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# Lesson 1.4: Units of Measurement

← Lesson 1.3: Lesson 1.5: →

## Example 1: For the following identify the name, prefix, base unit, and state how large the unit is relative to the base unit.

Abbreviation Name Prefix Base Unit How large
km kilometer k m 1 km = 103m
mg milligram m g 1 g = 103 mg
dag
ns

Prefixes are used to make larger or smaller units than the base unit. Any prefix at the top half of this chart will make a larger unit, and any unit at the bottom half will make a smaller unit. For example, the unit km has two parts. The letter k indicates a prefix. It is on the top half of the chart and thus makes a larger unit than the base unit which is indicated by the letter m for meter. How much larger is given in the chart. It is 1000 or 103 times as large as the base unit. We can write this as an equation.

1 km = 1000 m or 1 km = 103m

1 km means 1000 m, 1 kg means 1000 g, 1 ks means 1000 s, and 1 kmol means 1000 mol.

The second answer may seem more confusing. First, note that it uses the prefix m for milli. This is confusing only in that m can also be used to indicate the base unit m for meter. Keep in mind that when there are two letters in a unit the first indicates a prefix and the second the base unit. If a unit has a single letter that letter refers to the base unit. Second, it is conceptually less confusing for most students to say that there are 1000 mg per g than to say that 1 mg is 1/1000 of a gram. This way of writing unit comparisons always writes out how many of the smaller unit are in the larger, and the larger unit is the one higher in the chart. It is perfectly correct to write 1 mg = 10-3 g but on this site we will consistently write 1 g = 103 mg.

1 ms means 1/1000 of a second, and 1 mm means 1/1000 of a meter, but we will write 1 s = 1000 ms, and 1 m = 1000 mm.

## Example 2: For the following identify which unit is larger and how many of the smaller units are equivalent to 1 of the larger units.

Units Larger Comparison
km,cm km 1 km = 105 cm
km,hm km 1 km = 10 hm
m,hm
dm,cm
mg,kg

The unit that is higher in the chart is the larger unit. It is larger by one or more powers of ten. Compare the powers of the larger with the smaller to see how many powers of ten. For example, km means 103 m, and cm means 10-2 m, so km is larger than cm by five powers of ten, and we write 1 km = 105 cm.

## Example 3: For the following identify the conversion factor and complete the conversion.

12 cm = ? km 1 km = 105 cm 1.2 x 10-4 km
3.25 km = ? hm
1.25 x 103 cm = ? dm
3.4 x 104 mg = ? kg

Once we identify how many of the larger unit are equivalent to the smaller unit we can write out a conversion factor. For example, if 1 km = 105 cm we can write out a conversion factor in two ways.

$$\frac{1 \,km}{10^5 \,cm}$$ or $$\frac{10^5 \,cm}{1 \,km}$$

Which of these conversion factors we use depends on which unit we are trying to convert. If we wish to convert the following, 12 cm = ? km we would use the first and write

\begin{align*} 12 \, cm \times \frac{1 \,km}{10^5 \,cm} & = 12 \times 10^{-5} \, km \\\\ & = 1.2 \times 10 \times 10^{-5} \, km \\\\ & = 1.2 \times 10^{-4} \, km \\\\ \end{align*}

If the task had been to convert the following, 12 km = ? cm we would use the second and write

\begin{align*} 12 \, km \times \frac{10^5 \,cm}{1 \, km} & = 12 \times 10^{5} \, cm \\\\ & = 1.2 \times 10 \times 10^{5} \, cm \\\\ & = 1.2 \times 10^{6} \, cm \\\\ \end{align*}

## Example 4: For the following identify the conversion factor and complete the conversion.

Conversion Problem Length Conversion Factor Derived Conversion Factor Answer
12 m2 = ? cm2 1 m = 102 cm 1 m2 = 104 cm2 1.2 x 105 cm2
3.4 x 104 cm2 = ? hm2
1.25 x 103 dm3 = ? cm3
3.25 km3 = ? cm3

As before, we start by identifying how many of the smaller units are equivalent to one of the larger. However, since neither area not volume units are base units, we must derive them from the base units. Since 1 m = 102 cm, 1 m2 is equivalent to 102 cm x 102 cm or 104 cm2. Using this derived conversion factor we can proceed as before.

Using this derived conversion factor we can proceed as before.

\begin{align*} 12 \, m^2 \times \frac{10^4 \,cm^2}{1 \,m^2} & = 12 \times 10^4 \, cm^2 \\\\ & = 1.2 \times 10 \times 10^4 \, cm^2 \\\\ & = 1.2 \times 10^5 \, cm^2 \\\\ \end{align*}

A more general set of diagrams showing area relations is shown below.