Units of Measure and Dimensional Analysis

If someone came up to you and said they had "four", what would your reaction be? For most people the response would be: "Four what?" The what is called a unit or dimension and lets us know what a number means. The person talking could be referring to four inches, four feet, four dollars, four apples, and the list goes on. So if someone said they had four oranges the unit would be oranges.

The most important system of units of measure is the International System of Units which is typically abbreviated SI. This system is defined by seven fundamental units. Here are four of them:

More importantly, all physical quantities are made up of these fundamental quantities. For example:

This means that the units of energy, for example, are kilograms times mass-squared divided by seconds-squared. That's a lot to say so we just call them Joules after James Prescott Joule who studied the nature of heat energy and its relationship to mechanical work.

Outside of just communicating what we are talking about, units play a special role in how we describe the real world with Mathematics. Ask yourself a simple question: Is an apple the same as an orange? The answer, of course, is no. Similarly, for a mathematical equation to describe the real world the units on the left side of an equal sign must be the same as the units on the right side of the equal sign. The process of determining if the units on the left side of an equal sign are equal to the units on the right side of an equal sign is called dimensional analysis.

For example, the force of gravity between two objects is given by the equation:
\[
F=G\frac{m_1m_2}{r^2}
\]
where $r$ is the distance between the two masses $m_1$ and $m_2$. $G$ is the gravitational constant which has a value of $6.67\times10^{-11}m^3/kg\,s^2$. Finally, $F$ is the force in Newtons. To see if this equation is physically valid we will do a dimensional analysis. To accomplish this we don't worry about any numbers, but place each quantity's unit in the equation like this:
\[
kg\,m/s^2=\bigg\{\frac{m^3}{kg\cdot s^2}\bigg\}\frac{kg\cdot kg}{m^2}
\]
We now cancel terms according to our rules of algebra. (Notice that the $kg$ on the bottom will cancel with one of the $kg$'s in the top. Also, the two $m$'s ($m^2$) in the bottom will cancel with two on the top.) Doing this gives:
\begin{eqnarray}
kg\,m/s^2&=&\bigg\{\frac{m^3}{kg\cdot s^2}\bigg\}\frac{kg\cdot kg}{m^2}\\
kg\cdot m/s^2&=&kg\cdot m/s^2. \\
\end{eqnarray}

Both sides of the equal sign have the same unit of Newtons ($kg\cdot m/s^2$)! While dimensional analysis will not tell you if an equation is correct or not, it is always the first step to see if an equation is a candidate for modelling the real world.