Wikipedia helpfully has a list of dimensionless physical quantities. Many of these are dimensionless

*variables*, comparing ratios of quantities which

*do* have a dimension.

For example, the refractive index is the ratio of the speed of light in some material (in m/s) vs the speed of light in a vacuum. But there are many potential materials, which could be measured at many potential frequencies. You could measure the velocity in different units (eg miles/hour), but the ratio is dimensionless, and independent of the units used.

When you take a ratio, the dimensions cancel out, and you are left with a dimensionless number. See:

https://en.wikipedia.org/wiki/List_of_dimensionless_quantitiesdimensionless physical constants.

These are values which are considered basic to measurement.

Some of them are

*exact* quantities because they are

*defined* that way by the units we use - for example Avogadro's number of Carbon-12 atoms weighs

*exactly* 12 grams, by definition. If Chemists were measuring in ounces instead of grams, Avogadro's number would be different.

But we only know Avogadro's number to about 8 decimal places, while it actually has 23 digits. So it doesn't

*have to be* a whole number, but we may as well

*treat it* as if it is a whole number.

Another one is

*defined* to be irrational: the vacuum permeability constant µ

_{0}, whose numerical value is 4π×10

^{−7}.

On the other hand, once we knew the fine structure constant to 5 decimal places, it became clear that it wasn't a whole number.

There are some constants which are believed to be so fundamental that they are independent of the units we use. For a list, see:

https://en.wikipedia.org/wiki/Dimensionless_physical_constant