Imaginary Unit i in the Complex Plane
The imaginary unit i is √(−1), and any complex number a+bi sits on a 2D plane whose horizontal axis is the real part and whose vertical axis is the imaginary part.
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What is the imaginary unit i, and how do we picture it on a coordinate system?
- definition-of-i
- i is defined as a number whose square is −1, extending the reals to allow negative-square roots.
- powers-of-i
- Powers of i cycle with period 4: i, −1, −i, 1.
- complex-number-form
- A complex number is written a+bi, where a is the real part and b is the imaginary part.
- complex-plane
- The complex plane pairs each complex number a+bi with the point (a,b), giving the real axis horizontally and the imaginary axis vertically.
- plotting-and-reading
- Plotting a complex number means moving along the real axis by a and then along the imaginary axis by b; reading a point (a,b) gives back a+bi.
i is a mysterious made-up number with no real meaning.
Show that i is a precisely defined extension of the reals, and that it behaves predictably on a coordinate plane.
i and −1 are the same thing.
Demonstrate that i² = −1, so i is not −1 itself but a square root of −1, distinct from ±1.
The imaginary part of a complex number is imaginary in the everyday sense, so it isn't real.
Make clear that the imaginary part is an ordinary real number coefficient b attached to i; 'imaginary' is a historical label, not a judgment about reality.
- basic algebra
- familiarity with the Cartesian coordinate plane
- polar/exponential form
- modulus and argument as full topics
- complex arithmetic beyond simple identification
- roots of unity beyond a brief power-cycle observation
- Learner writes i², i³, i⁴ without hesitation.
- Learner identifies real and imaginary parts of a+bi.
- Learner plots a complex number on the complex plane and reads another back from coordinates.
- Use the complex plane representation to locate sums or opposites of simple complex numbers, or to interpret a given (x,y) point as a complex number in a new context.
Learners comfortable with basic algebra and the number line, but with no prior exposure to complex numbers.
- 01What Are We Trying to Extend?slideOrientation
Frame the problem: real numbers have no square root of −1, and complex numbers fix exactly that gap.
- Reals have no √(−1)
- We want a number whose square is −1
- i fills that gap
- 02Defining i by Its SquareinteractiveModel buildingObserve
Toggle i² on/off and see the value flip between i and −1, letting learners feel the defining property i² = −1.
- i² = −1
- i itself is not −1
- i behaves like a new number
- 03The Power Cycle of iinteractivePredictionPredict
Predict the next power of i in the sequence i, i², i³, i⁴, then watch the cycle reveal.
- Powers cycle every 4 steps
- i⁴ = 1
- Pattern: i, −1, −i, 1
- 04From i to a+bislideOrientationObserve
Introduce the general complex number form a+bi, naming the real and imaginary parts.
- a+bi is a complex number
- a = Re(z), b = Im(z)
- b is an ordinary real coefficient
- 05Label the PartsinteractivePracticeChoose
Drag sliders to change a and b, and watch the labels update on the point.
- Re(z) moves horizontally
- Im(z) moves vertically
- Both a and b are real numbers
- 06Meet the Complex PlaneinteractiveModel buildingConstruct
Plot a point and reveal how (a,b) connects to a+bi, with axes labeled real and imaginary.
- Real axis is horizontal
- Imaginary axis is vertical
- (a,b) ↔ a+bi
- 07Plot It YourselfinteractiveApplicationApply
Given target complex numbers like 2+3i or −1+4i, drag a marker to the matching point.
- Read the real part first
- Then move along the imaginary axis
- Check the signed directions
- 08Reading the Plane BackslideOrientationObserve
Show the reverse direction: starting from a point on the plane, write the complex number it represents.
- Point (a,b) becomes a+bi
- Purely real points sit on the real axis
- Purely imaginary points sit on the imaginary axis
- 09Point to Complex NumberinteractivePracticeConstruct
Given a highlighted point on the complex plane, type the matching complex number.
- Type a+bi form
- Sign of b matters
- Use Re and Im correctly
- 10Map the Number Line Into the PlaneinteractiveSynthesisExplain
See how ordinary real numbers sit along the real axis of the complex plane, anchoring the new picture to the familiar number line.
- Reals are a line inside the plane
- i is one unit up the imaginary axis
- −1, 0, 1 sit on the real axis
- 11Putting It TogetherslideSynthesisExplain
Recap i, a+bi, and the complex plane as one coherent picture.
- i is defined by i² = −1
- a+bi has real part a and imaginary part b
- The complex plane visualizes a+bi as (a,b)
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