The Argand plane and the conjugate

A number with an address

A real number lives on a line. A complex number needs two real numbers, so it needs a whole plane. 2 − 5i is simply the point (2, −5).

Use the x-axis as the real axis and the y-axis as the imaginary axis. This is the complex plane, or Argand plane, named after Jean Argand (1768–1822).

x + iy sits at the point (x, y). Its conjugate x − iy is its mirror image in the real axis.

Key terms

Argand plane

The xy-plane used to plot complex numbers.

Real axis, imaginary axis

Real numbers lie on the first; pure imaginary numbers on the second.

Position vector

The arrow that also stands for z.

Conjugate (z̄)

Same real part, opposite imaginary part.

Short questions with model answers

1. Q1. p. 9: write 2 − 5i, 8 and −7i as points.

   • 2 − 5i → (2, −5)
   • 8 = 8 + 0i → (8, 0)
   • −7i = 0 − 7i → (0, −7)

   (2, −5), (8, 0), (0, −7)

2. Q2. p. 10: find the conjugate of 1 + 2i and multiply it by 1 + 2i.

   • z̄ = 1 − 2i
   • (1 + 2i)(1 − 2i) = 1² − (2i)²
   • = 1 + 4

   5, a real number

Common mistakes

• ✗ 3 − 4i is the point (−4, 3).

  ✓ Real part first: (3, −4).

• ✗ The conjugate of 9 − 4i is −9 + 4i.

  ✓ Only the imaginary part changes sign: 9 + 4i.

• ✗ z̄ is z reflected in the imaginary axis.

  ✓ The real part stays, so the mirror is the real axis.

MCQs

1. 1. Conjugate of 9 − 4i is:

   1. (a) −9 − 4i
   2. (b) 9 + 4i
   3. (c) 9 + 9i
   4. (d) 4 − 9i
   Show answer

   (b) Review Q1(viii) (p. 21): change i to −i.

2. 2. The number −7i is plotted at:

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

   (c) x = 0 and y = −7 (p. 9).

3. 3. Multiplying a complex number by its conjugate gives:

   1. (a) A pure imaginary number
   2. (b) Always 0
   3. (c) Always 1
   4. (d) A real number
   Show answer

   (d) Review Q2(ii); p. 10: (1 + 2i)(1 − 2i) = 5.

4. 4. Geometrically, z̄ is z reflected in:

   1. (a) The real axis
   2. (b) The imaginary axis
   3. (c) The origin
   4. (d) The line y = x
   Show answer

   (a) 1.3.2 (p. 10): the real part stays the same.

5. 5. In Fig. 1.10, which number is in quadrant III?

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

   (d) Both parts negative means quadrant III.

Quick revision

• x + iy is the point (x, y) on the Argand plane, or the vector to it.
• The conjugate of x + iy is x − iy, its mirror image in the real axis.
• z times z̄ is always real, which is why conjugates help in division.