I'm having trouble understanding this passage.

12 Geometry and Comples Arithmetic

Figure $[8]$

Since we now know that $|Z(t)|$ remains equal to $1$ throughout the motion, it follows that the particle's speed $|V(t)|$ also remains equal to $1$. Thus after time $t=\theta$ the partial will have travelled a distance $\theta$ round the unit circle, and so the angle of $Z(\theta)=e^{i\theta}$ will be $\theta$. This is the geometric statement of Euler's formula.

I don't understand the last paragraph. Why is it that $|Z(t)|$ remains equal to 1 throughout the motion?

  • $\begingroup$ I don't see $|z(t)|$ anywhere. Edit: you mean in the third page? $\endgroup$ – Mark Fantini Jul 19 '14 at 7:25

He mentions that $Z(t) = e^{it}.$ You have $$|Z(t)| = |e^{it}| = \sqrt{\cos^2(t) + \sin^2(t)} = 1.$$

  • $\begingroup$ Oh thank you. Jebus math is hard. $\endgroup$ – Sam Sauce Jul 19 '14 at 7:33

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