Real and imaginary parts of powers and quotients

Hidden inside a power

What is the real part of (4 + 3i)⁻¹? It is not 4, and it is not 1/4. A power or a fraction hides the parts until you rewrite it as x + iy.

Two tools do all the work. Square a bracket with (a + b)² = a² + b² + 2ab. Clear a bottom with its conjugate.

Rewrite any power or quotient as x + iy first: expand squares, clear the bottom with its conjugate, then read off Re and Im.

Key terms

Reciprocal (z⁻¹)

The case n = −1.

Square (z²)

The case n = 2, using i² = −1.

Inverse square (z⁻²)

Square first, then take the reciprocal.

Quotient to a power

For n = −1, just turn the fraction upside down.

Short questions with model answers

1. Q1. Example 17 (p. 17): find the real and imaginary parts of (4 + 3i)⁻¹.

   • z = 1 ÷ (4 + 3i) = (4 − 3i) ÷ ((4 + 3i)(4 − 3i))
   • = (4 − 3i) ÷ (16 + 9) = 4/25 − (3/25)i

   Re(z) = 4/25, Im(z) = −3/25

2. Q2. Example 20 (p. 18): find the real and imaginary parts of z = ((1 + i) ÷ (1 − i))⁻².

   • z = ((1 − i) ÷ (1 + i))²
   • (1 − i) ÷ (1 + i) = (1 − i)² ÷ (1 − i²) = −2i ÷ 2 = −i
   • z = (−i)² = −1

   Re(z) = −1, Im(z) = 0

Common mistakes

• ✗ (4 + 3i)⁻¹ = 1/4 + (1/3)i.

  ✓ You cannot flip each part. Use the conjugate: 4/25 − (3/25)i.

• ✗ (8 − 3i)² = 64 − 9i².

  ✓ Square of a sum has a middle term: 2 × 8 × (−3i) = −48i.

• ✗ Im of (4 − 3i) ÷ 25 is 3/25.

  ✓ Keep the sign: Im = −3/25.

MCQs

1. 1. Re of (x + iy)⁻¹ is:

   1. (a) 1/x
   2. (b) x ÷ (x² + y²)
   3. (c) −y ÷ (x² + y²)
   4. (d) x
   Show answer

   (b) p. 17: (x − iy) ÷ (x² + y²).

2. 2. (4 + 3i)² =

   1. (a) 25 + 24i
   2. (b) 16 + 9i
   3. (c) 7 + 12i
   4. (d) 7 + 24i
   Show answer

   (d) 16 + 9i² + 24i = 16 − 9 + 24i (Example 18).

3. 3. Re of (5 + 3i)⁻¹ is (Exercise 1.4 Q1(ii)):

   1. (a) 5/34
   2. (b) 1/5
   3. (c) 5/16
   4. (d) −3/34
   Show answer

   (a) (5 − 3i) ÷ (25 + 9) = 5/34 − (3/34)i.

4. 4. In Example 19, ((4 + 3i) ÷ (3 + 2i))⁻¹ has Im =

   1. (a) 1/25
   2. (b) 18/25
   3. (c) −1/25
   4. (d) −17/25
   Show answer

   (c) (3 + 2i)(4 − 3i) ÷ 25 = (18 − i) ÷ 25.

5. 5. ((1 + i) ÷ (1 − i))⁻² equals (Example 20):

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

   (c) The fraction flips to −i, and (−i)² = −1.

Quick revision

• (x + iy)⁻¹ = (x − iy) ÷ (x² + y²): Re = x ÷ (x² + y²), Im = −y ÷ (x² + y²).
• For n = ±2, square with (a + b)² and remember i² = −1.
• A negative power of a fraction flips the fraction first.