IB AA HL | Complex Arguments & De Moivre

First understand the argument of 1 + i

Start with the point. Then build the right triangle. Only then introduce the formula.

Why is Arg(1 + i) = π/4?

Step 1 · Read rectangular form

In z = a + bi, a is the horizontal coordinate and b is the vertical coordinate. For z = 1 + i, we have (a,b) = (1,1).

Step 2 · Identify the quadrant

Both coordinates are positive: Quadrant I. The vector points up and right from the origin.

Step 3 · Use the right triangle

\tan θ = \frac{b}{a} = \frac{1}{1} = 1

Therefore θ = arctan(1) = 45°.

Step 4 · Convert to radians

45° × π/180° = π/4 radians.

The principal argument of 1 + i is π/4; its modulus is √(1²+1²)=√2.

Static graph: the actual point Z = (1,1)

Blue = vector from the origin to 1+i. Dotted teal = the two perpendicular legs. The angle is measured counterclockwise from the positive real axis.

Explanation: “Both legs are one unit long, so tangent is one. The first-quadrant angle is 45 degrees, which is π over 4 radians.”

Basic language you must know

Real part

a = Re(z)

Left–right position.

Imaginary part

b = Im(z)

Up–down position (the real coefficient of i).

Modulus

|z| = √(a²+b²)

Distance from the origin.

Argument

Directed angle from the positive real axis.

Principal argument

−π < Arg(z) ≤ π

The chosen angle in the principal interval.

Reference angle

α = arctan(|b/a|)

Acute angle for a point in a quadrant (a ≠ 0).

Important: Arg(0) is undefined. Do not apply arctan(b/a) at axis points with a = 0; use the axis angle. At the negative real axis, this edition chooses Arg = +π, not −π.

Eight rotations: powers of 1 + i

First study direction on a common circle. Then compare actual moduli in the table.

Static 45° direction map

Odd n: between axesEven n: on axes

Directions are drawn with equal ray lengths for clarity. Actual powers have increasing modulus (√2)ⁿ, shown exactly in the adjacent table.

Actual powers and their locations

n	Raw angle	(1 + i)ⁿ	Location	Modulus
1	45°	1 + i	Quadrant I	1.4142
2	90°	2i	Positive imaginary axis	2
3	135°	-2 + 2i	Quadrant II	2.8284
4	180°	-4	Negative real axis	4
5	225°	-4 − 4i	Quadrant III	5.6569
6	270°	−8i	Negative imaginary axis	8
7	315°	8 − 8i	Quadrant IV	11.3137
8	360°	16	Positive real axis	16

(1 + i)ⁿ = (√2)ⁿ cis(nπ/4)

n even: the point is on a coordinate axis. n odd: the point lies strictly inside a quadrant. n ≡ 0 (mod 4): real axis; n ≡ 2 (mod 4): imaginary axis.

Modulo reminder: n ≡ 0 (mod 4) is read “n is congruent to zero modulo four,” meaning n is divisible by four.

All 9 geometric cases, clearly separated

4 quadrants + 4 open half-axes + the origin. Learn this map before applying an inverse tangent.

Case 1 Positive real axis

4 → Arg = 0° (≈ 0.00000 rad)

Case 2 Quadrant I

3 + i → Arg = 18.4349° (≈ 0.32175 rad)

Case 3 Positive imaginary axis

3i → Arg = 90.0000° (≈ 1.57080 rad)

Case 4 Quadrant II

-1 + 3i → Arg = 108.4349° (≈ 1.89255 rad)

Case 5 Negative real axis

-2 → Arg = 180.0000° (≈ 3.14159 rad)

Case 6 Quadrant III

-3 − i → Arg = -161.5651° (≈ -2.81984 rad)

Case 7 Negative imaginary axis

−2i → Arg = -90.0000° (≈ -1.57080 rad)

Case 8 Quadrant IV

3 − i → Arg = -18.4349° (≈ -0.32175 rad)

Case 9 Origin

0 → Arg = undefined (origin)

Quadrant rules for the principal argument

Location	Condition	Principal argument, in radians
Quadrant I	a > 0, b > 0	α
Quadrant II	a < 0, b > 0	π − α
Quadrant III	a < 0, b < 0	−π + α
Quadrant IV	a > 0, b < 0	−α
Positive real / imaginary axis	b = 0, a > 0 / a = 0, b > 0	0 / +π/2
Negative real / imaginary axis	b = 0, a < 0 / a = 0, b < 0	+π / −π/2
Origin	a = b = 0	Undefined

α = arctan(|b/a|) is an acute reference angle only when both a and b are nonzero. Exact axis angles are separate cases.

Dynamic explorer A · Move z = a + bi

Drag on the complex plane or change a and b. Every readout updates with the point.

Real part a

Imaginary part b

Quick examples

Complex vectorRight-triangle legsArgument arc

Live mathematical readout

Case

Complex number and point

Modulus

Principal argument

The graph contains only O and Z as text labels. All long coordinates, formulas, and explanations appear here to prevent overlapping text.

Dynamic explorer B · De Moivre powers

Change the complex base and exponent. Watch the direction rotate while modulus follows its own rule.

Base real part

Base imaginary part

Exponent n · 1

Original z direction (dashed)zⁿ direction

Unit-direction diagram: all rays share a reference circle; actual coordinate and modulus are shown numerically to avoid misleading visual scaling.

Original base z

Actual coordinates of zⁿ

Powered modulus

Raw angle nθ

Principal argument of result

z = r cis θ → zⁿ = rⁿ cis(nθ)

Dynamic explorer C · All n-th roots

See exactly why the roots form an equally spaced regular polygon (when the target is nonzero).

Target real part

Target imaginary part

Root degree n (2–8)

Highlight root k = 0

Target direction (dashed)Other rootsSelected root k

Diagram normalizes root radius to show angular spacing; the actual root modulus and numerical coordinates are given beside the graph.

w_k = R^1/n cis((Θ + 2πk)/n), k = 0, …, n−1

k	Angle (°)	Approx. rectangular root

46 questions · Student Practice

All exercises are unanswered. Select a question, complete your working independently, and use the diagrams for conceptual exploration.

Practice mode. The exercise set contains no worked solutions, answer keys, or pre-filled exercise results. First classify the point, then show the required calculations and verification.

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 2 · Quadrant I

Q5–Q8 · 2 powers + 2 root-family questions

Q05Quadrant I: 2nd powerPowerEasy⌄

Question

Let z = $3 + i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q06Quadrant I: 5th powerPowerMedium⌄

Question

Let z = $3 + i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q07Quadrant I: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $3 + i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q08Quadrant I: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $3 + i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 3 · Positive imaginary axis

Q9–Q12 · 2 powers + 2 root-family questions

Q09Positive imaginary axis: 2nd powerPowerEasy⌄

Question

Let z = $3 i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q10Positive imaginary axis: 5th powerPowerMedium⌄

Question

Let z = $3 i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q11Positive imaginary axis: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $3 i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q12Positive imaginary axis: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $3 i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 4 · Quadrant II

Q13–Q16 · 2 powers + 2 root-family questions

Q13Quadrant II: 2nd powerPowerEasy⌄

Question

Let z = $-1 + 3 i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q14Quadrant II: 5th powerPowerMedium⌄

Question

Let z = $-1 + 3 i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q15Quadrant II: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $-1 + 3 i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q16Quadrant II: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $-1 + 3 i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 5 · Negative real axis

Q17–Q20 · 2 powers + 2 root-family questions

Q17Negative real axis: 2nd powerPowerEasy⌄

Question

Let z = $-2$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q18Negative real axis: 5th powerPowerMedium⌄

Question

Let z = $-2$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q19Negative real axis: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $-2$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q20Negative real axis: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $-2$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 6 · Quadrant III

Q21–Q24 · 2 powers + 2 root-family questions

Q21Quadrant III: 2nd powerPowerEasy⌄

Question

Let z = $-3 - i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q22Quadrant III: 5th powerPowerMedium⌄

Question

Let z = $-3 - i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q23Quadrant III: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $-3 - i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q24Quadrant III: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $-3 - i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 7 · Negative imaginary axis

Q25–Q28 · 2 powers + 2 root-family questions

Q25Negative imaginary axis: 2nd powerPowerEasy⌄

Question

Let z = $- 2 i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q26Negative imaginary axis: 5th powerPowerMedium⌄

Question

Let z = $- 2 i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q27Negative imaginary axis: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $- 2 i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q28Negative imaginary axis: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $- 2 i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 8 · Quadrant IV

Q29–Q32 · 2 powers + 2 root-family questions

Q29Quadrant IV: 2nd powerPowerEasy⌄

Question

Let z = $3 - i$ . Find z² using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q30Quadrant IV: 5th powerPowerMedium⌄

Question

Let z = $3 - i$ . Find z⁵ using De Moivre’s theorem. Give the polar result, the principal argument of the answer, and the rectangular result.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q31Quadrant IV: all 3 rootsRootsDifficult⌄

Question

Solve w³ = $3 - i$ . Find all complex solutions. State why there are exactly 3 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q32Quadrant IV: all 6 rootsRootsVery difficult⌄

Question

Solve w⁶ = $3 - i$ . Find all complex solutions. State why there are exactly 6 distinct roots and explain their geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Case 9 · Origin

Q33–Q36 · Concept and error analysis

Q33Origin: undefined argumentPowerEasy⌄

Question

Explain why Arg(0) is undefined and why the reference-angle formula cannot be used.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q34Origin: zero powersPowerMedium⌄

Question

Evaluate 0⁷. Should De Moivre’s theorem be used? Explain.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q35Origin: repeated rootRootsDifficult⌄

Question

Solve w³ = 0 over the complex numbers and explain why there are not three distinct roots.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q36Origin: diagnose the mistakeRootsVery difficult⌄

Question

A student writes 0 = 0 cis(π/3), then says Arg(0) = π/3, and uses De Moivre to claim that the fourth roots of zero are four different directions. Diagnose every error.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Mixed comprehensive capstones

Q37–Q46 · Classification, powers, quotients, roots, branch cuts, and proofs

Q37All 9 cases: exact principal argumentsConcept / mixedComprehensive⌄

Question

For each of 5, 2+2i, 7i, −3+i, −9, −1−i, −4i, 3−i, and 0, identify its geometric case and principal argument.

Suggested working order

Identify the mathematical idea or geometric case.
Write the correct general rule or formula.
Show your reasoning clearly and solve the question.
Check all cases and justify the conclusion.

Q38Branch cut: raw angle versus principal anglePowerComprehensive⌄

Question

Compute (−1+i)², show the raw De Moivre angle, then give the principal argument.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q39Complex quotient: Quadrants III and IVConcept / mixedComprehensive⌄

Question

Let z₁ = −3−i and z₂ = 3−i. Find z₁/z₂ in polar and rectangular form.

Suggested working order

Identify the mathematical idea or geometric case.
Write the correct general rule or formula.
Show your reasoning clearly and solve the question.
Check all cases and justify the conclusion.

Q40Fourth roots of −16iRootsComprehensive⌄

Question

Solve w⁴ = −16i and explain the root geometry.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q41Fifth roots of 32 cis(2π/3)RootsComprehensive⌄

Question

Solve w⁵ = 32 cis(2π/3).

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q42High power: (−3−i)⁷PowerComprehensive⌄

Question

Let z = −3−i. Find z⁷, showing every angle reduction step.

Suggested working order

Plot the complex number and identify its quadrant or axis.
Calculate the modulus and principal argument.
Apply De Moivre’s theorem to the given exponent.
Convert to the requested form and verify.

Q43Substitution: (z−i)³ = −8RootsComprehensive⌄

Question

Solve (z−i)³ = −8.

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q44Proof: why use |b/a| for the reference angle?Concept / mixedComprehensive⌄

Question

Derive the reference-angle formula from geometry and explain exactly why |b/a| is used.

Suggested working order

Identify the mathematical idea or geometric case.
Write the correct general rule or formula.
Show your reasoning clearly and solve the question.
Check all cases and justify the conclusion.

Q45Proof: n roots form a regular n-gonRootsComprehensive⌄

Question

Prove the geometric root pattern for a nonzero target, then apply it to w⁶ = 64 cis(−π/3).

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

Q46Full capstone: powers, quotient and fourth rootsRootsComprehensive⌄

Question

Let z = 3+i and w = −1+i. First find z³/w². Then solve u⁴ = z³/w².

Suggested working order

Sketch or classify the target on the complex plane.
Express the target in polar form and identify the root degree.
Find every required root and describe the geometry.
Check your results by substitution.

How to teach each question: ask the student to identify the geometric case first, then read the numbered steps. Use “Show this example on the interactive graph” to send its original target and exponent or root degree to the right live diagram. The formula and graph are separate from the written solution so that labels remain legible.

Exam strategies and self-check

Use these rules while working independently. They apply to both the practice and solutions editions.

Before calculating

Mark the point (a, b) on the complex plane.
Determine the case: quadrant, one of the four half-axes, or the origin.
Find the modulus using |z| = √(a²+b²).
Find the principal argument in (−π, π], using signs and the reference angle where appropriate.
Keep the exact form (fractions of π, radicals, inverse tangent) whenever possible.

After calculating

For powers: raise the modulus to n and multiply the angle by n.
For roots: include every k = 0, 1, …, n−1 for a nonzero target.
For the origin: Arg(0) is undefined; zⁿ = 0 has one distinct solution if n is positive.
Check: convert back to rectangular form or substitute into the original equation.
Report clearly: exact answer, units (degrees or radians), and any requested principal angle.

Principal-value convention: −π < Arg(z) ≤ π. Therefore the negative real axis has principal argument +π (not −π).