Locating a complex number by distance and angle

The algebraic form a + ib locates a complex number by its coordinates. But in the plane a point can also be located by its distance to the origin and the angle it makes with the horizontal axis. This second system is the one that reveals the true nature of multiplication.

The two coordinate systems

              ^
              |         • M(z)
              |        /|
              |    r  / | b
              |      /  |
              |     /θ  |
      --------+----------------->
              O      a

  Cartesian coordinates : (a ; b)      ->  z = a + ib
  polar coordinates     : (r ; θ)      ->  r = |z|, θ = arg(z)

• r = |z| = √(a² + b²) is the modulus: the distance OM, always positive;
• θ = arg(z) is the argument: the signed angle between the real axis and OM.

Trigonometric form

The relations between the two systems are those of right-triangle trigonometry:

a = r cos θ            r = √(a² + b²)
b = r sin θ            cos θ = a/r  and  sin θ = b/r

hence the so-called trigonometric form:

z = r (cos θ + i sin θ)

The argument is defined modulo 2π

A full turn returns to the same point: if θ is an argument of z, so are θ + 2π, θ - 2π, θ + 4π. We therefore write:

arg(z) = θ  [2π]        ("modulo 2π")

The principal argument is the one chosen in ]-π ; π]. Two practical consequences:

- the complex number 0 has NO argument (no direction);
- an equality of arguments is always read "up to 2π".

Converting between forms

Algebraic → trigonometric. Take z = 1 + i√3:

r = √(1 + 3) = 2

cos θ = 1/2   and   sin θ = √3/2      ->   θ = π/3

z = 2 (cos π/3 + i sin π/3)

Watch out for the trap: computing tan θ = b/a is not enough, since the tangent does not distinguish an angle from the one shifted by π. For z = -1 - i√3 we would also get tan θ = √3, whereas the angle is -2π/3, not π/3. You must always look at the signs of cos θ and sin θ, that is, at the quadrant the point is in.

Values worth knowing

z          |z|      arg(z)
---------  -------  --------
1           1        0
-1          1        π
i           1        π/2
-i          1        -π/2
1 + i       √2       π/4
1 + i√3     2        π/3
√3 + i      2        π/6

The unit circle — the set of z with modulus 1 — plays a central role: its points are exactly the cos θ + i sin θ.

Summary

• The modulus r = |z| is the distance to the origin; the argument θ is the angle with the real axis.
• Trigonometric form: z = r(cos θ + i sin θ).
• Conversion: a = r cos θ, b = r sin θ, r = √(a² + b²).
• The argument is defined modulo 2π; the number 0 has no argument.
• Never conclude from the tangent alone: check the quadrant via the signs of cos and sin.