Do dimensionless quantities have units or not? I am confused a bit about this concept.
The tricky examples are angles and percentages. The SI-brochure says these quantities are dimensionless, also termed quantities of dimension 1. These dimensionless quantities do have units: 1 rad is equal to 1; 1° is equal to 2π/360. The unit of percentage is %, which is equal to 0.01. Hence 1 rad = 1 = 100%.
The mere definition of a dimensionless quantity states that a dimensionless quantity is such a quantity to which no physical dimension can be assigned.They can't obviously have any units then.There is a difference between dimensions and units. A dimension is a measure of a physical variable (without numerical values), while a unit is a way to assign a number or measurement to that dimension. For example, length is a dimension, but it is measured in units of feet (ft),meters (m),etc.So when we define a quantity it is necessary to assign both a unit and a dimension to it.Leaving any one of them will render the quantity meaningless