# Euler's Formula, from Needham's Visual Complex Analysis

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?

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

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