Answer
The Mach number, Ma, is dimensionless.
Work Step by Step
For an airplane flying at velocity V in air at absolute temperature T, the Mach number is:
$Ma=\frac{V}{\sqrt kRT}$
Where (Using the International System of Units),
$V=\frac{m}{s}=LT^{-1}$
$K=dimensionless$ (Given)
$R=\frac{J}{KgK}=\frac{Nm}{KgK}=\frac{Kgm^{2}}{s^{2}KgK}=\frac{m^{2}}{s^{2}K}=L^{2}T^{-2}
Θ^{-1}$
$T=K= Θ $
Therefore,
$Ma=LT^{-1}(L^{2}T^{-2}Θ^{-1}\timesΘ)^{-1/2}=LT^{-1}(L^{2}T^{-2}Θ^{0})^{-1/2}=LT^{-1}(LT^{-1}Θ^{0})^{-1}$
$Ma=L^{0}T^{0}Θ^{0}$
Thus, it's proven that the Mach number, Ma, is dimensionless.
