Multiplying and dividing complex numbers

Where does i² go?

Multiplying two brackets like (3 − 4i)(5 − 6i) works like ordinary algebra. The only new move comes at the end: every i² turns into −1, so it joins the real part.

Dividing is harder, because i in the bottom is not allowed in the answer. Multiply the top and bottom by the conjugate of the bottom. The bottom then becomes a real number.

Multiply out and replace i² by −1. To divide, multiply top and bottom by the conjugate of the bottom.

Key terms

Product (z₁z₂)

Multiply every term by every term.

Conjugate of the bottom

The number that turns the bottom real.

Real denominator

What (x₂ + iy₂)(x₂ − iy₂) always gives.

Quotient (z₁ ÷ z₂)

Written as a real part plus an i part.

Short questions with model answers

1. Q1. Example 5(i) (p. 6): simplify (3 − 4i)(5 − 6i).

   • = 15 − 18i − 20i + 24i²
   • = 15 − 38i − 24

   −9 − 38i

2. Q2. Example 6 (p. 7): write (3 + 4i) ÷ (5 − 7i) as x + iy.

   • = (3 + 4i)(5 + 7i) ÷ ((5 − 7i)(5 + 7i))
   • = (15 + 21i + 20i + 28i²) ÷ (25 + 49)
   • = (−13 + 41i) ÷ 74

   −13/74 + (41/74)i

Common mistakes

• ✗ 15 − 38i + 24i² is the final answer.

  ✓ Always replace i² by −1: 15 − 38i − 24 = −9 − 38i.

• ✗ (3 − 4i)(5 − 6i) = 15 + 24i.

  ✓ Multiplying is not part by part. The cross terms −18i and −20i matter.

• ✗ Multiply only the bottom by its conjugate.

  ✓ Multiply top and bottom, so the value does not change.

MCQs

1. 1. Real part of (2 − 3i)(2 + 3i) is:

   1. (a) −3
   2. (b) 1
   3. (c) 4
   4. (d) 13
   Show answer

   (d) Review Q1(ii) (p. 20): 4 − 9i² = 4 + 9 = 13.

2. 2. Imaginary part of (2 − i)(2 + i) is:

   1. (a) 0
   2. (b) 1
   3. (c) 7
   4. (d) 9
   Show answer

   (a) Review Q1(iii): the product is 4 + 1 = 5, a real number.

3. 3. If z₁ = 2 + 3i and z₂ = 4 + 7i, z₁z₂ = (Example 5(ii))

   1. (a) 8 + 21i
   2. (b) 29 + 26i
   3. (c) −13 + 26i
   4. (d) −13 + 2i
   Show answer

   (c) (8 − 21) + (14 + 12)i.

4. 4. (5 − 2i)(3 − 4i) = (Exercise 1.2 Q1(vi))

   1. (a) 23 − 26i
   2. (b) 7 − 26i
   3. (c) 15 + 8i
   4. (d) 7 − 14i
   Show answer

   (b) 15 − 20i − 6i + 8i² = 15 − 26i − 8.

5. 5. To divide by 6 − 3i, multiply top and bottom by:

   1. (a) 6 − 3i
   2. (b) −6 + 3i
   3. (c) 3i
   4. (d) 6 + 3i
   Show answer

   (d) The conjugate of 6 − 3i makes the bottom 36 + 9 = 45.

Quick revision

• Multiply out every term, then use i² = −1.
• To divide, multiply top and bottom by the conjugate of the bottom.
• The bottom becomes x₂² + y₂²; write the answer as x + iy.