Multiplication and Powers in the Complex Plane (Grades 11-12)

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Multiplication and Powers in the Complex Plane

1. 
If [math]z = -4+2i[/math] is plotted in the complex plane, how can multiplying by [math]i[/math] be described geometrically?
  1. It rotates [math]z[/math] 90° counterclockwise about the origin.
  2. It rotates [math]z[/math] 90° clockwise about the origin.
  3. It reflects [math]z[/math] across the imaginary axis.
  4. There is no geometric interpretation.
2. 
If [math]z=-2+7i[/math] is plotted in the complex plane, how can multiplying by [math]-1[/math] be described geometrically? Choose all correct answers.
  1. It rotates [math]z[/math] 90° clockwise about the origin.
  2. It rotates [math]z[/math] clockwise 180° about the origin.
  3. It reflects [math]z[/math] across the imaginary and real axis.
  4. It reflects [math]z[/math] across the real axis.
3. 
Let [math]z_1 = 4 + 4i[/math] and [math]z_2 = -1 + 3i[/math].
A. 
What is the product of [math]z_1[/math] and [math]z_2 ?[/math]
  1. [math]8 + 8i[/math]
  2. [math]8 - 8i[/math]
  3. [math]-16 + 8i[/math]
  4. [math]3 + 7i[/math]
B. 
What is the modulus, [math]r[/math], and argument, [math]theta[/math], of [math]z_1?[/math] List them as [math](r, theta)[/math].
  1. [math](4, \ 90°)[/math]
  2. [math](2sqrt(2), \ 90°)[/math]
  3. [math](4, \ 45°)[/math]
  4. [math](4sqrt(2), \ 45°)[/math]
C. 
What is the modulus, [math]r[/math], and argument, [math]theta[/math], of [math]z_2?[/math] List them as [math](r, theta)[/math].
  1. [math](sqrt(2), \ -19.5°)[/math]
  2. [math](sqrt(10), \ 108.4°)[/math]
  3. [math](2, \ 161.4°)[/math]
  4. [math](sqrt(10), \ -71.5°)[/math]
D. 
What is the modulus, [math]r[/math], and argument, [math]theta[/math], of the product of [math]z_1[/math] and [math] z_2 ?[/math] List them as [math](r, \ theta)[/math].
  1. [math](8sqrt(2), \ 45°)[/math]
  2. [math](8sqrt(2), \ -45°)[/math]
  3. [math]sqrt(58), \ 66.8°)[/math]
  4. [math](8sqrt(5), \ 153.4°)[/math]
E. 
What is the relationship between the answers in parts b and c and the answer in part d?
  1. The moduli of [math]z_1[/math] and [math]z_2[/math] multiplied together equals the modulus of [math]z_1 z_2[/math], and the arguments of [math]z_1[/math] and [math]z_2[/math] added together equals the argument of [math]z_1 z_2[/math].
  2. The moduli of [math]z_1[/math] and [math]z_2[/math] multiplied together equals the modulus of [math]z_1 z_2[/math], but the arguments have no special relationship.
  3. The moduli have no special relationship, but the arguments of [math]z_1[/math] and [math]z_2[/math] added together equals the argument of [math]z_1 z_2[/math].
  4. There is no relationship.
4. 
For any [math]z in CC[/math], with [math]r, theta in RR[/math], and [math]n[/math] a positive integer, [math]z^n = r^n [cos(n theta) + i sin(n theta) ][/math].
A. 
For [math]n=2[/math], show that the equation is true. Hint: use trigonometric identities.







B. 
For [math]n=3[/math], show that the equation is true. Hint: use trigonometric identities and the result from the previous answer.







C. 
Which of the following gives the best reasoning of how, for [math]n>3[/math], this equation can be proved true?
  1. There are more complicated trigonometric identities that deal with higher powers.
  2. As was done in part b, by using the answer from the previous integer, each successive integer can be shown to be true using the same trigonometric identities (as used in part b).
  3. Having shown it true for two different cases, it can be assumed true for all other cases.
  4. For even powers of n, the process will be similar to [math]n=2[/math], while for odd powers of n, the process of showing this equation is true will be similar to [math]n=3[/math], but increasingly more complicated.

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