Imaginary Numbers Unlocked
The imaginary unit i extends the number line into the complex plane, and multiplying by i rotates any point 90 degrees counter-clockwise.
A complete interactive classroom, not just a preview.
Start when you are ready to enter this Stage's 9 scenes and explore, respond, and learn as you go.
Can a number exist that, when squared, gives -1?
- definition-of-i
- i is defined as a number whose square is -1, extending the real numbers.
- powers-of-i
- i^n cycles every four steps: i, -1, -i, 1, then repeats.
- complex-plane
- The complex plane pairs a real axis with a perpendicular imaginary axis to locate a+bi.
- rotation-by-i
- Multiplying any complex number by i rotates its position 90 degrees counter-clockwise around the origin.
i is not a real number, so it cannot exist or be used like other numbers.
Show that i is just as valid as -1 or 0 once you extend the number system, and that plotting it on the complex plane makes it as concrete as any point on a map.
Multiplying by i should change the size (magnitude) of a number, like scaling by 2 or 1/2.
Demonstrate that multiplication by i only rotates 90 degrees and leaves the distance from the origin unchanged.
Powers of i keep changing unpredictably and never repeat.
Reveal the four-step cycle i, -1, -i, 1 and show how it repeats forever.
- comfort with negative numbers and squaring
- basic familiarity with the xy-coordinate plane
- complex arithmetic in polar/exponential form
- Argand diagrams beyond basic plotting
- Euler's formula
- complex analysis theorems
- Learner computes i^n for any positive integer n by reducing mod 4.
- Learner plots a+bi on the complex plane given values of a and b.
- Learner predicts where a point ends up after multiplying it by i one or more times.
- Apply the 90-degree rotation rule to multiply any simple complex number by i and predict the result.
Curious learners comfortable with basic algebra (squares, negatives, the xy-coordinate plane). No prior complex-number experience needed.
- 01Why Imaginary Numbers?slideOrientationObserve
Open with the puzzle: what number squared gives -1? Introduce i as a new building block, not a fake number.
- No real number squares to -1
- We define i so that i^2 = -1
- Extending the number system is a normal move in math
- 02Powers of i ExplorerinteractiveModel buildingConstruct
Step through i^1, i^2, i^3, i^4 and beyond with a slider that highlights where in the 4-step cycle you are.
- i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
- The pattern repeats every four steps
- Use n mod 4 to find any i^n
- 03The Complex PlaneslideOrientationObserve
Introduce the real-Imaginary grid used to plot numbers like 2 + 3i, mirroring the familiar xy-plane.
- Horizontal axis: real numbers
- Vertical axis: imaginary numbers
- Point a + bi sits at (a, b)
- 04Plot Any Complex NumberinteractivePracticeConstruct
Drag the real and imaginary sliders; the point and its label update live on the complex plane.
- Slider for the real part
- Slider for the imaginary part
- See the dot snap into place at (a, b)
- 05Multiplying by i Rotates 90°slideOrientationObserve
Reveal the geometric secret: multiplying any complex number by i is the same as a quarter-turn around the origin.
- Distance from origin is preserved
- Direction turns 90 degrees counter-clockwise
- Repeated multiplication keeps stepping around the axes
- 06Rotation LabinteractiveMisconception repairPredict
Move a point on the complex plane, multiply by i, and watch it rotate. Test the belief that size should change.
- Drag a starting point
- Click multiply by i to rotate it
- Distance from origin stays the same
- 07Quarter-Turn ChallengeinteractiveApplicationApply
Action game: rotate the highlighted point to land on the target zone by choosing how many times to multiply by i.
- Read the target quadrant
- Choose the right number of i-multiplies
- Beat the clock with as few moves as possible
- 08Cycle Decoder DiagraminteractiveSynthesisExplain
A single concept-map diagram connecting i, powers, the complex plane, and rotation into one mental model.
- Definition node
- Powers node
- Plane node
- Rotation node
- Arrows show how each piece supports the next
- 09Recap: The Four Faces of islideSynthesisExplain
Summarize the four key ideas: definition, powers, plane, rotation, with a single unifying visual.
- i^2 = -1 by definition
- i^n cycles every four steps
- Complex plane plots a+bi like (x, y)
- Times i = rotate 90 degrees
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