Equality of complex numbers

One equation, two answers

With real numbers, one equation usually finds one unknown. With complex numbers, one equation can find two. The trick is that a complex number has two separate parts.

Two complex numbers are equal only when their real parts match and their imaginary parts match. So one complex equation splits into two real equations.

Equal complex numbers have equal real parts and equal imaginary parts. One complex equation gives two real equations.

Key terms

Equal complex numbers

Same real part and same imaginary part.

Compare real parts

The first real equation you get.

Compare imaginary parts

The second real equation you get.

Collect the terms

Group every real term together and every i term together before comparing.

Short questions with model answers

1. Q1. Example 2 (p. 4): if (2x − 1) + (y + 4)i = 5 + 7i, find x and y.

   • 2x − 1 = 5, y + 4 = 7
   • 2x = 6, y = 7 − 4

   x = 3, y = 3

2. Q2. Exercise 1.1 Q3(i) (p. 4): (2x + 5) + (y − 3)i = 1 + 2i.

   • 2x + 5 = 1, y − 3 = 2
   • 2x = −4, y = 5

   x = −2, y = 5

Common mistakes

• ✗ y + 4 = 7i.

  ✓ Compare the coefficients of i: y + 4 = 7. The parts are real numbers.

• ✗ In (2 + i)x + (1 − 2i)y, the real part is 2 + 1.

  ✓ Multiply out first. The real part is 2x + y and the imaginary part is x − 2y.

• ✗ (3x + 2) − (4 − y)i = 5 + 3i gives 4 − y = 3.

  ✓ The imaginary part is −(4 − y), so −(4 − y) = 3 and y = 7.

MCQs

1. 1. If α + iβ = −2 + 5i, then:

   1. (a) α = 5, β = −2
   2. (b) α = −2, β = 5i
   3. (c) α = −2, β = 5
   4. (d) α = 2, β = 5
   Show answer

   (c) p. 3: compare parts; β is real.

2. 2. Two complex numbers are equal when:

   1. (a) Their real parts are equal
   2. (b) Both real parts and imaginary parts are equal
   3. (c) Their imaginary parts are equal
   4. (d) Their sums of parts are equal
   Show answer

   (b) Review Q2(iii) (p. 21): both parts must match.

3. 3. (3x + 2) − (4 − y)i = 5 + 3i. Find y (Exercise 1.1 Q3(ii)).

   1. (a) 1
   2. (b) −7
   3. (c) 1/3
   4. (d) 7
   Show answer

   (d) −(4 − y) = 3, so y − 4 = 3 and y = 7.

4. 4. Same equation: find x.

   1. (a) 1
   2. (b) 7/3
   3. (c) −1
   4. (d) 5
   Show answer

   (a) 3x + 2 = 5, so x = 1.

5. 5. Expand (2 + i)x + (1 − 2i)y. Its imaginary part is:

   1. (a) 2x + y
   2. (b) x − 2y
   3. (c) x + 2y
   4. (d) −2y
   Show answer

   (b) The i terms are ix and −2iy.

Quick revision

• z₁ = z₂ means equal real parts and equal imaginary parts.
• Collect each side into (real) + i(real) before comparing.
• One complex equation gives two real equations, so it can find two unknowns.