This concept is the track start.

See next concept

Short explanation

What the system is doing

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.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current plane state rather than a detached worksheet.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

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

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. 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

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.

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.

Prediction prompt

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

Check your reasoning

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?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Carry the plane language into motion traced from equationsComplex Numbers and Parametric Motion

Complex Numbers on the Plane

See each variable before you move it.

Head-to-tail guide

Rotate onto the positive imaginary axis

Complex and Parametric Motion

What the system is doing

Read the full frozen walkthrough.

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

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

Keep moving through the concept map.

Parametric Curves / Motion from Equations

Vectors in 2D

Graph Transformations