The complex plane

Complex numbers stopped being suspect the day we learned to draw them. Being described by two real numbers, a complex number naturally sits in a plane.

Affix and image point

Equip the plane with an orthonormal frame. To the complex z = a + ib associate the point M with coordinates (a ; b):

              ^ imaginary axis
              |
            b +--------• M(z)      z = a + ib
              |        |
              |        |           M is the IMAGE of z
      --------+--------+------->   z is the AFFIX of M
              O        a   real axis

This plane is called the complex plane. The horizontal axis carries the reals, the vertical one the purely imaginary numbers. Each complex corresponds to one point, and only one — and conversely.

The modulus is a distance

This is the bridge between algebra and geometry:

|z| = √(a² + b²) = OM        (by Pythagoras' theorem)

The modulus of a complex number is the distance from its image to the origin. More generally:

|z - z'| = distance between the points with affixes z and z'

This last formula turns geometric conditions into equations, and back:

|z| = 3           ->  circle centred at O with radius 3
|z - 2i| = 1      ->  circle centred at (0 ; 2), radius 1
|z - 1| = |z + 1| ->  perpendicular bisector of the segment (1;0)-(-1;0)
                      that is, the imaginary axis

The operations, seen geometrically

z + z'  ->  VECTOR ADDITION (parallelogram rule)
-z      ->  symmetry about the ORIGIN
z̄       ->  symmetry about the REAL AXIS
k·z     ->  scaling by a factor k (k real)

The conjugate is especially expressive:

              ^
            b +----• M(z)
              |    |
      --------+----+------>
              |    |
           -b +----• M'(z̄)      reflection in the mirror of the real axis

A plane, not a line

One reflex has to go: two complex numbers cannot be compared. Writing z < z' is meaningless, because there is no order on C compatible with the operations. That is the price of the second dimension: we gain the roots of every polynomial, we lose the ability to rank numbers.

Their moduli, however, are positive reals and can always be compared.

What comes next

Locating a point by its coordinates (a ; b) is not always the handiest way. It can also be located by its distance to the origin and by the angle OM makes with the real axis: these are the modulus and the argument, the subject of the course on trigonometric form. That notation reveals that multiplying by a complex number amounts to a rotation.

Summary

• The complex plane represents z = a + ib by the point M(a ; b).
• z is the affix of M, M the image of z.
• |z| = OM: the modulus is a distance; |z - z'| is the distance between two points.
• z + z' corresponds to vector addition, z̄ to a reflection in the real axis.
• Conditions like |z - a| = r describe circles.
• Complex numbers cannot be compared: C is not ordered.