1. **State the problem:** We need to find the number of radians in a central angle whose arc length is 23 of a circle, expressed in terms of $\pi$ radians and rounded to three significant digits. 2. **Recall the formula:** The central angle $\theta$ in radians is related to the arc length $s$ and radius $r$ by the formula: $$\theta = \frac{s}{r}$$ 3. **Important rule:** The circumference $C$ of a circle is $2\pi r$. If the arc length $s$ is a fraction of the circumference, say $\frac{23}{100}$ of the circle, then: $$s = \frac{23}{100} \times 2\pi r$$ 4. **Calculate the central angle:** Substitute $s$ into the formula for $\theta$: $$\theta = \frac{s}{r} = \frac{\frac{23}{100} \times 2\pi r}{r}$$ 5. **Simplify the expression:** Cancel $r$ in numerator and denominator: $$\theta = \frac{\cancel{r} \times \frac{23}{100} \times 2\pi}{\cancel{r}} = \frac{23}{100} \times 2\pi = \frac{46}{100} \pi = 0.46\pi$$ 6. **Express in terms of $\pi$ radians:** $$\theta = 0.46\pi \text{ radians}$$ 7. **Calculate numerical value:** $$\theta \approx 0.46 \times 3.1416 = 1.445 \text{ radians}$$ 8. **Round to three significant digits:** $$\theta \approx 1.45 \text{ radians}$$ **Final answer:** The central angle is $0.46\pi$ radians or approximately $1.45$ radians rounded to three significant digits.