Definition of Dimensionless Quantity
Definition
In physics, a dimensionless quantity is a quantity whose dimension is $1$.
Explanation
Although physics was mentioned just above, the most representative and familiar dimensionless quantity is the radian, which measures angles. It is a perfect example for understanding dimensionless quantities, but first consider velocity as a contrasting example. $$ v = {\frac{ \Delta s }{ \Delta t }} \left[ {\frac{ \mathrm{m} }{ \mathrm{s} }} \right] $$ Velocity is displacement divided by the elapsed time; dividing meters (the unit for measuring length) by seconds (the unit of time) gives the unit $m/s$. Regardless of whether its numerical value is 10 or 1, because it fundamentally divides length by time, it retains its dimension.
By contrast, the radian is a unit of angle defined as the ratio of an arc length to the radius of a circle. In other words, since the numerator and denominator are both length units, the length units cancel out, so the radian is a dimensionless quantity with unit 1.