# 1.8: Measured Equalities and Prefix Modifiers

Learning Objectives
• Identify measured equalities.
• Describe how prefixes are used in the metric system.
• Develop prefix modifier equalities.

While temperature units are converted using equations, units for mass, volume, length, and time are changed using conversion factors, which are based on equalities.  Conversion factors will be discussed in the next section of this chapter.  Equalities do not contain variables and instead include a number and a unit on both sides of an equal sign.  There are two types of equalities:  Measured equalities and prefix modifier equalities.

## Measured Equalities

Measured equalities are developed by taking the same measurement with two different tools, where the tools each have different units.  For example, a block of wood could be measured in centimeters, using the metric side of a ruler, and in inches, using the U.S. side of a ruler.  Examples of measured equalities are shown below in Table $$\PageIndex{1}$$.

Time Mass Length
Table $$\PageIndex{1}$$: Measured Equalities
1 week (wk) = 7 days (d) 1 pound (lb) = 16 ounces (oz) 1 mile (mi) = 5,280 feet (ft)
1 d = 24 hours (h) 1 kilogram (kg) = 2.2 lb 1 yard (yd) = 3 ft
1 h = 60 minutes (min)    1 ft = 12 inches (in)
1 min = 60 seconds (s)   1 in = 2.54 centimeters (cm)

Note that each of these equalities is exact, so its numbers are considered to have infinitely-many significant figures.  These equalities will ultimately be used in conversion factors, which will involve multiplication and division.  Since the answer must be limited to the lesser count of significant figures when multiplying and dividing, these equalities will never limit the number of significant figures in a calculated answer.

## Prefix Modifiers

A prefix modifier is used to change a base unit by a power of 10.  The base units for mass, volume, length, and time are given below in Table $$\PageIndex{2}$$.

Measurable Quantity Base Unit
Table $$\PageIndex{2}$$: Base Units
Mass gram (g)
Volume liter (L)
Length meter (m)
Time second (s)

Remember that it is important to use appropriate capitalization for abbreviations!  Each of these base units can be changed by a power of 10 to be made larger or smaller.  A positive power is used to make a unit larger, and a negative power is used to make a unit smaller.  This system parallels the rules for powers when writing numbers in scientific notation.  Table $$\PageIndex{3}$$ lists the most common prefix modifiers and their meanings.  Note that all prefix modifiers are exact, by definition.

Prefix Abbreviation Meaning
Table $$\PageIndex{3}$$: Prefix Modifiers
"Larger" Prefix Modifiers
tera $$\text{T}$$ 1012
giga $$\text{G}$$ 109
mega $$\text{M}$$ 106
kilo $$\text{k}$$ 103
"Smaller" Prefix Modifiers
deci $$\text{d}$$ 10-1
centi $$\text{c}$$ 10-2
milli $$\text{m}$$ 10-3
micro $$\mu$$ 10-6
nano $$\text{n}$$ 10-9
pico $$\text{p}$$ 10-12

Each of these prefix modifiers can be inserted before a base unit to make a new unit.  For example, combining the prefix modifier "kilo" with the base unit "gram" would create the new unit "kilogram."

Example $$\PageIndex{1}$$

Give the abbreviation for each of the following units.

1. teraliter
2. microsecond

Solutions

Explanation:  Each of these units contains a prefix modifier, followed by a base unit.  The prefix modifier "tera" is abbreviated with a capital "T", and the base unit "liter" is abbreviated with a capital "L".

Explanation:  The prefix modifier "micro" is abbreviated with a Greek lower-case letter called mu, "µ", and the base unit "second" is abbreviated with a lower-case "s."

Exercise $$\PageIndex{1}$$

Give the full unit for each of the following abbreviations.

1. mm
2. pg
Each of these units contains a prefix modifier, followed by a base unit.  Understanding this is important, as "m" represents many different quantities in science!  The prefix modifier represented by "m" is milli, and the base unit abbreviated by "m" is meter.  Therefore, the full unit is the "millimeter".
The prefix modifier represented by "p" is pico, and the base unit abbreviated by "g" is gram. Therefore, the full unit is the "picogram".

Finally, each new unit can be related back to its corresponding base unit by replacing the prefix modifier with its meaning.  For example, the kilogram (kg) can be related to the gram (g), as shown below.

kg = 103 g

Since the prefix modifier "kilo" (k) means "103," these two quantities can be interchanged.  The base unit, gram (g), is written with both the prefix modifier and its meaning.  Since their base units (grams) match, and the prefix modifier and its meaning are equivalent, these two quantities are equal to one another.  If no numerical value is explicitly-written in an equality, an unwritten "1" is understood to be present.  Note that, while not absolutely necessary, a prefix modifier equality can be simplified by rewriting the numerical value in decimal format.

kg = 1,000 g

Example $$\PageIndex{2}$$

Develop a prefix modifier equality that relates centimeters and meters.

Solution

The prefix modifier "centi" (c) means "10-2".  Therefore,

cm = 10-2 m

This equality can be simplified to become

cm = 0.01 m

The decimal value can be eliminated by dividing both quantities by 0.01.  Note that if no number is explicitly written on one side of an equality, an unwritten "1" is understood.  Therefore,

100 cm = m

Exercise $$\PageIndex{2}$$

Develop a prefix modifier equality that relates nanoseconds and seconds.