{ "format": "recallweave-deck/1", "title": "Complex multiplication: turn, scale, repeat", "attribution": "Original questions, worked explanations and transfer prompts authored for RecallWeave by chatgpt-31366547c1f5 under Jacob Metoyer's project direction, 2026-10-08. Mathematical background checked against OpenStax Precalculus 2e, 3.1 Complex Numbers and 8.5 Polar Form of Complex Numbers: https://openstax.org/books/precalculus-2e/pages/3-1-complex-numbers and https://openstax.org/books/precalculus-2e/pages/8-5-polar-form-of-complex-numbers . No source exercises, prose or figures are reproduced.", "license": "Original lesson content is dedicated under CC0 1.0 Universal: https://creativecommons.org/publicdomain/zero/1.0/ . Referenced publications retain their own terms. This lesson makes no claim of measured learning efficacy.", "concepts": [ "Points and the imaginary unit", "Distance and direction", "Multiplication as a transformation", "Repeated products" ], "items": [ { "id": "cp-01", "concept": "Points and the imaginary unit", "prerequisites": [], "prompt": "A point represents z = 3 − 2i. The horizontal coordinate is Re(z) and the vertical coordinate is Im(z), the coefficient of i. Which ordered pair locates z?", "options": [ "(3, −2)", "(−2, 3)", "(3, 2)", "(−3, −2)" ], "answer": 0, "explanation": "The real coefficient is 3 and the imaginary coefficient is −2. Plot three units right and two units down. Im(z) is the real number −2, not the expression −2i.", "transfer": "Describe where −3 + 2i lies, and explain which two signs determine its location." }, { "id": "cp-02", "concept": "Points and the imaginary unit", "prerequisites": [], "prompt": "The imaginary unit i is defined by i² = −1. What is (4i)(3i)?", "options": [ "12i", "−12", "12", "−12i" ], "answer": 1, "explanation": "Multiply the coefficients to get 12, then multiply i by i. Since i² = −1, the product is 12(−1) = −12, which lies on the real axis.", "transfer": "Explain why multiplying two nonzero purely imaginary numbers can produce a real number." }, { "id": "cp-03", "concept": "Points and the imaginary unit", "prerequisites": [], "prompt": "Complex conjugation reflects a point across the real axis: it leaves the real coefficient unchanged and reverses the imaginary coefficient. What is the conjugate of −4 + 3i?", "options": [ "4 + 3i", "4 − 3i", "−4 − 3i", "−3 + 4i" ], "answer": 2, "explanation": "Only the imaginary coefficient changes sign. The point (−4, 3) becomes (−4, −3), so the conjugate is −4 − 3i. Its distance from the origin is unchanged.", "transfer": "Choose a point below the real axis and explain where its conjugate appears." }, { "id": "cp-04", "concept": "Distance and direction", "prerequisites": [ "Points and the imaginary unit" ], "prompt": "The modulus |a + bi| is the distance √(a² + b²) from the origin. What is |6 + 8i|?", "options": [ "14", "100", "2", "10" ], "answer": 3, "explanation": "The squared distance is 6² + 8² = 36 + 64 = 100. Taking its nonnegative square root gives 10. The sum 6 + 8 is not the Euclidean distance.", "transfer": "Find a different complex number with modulus 10 and explain why it has the same distance." }, { "id": "cp-05", "concept": "Distance and direction", "prerequisites": [ "Points and the imaginary unit" ], "prompt": "Angles are measured counterclockwise from the positive real axis, and the principal argument is chosen in [−180°, 180°). What is the principal argument of −3 − 3i?", "options": [ "−135°", "−45°", "45°", "135°" ], "answer": 0, "explanation": "Both coefficients are negative, so the point lies in the lower-left quadrant. Its direction is 225° counterclockwise, equivalent to −135°. The chosen interval records −135°, while −45° would point into the lower-right quadrant.", "transfer": "Explain why 225° and −135° describe the same direction but only one belongs to the declared principal interval." }, { "id": "cp-06", "concept": "Distance and direction", "prerequisites": [ "Points and the imaginary unit" ], "prompt": "The modulus of a complex number is its distance from the origin. For a nonzero complex number, an argument describes the direction of its point from the origin. Which statement correctly describes z = 0 + 0i?", "options": [ "Its modulus and argument are both 0.", "Its modulus is 0, and its argument is undefined.", "Its modulus is 1, and its argument is undefined.", "Its modulus is undefined, and its argument is 90°." ], "answer": 1, "explanation": "The origin is at distance 0 from itself, so the modulus is 0. It has no particular direction: every angle with radius 0 describes the same point. Assigning it a direction of 0° would introduce a convention that is not its mathematical argument.", "transfer": "Explain why a direction arrow for a point at the origin would be misleading even though its distance is known." }, { "id": "cp-07", "concept": "Multiplication as a transformation", "prerequisites": [ "Points and the imaginary unit", "Distance and direction" ], "prompt": "Using i² = −1 and ordinary distribution, what is (2 + i)(−1 + 2i)?", "options": [ "−2 + 4i", "4 + 3i", "−4 + 3i", "−4 + 5i" ], "answer": 2, "explanation": "The four terms are −2, 4i, −i and 2i². Since 2i² = −2, the real terms sum to −4 and the imaginary terms to 3i. The product is −4 + 3i.", "transfer": "Explain why multiplying real coefficients together and imaginary coefficients together is not the complete complex-product rule." }, { "id": "cp-08", "concept": "Multiplication as a transformation", "prerequisites": [ "Points and the imaginary unit", "Distance and direction" ], "prompt": "Let z be any nonzero complex number, and measure positive angles counterclockwise from the positive real axis. Multiplication by a nonzero w multiplies distance by |w| and adds an argument of w to the direction, modulo 360°. The multiplier −i has coordinates (0, −1). What does multiplying z by −i do?", "options": [ "Reflect it across the real axis, with unchanged distance.", "Rotate it 90° counterclockwise and double its distance.", "Rotate it 180° and halve its distance.", "Rotate it 90° clockwise, with unchanged distance." ], "answer": 3, "explanation": "The multiplier −i has modulus 1 and argument −90°. Thus the distance is multiplied by 1 and the direction changes by −90°, a clockwise quarter-turn. Conjugation is a different operation: its effect depends on the starting direction.", "transfer": "Apply multiplication by −i to 1 + 2i and explain how the coordinates confirm the clockwise quarter-turn." }, { "id": "cp-09", "concept": "Multiplication as a transformation", "prerequisites": [ "Points and the imaginary unit", "Distance and direction" ], "prompt": "For r > 0, the number r(cos θ + i sin θ) has modulus r and direction θ; angles here are in degrees and positive counterclockwise. Let z ≠ 0 and w = 3(cos 120° + i sin 120°). A complex product multiplies moduli and adds arguments modulo 360°. How does zw compare with z?", "options": [ "Its distance is three times as large; its direction is 120° counterclockwise from z.", "Its distance increases by 3 units; its direction is 120° counterclockwise from z.", "Its distance is three times as large; its direction is always 120° from the positive real axis.", "Its distance is divided by 3; its direction is 120° clockwise from z." ], "answer": 0, "explanation": "The modulus of w is 3, so |zw| = 3|z|. Its argument contributes a 120° counterclockwise rotation relative to the original point, rather than setting an absolute direction of 120°. Equivalent directions can differ by 360°.", "transfer": "Explain how this rule changes when the starting point is zero: which scaling quantity remains defined, and which direction comparison is unavailable?" }, { "id": "cp-10", "concept": "Repeated products", "prerequisites": [ "Multiplication as a transformation" ], "prompt": "Use i² = −1. Define P₀ = z and Pₖ₊₁ = Pₖw. Take z = 1 + 2i and w = −2 + i. After the first multiplication P₁ = −4 − 3i. What is P₂ after one more multiplication by the same w?", "options": [ "−7 − 6i", "11 + 2i", "−8 + 6i", "−4 − 3i" ], "answer": 1, "explanation": "Multiply (−4 − 3i)(−2 + i): the real part is 8 − (−3) = 11, and the imaginary part is −4 + 6 = 2. P₂ is 11 + 2i. This repeats the multiplier w; it does not square P₁ or replace the starting point with w.", "transfer": "Explain the difference between this repeated-product sequence and repeatedly squaring the previous point." }, { "id": "cp-11", "concept": "Repeated products", "prerequisites": [ "Multiplication as a transformation" ], "prompt": "Use i² = −1. A table retains every indexed point P₀ through Pₙ, with P₀ = 2 − i and Pₖ₊₁ = Pₖi. After N = 4 multiplications, how many rows does it contain and what is its last point?", "options": [ "4 rows; the last point is 2 − i.", "5 rows; the last point is −2 + i.", "5 rows; the last point is 2 − i.", "4 rows; the last point is −1 − 2i." ], "answer": 2, "explanation": "The table includes the starting row P₀ plus four multiplication results, so it has five rows. Four factors of i have product 1, returning the final point to 2 − i. The equal coordinates of P₀ and P₄ do not erase their distinct step indices.", "transfer": "Describe what a plot alone could hide when a sequence returns to an earlier coordinate, and how the indexed table resolves that ambiguity." }, { "id": "cp-12", "concept": "Repeated products", "prerequisites": [ "Multiplication as a transformation" ], "prompt": "Define P₀ = −2 + i and Pₖ₊₁ = Pₖw, with w = 0. The table includes P₀ through P₃. Which statement describes its later points and their arguments?", "options": [ "P₁, P₂ and P₃ stay at −2 + i, with its original argument.", "P₁, P₂ and P₃ are zero, each with argument 0°.", "P₁ is zero, but P₂ and P₃ are undefined because the sequence uses 0⁰.", "P₁, P₂ and P₃ are zero, each with undefined argument." ], "answer": 3, "explanation": "Each recurrence multiplies the previous point by zero, so all three later points are zero. Their arguments are undefined because the origin has no direction. P₀ is assigned directly; the recurrence never needs 0⁰. Multiplication by zero is a collapse, not a rotation with a known angle.", "transfer": "Explain the difference between using zero as the starting point and using zero as the multiplier. Which row can remain nonzero?" } ] }