## 2.3 Converting Units: Prefix Conversions

### Learning Objectives

By the end of this section, you will be able to:

• Be able to write prefix conversion factors.
• Use dimensional analysis to carry out prefix unit conversions.

Sometimes we use units that are fractions or multiples of a base unit. Ice cream is sold in quarts (a familiar, non-SI base unit), pints (0.5 quart), or gallons (4 quarts). We also use fractions or multiples of units in the SI system, but these fractions or multiples are always powers of 10. Fractional or multiple SI units are named using a prefix and the name of the base unit. For example, a length of 1000 meters is also called a kilometer because the prefix kilo means “one thousand,” which in scientific notation is 103 (1 kilometer = 1000 m = 103 m). The prefixes used and the powers to which 10 are raised are listed in Table 1.

Table 1. Common Unit Prefixes
Prefix Symbol Factor
femto f 10−15
pico p 10−12
nano n 10−9
micro µ 10−6
milli m 10−3
centi c 10−2
deci d 10−1
kilo k 103
mega M 106
giga G 109
tera T 1012

## Deriving Prefix Conversion Factors

Suppose we are asked to convert 50.0 mg of ibuprofen to grams.  How do we go about this conversion? The first thing to recognize is that we are dealing with a prefix conversion problem and in order for us to successfully convert milligrams to grams, we need to know what exactly does the prefix milli- mean.  From Table 1, we see milli- is 10-3. How do we express this as a conversion factor? The relationship between mg and g can be expressed two ways (shown as equalities):

$\large (1 \text{ mg}=10^{-3}\text{ g}) \text{ or }(10^{3}\text{ mg}=1\text{ g})$

We use the equality to write the conversion factor needed to convert whatever unit we wish to convert, by showing the equality as a fraction, as shown below:

$\large (\frac{10^{-3}{\text{ g }}}{1\text{ mg}})\text{ or }(\frac{1{\text{ mg }}}{10^{-3}{\text{ g }}})\text{ ; }(\frac{10^{3}{\text{ mg }}}{1\text{ g}})\text{ or }(\frac{1{\text{ g }}}{10^{3}{\text{ mg }}})$

Note: When writing the conversion factor, if you place the 10 to the power (10x) opposite the prefix unit, the sign of the power stays exactly the same as presented in Table 1. However, if you place the 10 to the power (10x) with the prefix unit, the sign of the power needs to be switched.

## Prefix to Base Unit Conversions

Now that we know the relationship between grams and milligrams we use dimensional analysis to determine how many grams are in 50.0 mg of ibuprofen. In dimensional analysis, we multiply what is given by one or more conversion factors. Conversion factors are applied in a manner to which cancel units until you arrive at the desired unit.  In this example, we start with 50.0 mg and use the conversion factor derived earlier to cancel the mg:

$\large 50.0 \cancel{\text{ mg}}\times\frac{10^{-3}{\text{ g}}}{1\cancel{\text{ mg}}}= 5.00\times 10^{-2}\text{ g}$

or

$\large 50.0 \cancel{\text{ mg}}\times\frac{1{\text{ g}}}{10^{3}\cancel{\text{ mg}}}= 5.00\times 10^{-2}\text{ g}$

### Example 1: Prefix to base Unit Conversions

How many femtoliters are in $\large 2.00\times 10^{2}$ liters?

How many seconds are in 5.0 gigaseconds (Gs)?

## Prefix to Prefix Conversions

Sometimes there will be a need to convert from one unit with one numerical prefix to another unit with a different numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or you can go the easier route: first convert the quantity to the base unit, the unit with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You can do the conversion in two separate steps or as one long algebraic step. For example, to convert 2.77 kg to milligrams:

First convert to the base unit of grams:

$\large 2.77 \cancel{\text{ kg}}\times\frac{10^{3}{\text{ g}}}{1\cancel{\text{ kg}}}= 2.77\times 10^{3} \text{ g}$

Then convert the base unit to milligrams:

$\large 2.77\times 10^{3}\cancel{\text{ g}}\times\frac{1{\text{ mg}}}{10^{-3}\cancel{\text{ g}}}= 2.77\times 10^{6} \text{ mg}$

Alternatively, it can be done in a single multistep process:

$\large 2.77 \cancel{\text{ kg}}\times\frac{10^{3}\cancel{\text{ g}}}{1\cancel{\text{ kg}}}\times\frac{10^{3}{\text{ mg}}}{1\cancel{\text{ g}}}= 2.77\times 10^{6} \text{ mg}$

You get the same answer either way.

You can also combine the two conversion factors into one, and convert directly from one prefix to another prefix:

$\large 2.77 \cancel{\text{ kg}}\times\frac{10^{3}\cancel{\text{ mg}}}{10^{-3}\cancel{\text{ kg}}}= 2.77\times 10^{6} \text{ mg}$

Note: When writing the prefix to prefix conversion factor, you place the 10 to the power (10x) opposite the prefix unit. In this case we have prefixes in the numerator and the denominator, so we end up with a 10x in both spots, with the 103 going opposite of the kilo prefix and the 10-3 going opposite the milli prefix.

When considering the significant figures of a final numerical answer in a conversion, there is one important case where a number does not impact the number of significant figures in a final answer—the so-called exact number. An exact number is a number from a defined relationship, not a measured one. For example, the prefix kilo- means 1,000—exactly 1,000, no more or no less. Thus, in constructing the conversion factor

$\large \frac{1000{\text{ g}}}{\text{ kg}}$

neither the 1,000 nor the 1 enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly by what the prefix kilo- means. Another way of thinking about it is that these numbers can be thought of as having an infinite number of significant figures, such as

$\large \frac{1000.0000000000 . . .{\text{ g}}}{1.0000000000 . . .\text{ kg}}$

The other numbers in the calculation will determine the number of significant figures in the final answer.

### Example 2: Prefix to Prefix Conversion

How many nanoseconds are in 368.09 μs?

How many milliliters are in 607.8 kL?

### Exercises

1. Perform each of the following conversions:

a.  3.4 m to nm

b.  72.6 mL to L

c.  0.50 g to mg

d.  60.0 s to Ms

2. Perform each of the following conversions:

a.  5.83 km to m

b.  5 × 10−4 L to cL

c.  10.0 kg to g

d.  3.6 × 103 s to ms

3. Perform each of the following conversions:

a.  1.2 × 10−3 mL to pL.

b.  200.0 cm to Tm.

c.  1.2 × 106 Ms to ks

d.  0.674 kL to mL

4. Perform each of the following conversions:

a.  588.2 dg to Mg.

b.  200.0 km to fm.

c.  0.150 ms to Gs

d.  3.0 × 109 nm to Tm