courses › Beyond College

Numbers with an imaginary part: arithmetic, size, and powers.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 53.

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Builds on: Quadratics (not open yet) · Identities & equations (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Powers of i par 10 s not tried yet

  worked example

  What is i⁷¹?

  1. +i
  2. −i
  3. −1
  4. 1

  Answer: −i

  1. Powers of i repeat every 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
  2. 71 = 4 × 17 + 3, so i⁷¹ = i³ = −i.

• Multiply par 25 s not tried yet

  worked example

  (3 − 5i)(4 − 3i) = a + bi, where a and b are real. Enter a, b.

  Answer: (-3, -29)

  1. Multiply out: 12 − 9i − 20i + 15i².
  2. Since i² = −1, 15i² = −15.
  3. Real part: 12 − 15 = −3. Imaginary part: −9 − 20 = −29.
  4. So the product is −3 − 29i: enter −3, −29.

• Modulus par 12 s not tried yet

  worked example

  What is the modulus |−8 + 6i|?

  Answer: 10

  1. |a + bi| = √(a² + b²), the distance from 0 to the point (a, b).
  2. √((−8)² + 6²) = √(64 + 36) = √100 = 10.

• Powers by angle par 30 s not tried yet

  worked example

  (1 − i)¹² is a real number. What is it?

  Answer: -64

  1. 1 − i has length √2 and angle −45°.
  2. The 12th power has length (√2)¹² = 64 and angle 12 × (−45°) = −540°.
  3. −540° = −3 × 180° points along the negative real axis, so (1 − i)¹² = −64.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

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