Flashcards and recall.

Try this with Mathematics Specialist

1. Write one prompt per card. Use definitions, explanations or examples from your current topic.
2. Answer aloud before turning the card. Split a card if it asks for several unrelated facts.
3. Set aside missed cards and revisit them in a later session.

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Recall topics

• What are Re(z) and Im(z) for z = 4 − 7i?
• Express (3 + 2i)(1 − 4i) in Cartesian form.
• How do you write (2 + i)/(1 − 2i) in Cartesian form?
• What is z z̄ for z = a + bi, and why is it useful?
• If z = 2 − 3i, what is z⁻¹ in Cartesian form?
• What is the principal argument Arg(z) and its range?
• State |z| and Arg(z) for z = −2 − 2i.
• Write 4 cis(2π/3) in Cartesian form.
• What is Arg(−5)? What is Arg(3i)?
• Why can't you just use tan⁻¹(b/a) for Arg(a + bi)?
• State the modulus identities for a product and a quotient.
• How does Arg(z₁z₂) relate to Arg z₁ and Arg z₂?
• Prove that |z̄| = |z|.
• If Arg z = θ with 0 < θ < π, what is Arg(z̄) and Arg(−z)?
• Simplify 6 cis(5π/6) ÷ 2 cis(−π/3).
• State de Moivre's theorem for integral powers.
• Evaluate (cis(π/12))⁸.
• Evaluate (1 + i)⁻⁴.
• How do you use de Moivre to express cos 3θ in terms of cos θ?
• Geometrically, how is z₁ + z₂ found on an Argand diagram?
• What transformation is z ↦ iz?
• What transformation is z ↦ 2 cis(−π/3) z?
• Square ABCD is labelled anticlockwise with A at z_A and B at z_B. How do you find D?
• Describe the locus |z − (1 − 2i)| = 4.
• Describe the locus |z − 3| = |z + i|.
• What does Arg(z + 2) = π/3 represent?
• Sketch-wise, what is the region Re(z) > Im(z)?
• What is the locus |z − 1| = 2|z − i|?
• Find the maximum value of |z| if |z − 3 − 4i| ≤ 2.
• How many nth roots does a non-zero complex number have, and where do they lie?
• Solve z⁴ = 16.
• What is the sum of the nth roots of unity for n ≥ 2?
• If ω is a non-real cube root of unity, simplify 1 + ω + ω².
• Solve z² = −8i in Cartesian form.
• State the remainder theorem.
• Find the remainder when P(z) = z³ − 2z + 5 is divided by z + 2.
• State the conjugate root theorem.
• Which real quadratic has roots 3 ± i?
• Why must a real cubic have at least one real root?
• When is the composite f(g(x)) defined?