Modulus of a complex number

How far from zero?

|−5| = 5 tells you how far −5 is from 0 on the number line. A complex number sits on a plane, so its distance from the origin is the slanted side of a right triangle.

The two short sides are x and y (Fig. 1.13). Pythagoras gives the distance: √(x² + y²). This distance is the modulus.

|z| = √(x² + y²) is the distance of z from the origin. Changing signs does not change it, and z z̄ = |z|².

Key terms

Modulus (|z|)

The distance of z from the origin.

Right triangle (Fig. 1.13)

The modulus is its hypotenuse.

|z| = |z̄| = |−z|

Reflections keep the distance from the origin.

z z̄ = |z|²

A number times its conjugate is its modulus squared.

Short questions with model answers

1. Q1. Example 15 (p. 14): z = 5 + 4i. Show |z| = |z̄|.

   • |z| = √(5² + 4²) = √41
   • z̄ = 5 − 4i, |z̄| = √(5² + (−4)²) = √41

   |z| = |z̄| = √41

2. Q2. Example 16 (pp. 15–16): z = 5 + 3i. Show z z̄ = |z|².

   • z z̄ = (5 + 3i)(5 − 3i) = 25 − 9i² = 25 + 9 = 34
   • |z|² = 5² + 3² = 25 + 9 = 34

   Both are 34.

Common mistakes

• ✗ |5 + 4i| = 41.

  ✓ 41 is |z|². The modulus is √41.

• ✗ |5 + 4i| = 5 + 4 = 9.

  ✓ Square, add, then take the root: √(25 + 16).

• ✗ |3 − 5i| = √(9 − 25).

  ✓ Use y = −5, not −5i: y² = 25, so √(9 + 25) = √34.

MCQs

1. 1. If z = 4 − 3i, then z z̄ =

   1. (a) 3
   2. (b) 9
   3. (c) 16
   4. (d) 25
   Show answer

   (d) Review Q1(vii) (p. 21): 16 + 9 = 25.

2. 2. |i| =

   1. (a) 0
   2. (b) 1
   3. (c) −1
   4. (d) i
   Show answer

   (b) p. 14: √(0² + 1²) = 1. A modulus is never negative.

3. 3. |4 + 3i| = (Exercise 1.3 Q1(ii))

   1. (a) 5
   2. (b) 7
   3. (c) 25
   4. (d) √7
   Show answer

   (a) √(16 + 9) = √25 = 5.

4. 4. |−5 − 4i| = (Exercise 1.3 Q1(iii))

   1. (a) −√41
   2. (b) 9
   3. (c) √41
   4. (d) 41
   Show answer

   (c) √(25 + 16) = √41; the signs do not matter after squaring.

5. 5. Which is NOT always true?

   1. (a) |z| = |z̄|
   2. (b) |−z| = |z|
   3. (c) z z̄ = |z|²
   4. (d) |z| = Re(z) + Im(z)
   Show answer

   (d) Properties 1–3 (pp. 14–15) hold; the modulus uses squares and a root.

Quick revision

• |z| = √(x² + y²), the distance of z from the origin.
• z, z̄, −z and −z̄ all have the same modulus.
• z z̄ = |z|², always a real number.