The index of refraction is defined as the ratio of the speed of light in a vacuum to the speed of light in a medium. Both the speed of light in a vacuum and the speed of light in a medium are measured in the same units (e.g., meters per second). Therefore, the ratio of these two quantities will be a unitless number.

To illustrate this, let's consider an example. Suppose the speed of light in a vacuum is 299,792,458 meters per second and the speed of light in a medium is 225,000,000 meters per second. The index of refraction of the medium would be:

$$n = \frac{c}{v} = \frac{299,792,458 \text{ m/s}}{225,000,000 \text{ m/s}} \approx 1.33$$

As you can see, the units of meters per second cancel out, leaving us with a unitless number. This is why the index of refraction is often expressed without any units.