The Argand Plane, Modulus and Argument · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Points, distances, and turns

Mathematics · Algebra · ages 17-18
Name ______________________   Date ____________

  1. What is the modulus of 6 + 8i?

    Answer: ______________

  2. Kim says the point (0, -4) on the Argand plane stands for the complex number -4i. Is Kim right?

    Circle one:   True   False

  3. What is the modulus of -9 + 12i?

    Answer: ______________

  4. Which point represents -2 + 5i on the Argand plane?

    • (5, -2)
    • (-2, 5)
    • (2, 5)

  5. Leo says -4 + 4i has argument -45 degrees, because tan θ = -1. Is Leo right?

    Circle one:   True   False

  6. Which is 4(cos 90° + i sin 90°) in a + bi form?

    • 4
    • 4i
    • -4
    • -4i

  7. What is the argument of 1 - i in degrees, between -180 and 180?

    Answer: ______________

  8. Which number has modulus 3 and argument 180 degrees?

    • 3
    • 3i
    • -3

  9. What is (1 - i) squared?

    • 2i
    • -2i
    • -2

  10. Multiplying a complex number by i does not change its modulus.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Points, distances, and turns W1-mt_Wq01c5UN9e-s1

  1. 10 · Root of 36 + 64 is root of 100, which is 10.
  2. True · Real part 0 and imaginary part -4 give 0 - 4i, which is -4i.
  3. 15 · 81 + 144 = 225, and root of 225 is 15.
  4. (-2, 5) · Real -2 goes across to the left and imaginary 5 goes up: (-2, 5).
  5. False · tan θ = -1 fits both -45 and 135 degrees. The point (-4, 4) is up and to the left, so its argument is 135 degrees, not -45.
  6. 4i · cos 90° = 0 and sin 90° = 1, so the number is 4(0 + i) = 4i.
  7. -45 · The point (1, -1) is right and down, in the fourth quadrant, at 45 degrees below the real axis: -45 degrees.
  8. -3 · Three units along the negative real axis is -3.
  9. -2i · Expanding gives 1 - 2i + i squared, and i squared cancels the 1.
  10. True · Multiplying by i is a quarter turn about the origin, and a turn keeps every distance from the origin the same.
Worksheet · LightMySky