Using any unit
We do not need to use the SI units for the physical quantities involved in an equation in order to show that the equation is homogeneous. Consider the case of speed: $$ \begin{equation}\begin{aligned} speed=\frac{distance}{time}\\ \end{aligned}\end{equation} $$
We can use $m/s$, $m$ and $s$ respectively for the units of speed, distance and time: $$ \begin{equation}\begin{aligned} m/s&=\frac{m}{s}\\ m/s&=m/s\\ \end{aligned}\end{equation} $$
We can also use $km/h$, $km$ and $h$ respectively for the units of speed, distance and time: $$ \begin{equation}\begin{aligned} km/h&=\frac{km}{h}\\ km/h&=km/h\\ \end{aligned}\end{equation} $$
Using dimensional analysis notation
We can use the following symbols as placeholders for any of the units, be it standard or not, in order to show that a formula is homogeneous:
- $[L]$ - length
- $[T]$ - time
- $[M]$ - mass
- $[\theta]$ - temperature
The first $3$ are the dimensions of the physical quantities associated with mechanics.
The dimension of a number (a numerical constant is exactly one ($[1]$)).