The Argand Plane, Modulus and Argument
A complex number is a point: real part across, imaginary part up. Its distance from the origin is the modulus and its angle is the argument, so multiplying becomes scaling and turning.
What a learner can do afterwards
- Plot complex numbers on the Argand plane and read real and imaginary parts back
- Compute modulus and argument and convert between a + bi and modulus-argument form
- Show geometrically that multiplying by i is a quarter turn
1 · Read
Plot each complex number as a point: real part across, imaginary part up. So 3 + 4i lands on (3, 4). Read them back the same way: 4 - 3i sits right and down, which is the fourth quadrant. Plotting first and calculating second helps you keep the signs right.
Modulus means distance from the origin, found by Pythagoras. For 3 + 4i: 9 + 16 is 25, whose root is 5. The same recipe gives 13 for 5 - 12i and 17 for 8 - 15i, since minus signs vanish on squaring. Argument means direction: 1 + i sits on the diagonal at 45 degrees, and modulus 2 with argument 90 degrees is the point 2i.
Multiplying by i turns the plane a quarter turn anticlockwise. Check it: (1 + i) times i is i + i squared, which is -1 + i, so (1, 1) moves to (-1, 1). The same turn explains (1 + i) squared: expanding gives 1 + 2i + i squared, which is 2i. Twice by i makes a half turn, sending any z to minus z.
To write a + bi in modulus-argument form, find r = √(a² + b²) and an angle θ with tan θ = b/a, then sketch the point to pick the right quadrant. For 1 + √3i, r = √(1 + 3) = 2 and tan θ = √3, so θ = 60 degrees and 1 + √3i = 2(cos 60° + i sin 60°). Going back, r(cos θ + i sin θ) is r steps in direction θ. Arguments are often given in radians: 180 degrees is π, so 60 degrees is π/3 and 90 degrees is π/2.
Plot across and up, measure distance and direction, and read multiplication by i as a quarter turn.
Readings 'Points with an address', 'Distance plus bearing' and 'Where the argument jumps' are original to this dataset. All 10 extra questions are original to this dataset.
2 · Watch
En anglaisTake it off screen
Where it sits
Learn first
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.