Article

09 April 2004

Secrets of a Theorist Part 2: Units - How to Check the Validity of an Equation

George E. Hrabovsky, President, MAST

Introduction

In the last installment I showed how plotting an equation can lead to a better understanding of the variables involved in the equation. I also showed that equations with similar structures will have similar properties, and similar graphs describe similar equations.

In theory we often derive new equations based on patterns that we encounter in sets of data. These equations are always suspect until we can validate them. One such validation uses the units of measurement themselves.

Units

Before we get into equation validation we have to get some terms and ideas in mind. The first of these comes from the need to adopt standards of measurement. It is a fairly intuitive notion that if our measurements are to be meaningful to anyone else they must be made using common methods. If your inch is five of my inches, and I am not aware of that fact, then your results will never match mine.

It turns out that there are only seven units of measurement we need adopt as standards; length, mass, time, temperature, electric current, chemical quantity, and light intensity. All other kinds of measurement are derived from these seven. Here is a table with the standard units in the English system of measurement (ENG), the International System of measurement (SI), and the centimeter-gram-second scale (CGS).

Standard	Symbol	ENG	SI	CGS
Length	[L]	feet (ft)	meter (m)	centimeter (cm)
Mass	[m]	pound (lb)	kilogram (kg)	gram (gm)
Time	[t]	seconds (s)	seconds (s)	seconds (s)
Temperature	[T]	degrees Fahrenheit (°F)	kelvin (K)	kelvin (K)
Electric Current	[I]	ampere (A)	ampere (A)	ampere (A)
Chemical Quantity	[M]	mole (mol)	mole (mol)	mole (mol)
Light Intensity	[l]	candela (cd)	candela (cd)	candela (cd)

By understanding the quantities under study, we can understand the units of measurement involved. If we define velocity as distance traveled (length) divided by the time to reach that distance, we have,

Thus, the units for velocity would be,

Units to Check the Validity of an Equation

Whenever we have an equation, since the two sides must have the same numerical value, they must also have the same units. We can thus check an equation to see if the units are correct; if not a mistake has been made and the equation is invalid. For example an expression for velocity,

where is mass would have the following units,

From above we know that , so

which is clearly not true. The expression for velocity is nonsense. Again, only if our units match on each side of the equation is the equation valid.

Created by Mathematica (April 9, 2004)