I have learnt that the dimensionless quantities have no unit. Such as the refractive index have no unit. But plane angles and solid angles have their radian and steradian units, although they are dimensionless quantities.
Could anyone explain to me why is that?

  • $\begingroup$ Radians are also dimensionless (as are degrees). $\endgroup$
    – Triatticus
    Commented Apr 30, 2021 at 13:10
  • $\begingroup$ We call them "units" because radians and degrees represent different magnitudes. They have no dimension in the space-time sense. You will see reference to the phase (angle) as a dimensionless unit in equations involving complex numbers. $\endgroup$ Commented Apr 30, 2021 at 13:24

2 Answers 2


I have learnt that the dimensionless quantities have no unit.

Whoever you learnt this from is incorrect . Clearly, dimensionless quantities CAN have units, as you have figured out in the case of angles.
Another example of dimensionless quantity with units is the relative abundance of particles which has units of ppm(parts per million) , ppb(parts per billion) etc.

  • 5
    $\begingroup$ I'm not convinced ${\rm ppm}$ etc are actually units instead of simply a shorthand for the dimensionless number $10^{-6}$. Similarly a degree would be equal to the number $\pi/180$. $\endgroup$
    – jacob1729
    Commented Apr 30, 2021 at 14:13
  • 4
    $\begingroup$ @jacob1729 is correct. An angle expressed in radians is just the ratio of the arc length to the length of the radial arm. It's just a pure number. $\endgroup$ Commented Apr 30, 2021 at 15:21

From Guide for the Use of the International System of Units (SI) (NIST site).

7.10 Values of quantities expressed simply as numbers: the unit one, symbol 1\
Certain quantities, such as refractive index, relative permeability, and mass fraction, are defined as the ratio of two mutually comparable quantities and thus are of dimension one (see Sec. 7.14). The coherent SI unit for such a quantity is the ratio of two identical SI units and may be expressed by the number 1. However, the number 1 generally does not appear in the expression for the value of a quantity of dimension one. For example, the value of the refractive index of a given medium is expressed as $n = 1.51 \times 1 = 1.51$.
On the other hand, certain quantities of dimension one have units with special names and symbols which can be used or not depending on the circumstances. Plane angle and solid angle, for which the SI units are the radian (rad) and steradian (sr), respectively, are examples of such quantities...

The rationale beyond units for dimensionless quantities is to keep memory of the quantity used as the denominator of the ratio. NIST and BIPM convention of allowing to use or not the symbol f dimensionless units is consistent with the original choice.


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