Editing Circular motion (section)

==== Using complex numbers ====
Circular motion can be described using [[complex number]]s. Let the {{mvar|x}} axis be the real axis and the <math>y</math> axis be the imaginary axis. The position of the body can then be given as <math>z</math>, a complex "vector":
<math display="block">z = x + iy = R\left(\cos[\theta(t)] + i \sin[\theta(t)]\right) = Re^{i\theta(t)}\,,</math>
where {{math|''i''}} is the [[imaginary unit]], and <math>\theta(t)</math> is the argument of the complex number as a function of time, {{mvar|t}}.

Since the radius is constant:
<math display="block">\dot{R} = \ddot R = 0 \, ,</math>
where a ''dot'' indicates differentiation in respect of time.

With this notation, the velocity becomes: 
<math display="block">v = \dot{z}
  = \frac{d}{dt}\left(R e^{i\theta[t]}\right)
  = R \frac{d}{dt}\left(e^{i\theta[t]}\right)
  = R e^{i\theta(t)} \frac{d}{dt} \left(i \theta[t] \right)
  = iR\dot{\theta}(t) e^{i\theta(t)}
  = i\omega R e^{i\theta(t)} = i\omega z
</math>
and the acceleration becomes:
<math display="block">\begin{align}
  a &= \dot{v} = i\dot{\omega} z + i\omega\dot{z} = \left(i\dot{\omega} - \omega^2\right)z \\
    &= \left(i\dot{\omega} - \omega^2 \right) R e^{i\theta(t)} \\
    &= -\omega^2 R e^{i\theta(t)} + \dot{\omega} e^{i\frac{\pi}{2}} R e^{i\theta(t)} \, .
\end{align}</math>

The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before.