Definition of Dimensionless Quantity
Definition
In physics, a Dimensionless quantity is a physical quantity whose dimension is $1$.
Explanation
Although this brings us to physics, the most representative and familiar dimensionless quantity is the radian, which denotes an angle. It is a perfect example for understanding dimensionless quantities; first, consider velocity as a contrasting example.
$$ v = {\frac{ \Delta s }{ \Delta t }} = \dfrac{\text{displacement}}{\text{time}} \implies \dim v = \dfrac{\mathsf{L}}{\mathsf{T}} = \mathsf{LT}^{-1} $$
Velocity is displacement divided by the elapsed time, so dividing length by time yields the dimension $\mathsf{LT}^{-1}$. Regardless of whether its numerical value is 10 or 1, because fundamentally length is divided by time that dimension remains.
▯eq2◀
In contrast, the radian is a unit expressing the magnitude of an angle, defined as the ratio of an arc length to the radius of a circle. In simple terms, both numerator and denominator have the dimension of length $\mathsf{L}$, so $\mathsf{L}$ cancels out, and therefore the radian is a dimensionless quantity of dimension 1.
$$ \theta = \dfrac{\ell}{r} = \dfrac{\text{arc length}}{\text{radius}} \implies \dim \theta = \dfrac{\mathsf{L}}{\mathsf{L}} = 1 $$
When assigning values to dimensionless quantities, it is very important to unify 🔒(26/04/19)units. Using the radian example again: you must not measure the radius in meters and the arc length in inches when computing the ratio. You need to make the units compatible so they can cancel out.
$$ \begin{align*} (\text{Incorrect})\quad &\dfrac{59.0551 \text{ inch}}{1 \text{ m}} = \xcancel{59.0551 \text{ rad}} \\[1em] (\text{Correct})\quad &\dfrac{59.0551 \text{ inch}}{1 \text{ m}} = \dfrac{1.5 \text{ m}}{1 \text{ m}} = \dfrac{1.5\ \cancel{\text{m}}}{1\ \cancel{\text{m}}} = 1.5 \text{ rad} \end{align*} $$