Copyright © 2008 jsd

Units of Measurement
John Denker

*   Contents

1  Expressing Units, and Converting Units

As a first example, suppose the length of a certain hallway is L. This length is a thing unto itself, a mathematical thing, indeed a physical thing. It exists whether you measure it or not, and it exists independently of whatever units – if any – you decide to use. Therefore the length is not «L feet» or «L yards» or anything like that; the length is simply L.

Now suppose we measure the hallway and find that it is 10 yards long. We can write that as an equation:

L = 10 yd

Note the structure on the RHS: 10 is a pure number, while yd is a unit of measurement. L is not equal to 10, and L is not equal to yd; rather, L is equal to the product, 10·yd.

As a separate matter, we know:

1 yd = 3 ft   (by definition)

Therefore we can write:

3 ft
1 yd

 = 1

Equation 3 comes from equation 2 by direct application of the axioms of algebra, namely the definition of division, and the definition of equality.

The axioms also say we can multiply the RHS of equation 1 by unity. This gives us:

L = 10 yd · (1)      (4a)
 = 10 yd ·      (4b)

which simplifies to

L = 30 ft

which is true and useful.

This whole discussion illustrates the real-world power and generality of algebra. Units of measurement (such as ft and yd) play by the same rules as algebraic variables (such as x and y).

This is the way smart people have been doing it for thousands of years. In King Tut’s time, the royal cubit was equal to 7 palms, and a palm was equal to 4 fingers. Nowadays kids learn in first grade that a quarter is equal to 5 nickels, and a nickel is equal to 5 pennies; from this you can infer that a quarter is equal to 25 pennies.

The technique of writing equations to express the relationships between units is used even in non-scientific settings such as cooking; see e.g. reference 1. This is the smart way to do it. (There are plenty of dumber ways, but let’s not get into that.)

The tactic of multiplying by unity is heavily used in science and engineering at all levels, from high-school chemistry on up. It even has a name: The Factor Label method.

In my experience, this is the #1 best response to those who claim they have never had occasion to apply anything they learned about algebra.

In simple cases, the Factor Label method can be carried out mechanically, even by people who have no clue about algebra. However, in more complicated cases – involving (say) square feet and cubic yards – it helps to know the axioms of algebra. In all cases, if you want to understand what’s going on, algebra helps a lot.

Some pretty smart mathematicians have looked into the algebraic structure of units. In the metric system, a kilogram is 1000 grams. This was started by a committee that included de Borda, Laplace, Monge, Condorcet, Legendre, and others. Later Gauss had a few things to say about units.

On various odd occasions, it may be expedient to strip the unit out of a quantity such as L. In such cases, the algebraically reasonable way to write such things is L/yd and L/ft ... with the units in the denominator. This leads to the inelegant but correct equation:

L/ft = L/yd

This inelegant tactic is useful if you are using a calculator that can multiply pure numbers but is too stupid to keep track of the units. Such situations are becoming rare; any computer worthy of the name can run a computer algebra system such as Macsyma that can easily keep track of units, directly using equations such as L=10yd.

In any case, it must be emphasized that the LHS of equation 6 is not «L feet», and the RHS is not «L yards». The units belong in the denominator, because we are stripping the units out of L by dividing them out.

It must also be emphasized that carrying around a pure number, stripped of its units, is bad practice. You run the risk of forgetting which units go with which number. Sometimes the penalty for getting this wrong is on the order of three hundred million dollars, as in the case of the Mars Climate Orbiter (reference 2 and reference 3).

Figure 1: Mars Climate Orbiter mission logo

As hinted in item 1, it is entirely possible to measure something using no units at all. On more than a few occasions I have been miles away from the nearest ruler, so I recorded in my notebook that something was |——| long. That’s an analog measurement.

2  Some Other Examples

There is a right way and a wrong way to express the units associated with a given quantity.

2.1  Length and Area

Some simple yet useful examples include:

To a good approximation you can make a yardstick by gluing three one-foot rulers together end-to-end, as shown in figure 2. This gives us a concrete, physical basis for saying that 1 yard “equals” three feet.

Figure 2: One Yard = Three Feet

This physical interpretation is captured in the algebra, if we do things right. For example, if we multiply two quantities together, the numerical magnitudes get multiplied and the units get multiplied. For example, if the length L is 30 feet and the width W is 2 feet, the area A is given by:

A = L · W
  = (30 ft) · (2 ft) 
  = 60 ft2

It would be quite improper to say «the area was 60», or that «the area was 60 feet». The area is 60 square feet.

2.2  Deprecated Example: C versus F

Here is an example of what not to do. This is taken from page 13 of reference 4.

     C = 5/9 (F − 32)   
     F = 9/5 C + 32   

In this document, we put equations and tables on a red background to warn you about things that are not recommended. Similarly we sometimes use «⋯» as scare-quotes around improper expressions.

This equation uses the unit of measurement (C or F) as a stand-in for the thing being measured. This is a Bad Idea. It is diametrically inconsistent with modern notions of unit-analysis.

There is a very important difference between the thing being measured and the unit used to measure it, as discussed in section 3.2.

2.3  Deprecated Example: Motor Data Plate

Here’s another example: Recently I was looking at the data plate on an electric motor. It said:

HP:  1              Type:  C
RPM:  1725              SF:  1.0
A:  12.3              PH:  1
V:  115              HZ:  60
Table 1: Motor Data Plate

The colons on the data plate were meant to be interpreted as equal signs.

If you find this confusing, good for you. This usage is considered perverse, as indicated by the red background. This usage is exactly backwards relative to how it should be done. However, you still see it from time to time, so it is worth learning to recognize it. Here’s how it should have been done:

rated power=  1 HP              type=  capacitor-start
rotation rate=  1725 RPM              service factor=  1.0
max current=  12.3 A              arrangement=  1 phase
voltage=  115 V              powerline frequency=  60 Hz
Table 2: Recommended Notation for Motor Data

If you are going to use the proper system you must strictly avoid mixing it with the perverse system. This leaves us with the question, what to do if you encounter something expressed the bad way, such as the motor data plate mentioned above.

The simplest answer is to recognize that the RPM mentioned on the data plate belongs on the RHS of the equation. Another option is to say that the RPM belongs in the denominator on the LHS; this is inelegant but not wrong:

rated power / HP=  1              Type=  capacitor-start
rotation rate / RPM=  1725              service factor=  1.0
max current / A=  12.3              arrangement / phase=  1
voltage / V=  115              powerline frequency / Hz=  60
Table 3: Another Good Notation for Motor Data

Note: Whenever you see the words “measured in” you can replace them by “divided by”. This is a trustworthy rule for translating word problems into equations. (It is analogous to the rule that says in expressions like “one tenth of thirty” the “of” gets replaced by “multiplied by”.)

2.4  Deprecated Example: Indicated Power of a Steam Engine

Here’s a trickier example of bad practice, from page 190 of reference 4. For a steam engine:

    IHP =  

I wish to make two different points about this. Please let’s not confuse the two points.

  1. First, I call attention to the LHS, which is “indicated horsepower”. This is another an example of using the unit of measurement as a stand-in for the thing being measured. This is a Bad Idea, as discussed in section 3.2.

    The thing that is really interesting here is that the way units are used on the LHS of equation 9 is inconsistent with the way units are used on the RHS, and in particular with the way the quantities P, L, A, and N are defined on the RHS of equation 10.

    This is relevant because if the people who use backwards units would use them consistently backwards, we might chalk it up to a difference of taste, not worth arguing about. But when they are inconsistent about it, it’s more than a question of taste; it’s an out-and-out blunder.

    The best way to remove the inconsistency is to standardize on the modern approach.

  2. Secondly, let us turn to the RHS. The book says:
    average pressure = P pounds per square inch    
    stroke = L ft    
    area = A sq ft    
    there are   N power strokes per minute

    which is interesting because in some sense those four statements are dimensionally correct, if we consider P, L, A, and N to be “pure numbers” (unitless and dimensionless) standing in front of explicit units.

    This is, however, still not recommended. Even though it is not wrong in any deep sense, it is not recommended. The problem is that the numerical value of (say) A is tied to the units in use.

In contrast, modern practice is to do algebra using variables that have not had units factored out, so that the area-variable expresses the concept of area, no matter what units are used. For example:

yes: A =
(in whatever units)
yes: area =
(in whatever units)
no: area =
A sq. ft

At this point you may be wondering about an interesting issue:

So you could argue that a formula involving pure numbers has some value, when you are using not-so-fancy electronic aids.

That is a valid argument, but there is a stronger counterargument. The computer’s need for pure numbers can be accommodated within the formalism of modern unit analysis. One could write:

indicated power/HP = 
(P/psi) (L/ft) (A/ft2) (N/spm)

(where spm means power strokes per minute).

which has the advantage of being absolutely true no matter what units the variables are measured in. If L is measured in (say) meters instead of feet, then the problem is obvious: You will wind up with a factor of (m/ft) left over. More importantly the solution is obvious if you follow the factor-label method, because 1 (m/ft) is a number with a well-known value.

3  Discussion

3.1  Units versus Dimensions

Units are not the same as dimensions. A foot and a yard have the same dimensions, namely dimensions of length. But a foot and a yard are very different units; one is smaller than the other by a factor of three. Length and width have the same dimensions. Sometimes length and width are measured in the same units, and sometimes in different units. The distinction between units and dimensions comes into sharp focus when we consider dimensionless units, as in reference 5.

See reference 6 for an introduction to dimensional analysis. All notions of units and dimensions rest on deeper notions of scaling, as discussed in reference 7.

3.2  Units of Measurement versus Thing Being Measured

There is a profound difference between the unit of measurement and the thing being measured. For example, if you are measuring a piece of fabric with a yardstick, the fabric is not to be confused with the yardstick. The fabric is the thing being measured, and the yardstick embodies the unit of measurement.

Another example can be found in equation 8. Temperature is the thing being measured; degrees C or degrees F are the units of measurement.

Yet another example is the rotation rate in table 2. The rate is the thing being measured, and RPM is the unit of measurement. Just as the rulers in figure 2 have physical significance, the RPM has physical significance. If the rate changes from 1725 RPM to 1730 RPM, the RPM is not what changes; the rate is what changes. It makes absolutely no sense to write RPM = 1725. Writing such a thing is super-sloppy.

3.3  Dimensions Named After Units

Sometimes, the thing being measured is named after a unit. Examples include

It must be emphasized that the physical quantity is not required to be measured in the named units.

4  References

“Baking Conversion Table”
Douglas Isbell, Mary Hardin, Joan Underwood,
“Mars Climate Orbiter Fact Sheet”
Reno C. King Jr.,
Marine Engineering
Prentice-Hall (1948).
John Denker,
“Dimensionless Units”
John Denker
“Dimensional Analysis”
John Denker,
“Scaling Laws”
Copyright © 2008 jsd