SI units are part of the metric system, which is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 2 lists the metric prefixes and symbols used to denote various factors of 10 in SI units. For example, a centimeter is one-hundredth of a meter (in symbols, 1 cm = 10–2 m) and a kilometer is a thousand meters (1 km = 103 m). Similarly, a megagram is a million grams (1 Mg = 106 g), a nanosecond is a billionth of a second (1 ns = 10–9 s), and a terameter is a trillion meters (1 Tm = 1012 m).
Table 2 Metric Prefixes for Powers of 10 and Their Symbols
The only rule when using metric prefixes is that you cannot “double them up.” For example, if you have measurements in petameters (1 Pm = 1015 m), it is not proper to talk about megagigameters, although 106 × 109 = 1015. In practice, the only time this becomes a bit confusing is when discussing masses. As we have seen, the base SI unit of mass is the kilogram (kg), but metric prefixes need to be applied to the gram (g), because we are not allowed to “double-up” prefixes. Thus, a thousand kilograms (103 kg) is written as a megagram (1 Mg) since
103 kg = 103 × 103 g = 106 g = 1 Mg.
Incidentally, 103 kg is also called a metric ton, abbreviated t. This is one of the units outside the SI system considered acceptable for use with SI units.
As we see in the next section, metric systems have the advantage that conversions of units involve only powers of 10. There are 100 cm in 1 m, 1000 m in 1 km, and so on. In nonmetric systems, such as the English system of units, the relationships are not as simple—there are 12 in. in 1 ft, 5280 ft in 1 mi, and so on.
Another advantage of metric systems is that the same unit can be used over extremely large ranges of values simply by scaling it with an appropriate metric prefix. The prefix is chosen by the order of magnitude of physical quantities commonly found in the task at hand. For example, distances in meters are suitable in construction, whereas distances in kilometers are appropriate for air travel, and nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications. Instead, we rescale the units with which we are already familiar.
Restate the mass 1.93 × 1013 kg using a metric prefix such that the resulting numerical value is bigger than one but less than 1000.
Strategy
Since we are not allowed to “double-up” prefixes, we first need to restate the mass in grams by replacing the prefix symbol k with a factor of 103 (see Table 2). Then, we should see which two prefixes in Table 2 are closest to the resulting power of 10 when the number is written in scientific notation. We use whichever of these two prefixes gives us a number between one and 1000.
Solution
Replacing the k in kilogram with a factor of 103, we find that
1.93 × 1013 kg = 1.93 × 1013 × 103 g = 1.93 × 1016 g.
From Table 2, we see that 1016 is between “peta-” (1015) and “exa-” (1018). If we use the “peta-” prefix, then we find that 1.93 × 1016 g = 1.93 × 101 Pg, since 16 = 1 + 15. Alternatively, if we use the “exa-” prefix we find that 1.93 × 1016 g = 1.93 × 10−2 Eg, since 16 = −2 + 18. Because the problem asks for the numerical value between one and 1000, we use the “peta-” prefix and the answer is 19.3 Pg.
Significance
It is easy to make silly arithmetic errors when switching from one prefix to another, so it is always a good idea to check that our final answer matches the number we started with. An easy way to do this is to put both numbers in scientific notation and count powers of 10, including the ones hidden in prefixes. If we did not make a mistake, the powers of 10 should match up. In this problem, we started with 1.93 × 1013 kg, so we have 13 + 3 = 16 powers of 10. Our final answer in scientific notation is 1.93 × 101 Pg, so we have 1 + 15 = 16 powers of 10. So, everything checks out.
If this mass arose from a calculation, we would also want to check to determine whether a mass this large makes any sense in the context of the problem.
