Adding, subtracting and multiplying complex numbers

About this lesson

Precalculus › Polynomial and rational functions

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Adding, subtracting and multiplying complex numbers

Warm-up

Three quick questions from earlier lessons.

Discover

(2+3i)+(4+5i)(2 + 3i) + (4 + 5i). Treat ii like a variable. What do you get?

  1. 6+8i6 + 8i
  2. 1414

Key idea

Calculate as with polynomials in ii, then use i2=−1i^2 = -1.

(a+bi)±(c+di)=(a±c)+(b±d)i(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i. (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i, since bd i2=−bdbd\,i^2 = -bd. The conjugate of a+bia + bi is a−bia - bi, and (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2.

Worked example

(2+3i)(1−4i)(2 + 3i)(1 - 4i).

2−8i+3i−12i22 - 8i + 3i - 12i^2
Distribute.
2−5i+122 - 5i + 12
i² = −1.
14−5i14 - 5i
Combine.

Your turn, guided

Your turn: (3+2i)(3−2i)(3 + 2i)(3 - 2i).

1313
9 + 4.

Practice ★

  1. Add: (9+8i)+(2+1i)(9 + 8i) + (2 + 1i).
  2. Add: (3+4i)+(9+2i)(3 + 4i) + (9 + 2i).
  1. Multiply: (2+8i)(2−8i)(2 + 8i)(2 - 8i).
  2. Multiply: (7+2i)(7−2i)(7 + 2i)(7 - 2i).

Watch out!

Practice ★★

  1. Expand (3+6i)2(3 + 6i)^2.
  2. Expand (1+6i)2(1 + 6i)^2.

Exit ticket

Three questions to check you have got it.

  1. Multiply: (2+8i)(2−8i)(2 + 8i)(2 - 8i).
  2. Multiply: (7+2i)(7−2i)(7 + 2i)(7 - 2i).
  3. Multiply: (8+4i)(8−4i)(8 + 4i)(8 - 4i).

Summary card

Complex arithmetic

Add parts separately; multiply out and use i2=−1i^2 = -1; conjugates give a real product.

Watch out: i2i^2 is −1-1, not 11.

End of the lesson

Next lesson: Quadratic equations with complex solutions Back to the chapter

Answers: Adding, subtracting and multiplying complex numbers

A

  1. 11+9i11 + 9 i
  2. 12+6i12 + 6 i

B

  1. 6868
  2. 5353

C

  1. −27+36i-27 + 36 i
  2. −35+12i-35 + 12 i

D

  1. 6868
  2. 5353
  3. 8080

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