Metric Prefixes

Even scientific notation can get to be a hassle, and so there are a number of abbreviations that scientists and others use, called metric prefixes. For example, the symbol k (which stands for kilo) is an abbreviation for $\times {10}^3$, so when you read

1. 3km ("6.3 kilometers") you can think of it as $6.3\times {\mathbf{10}}^{\mathbf{3}}\mathrm{m}$.

There are metric prefix abbreviations for every thousands place. These are the ones we will see most frequently in class:

1. If the unit is squared or cubed, then the metric prefix is also squared or cubed. For example, the expression $5{\mathrm{cm}}^2$ is equivalent to

   $$5(\mathrm{cm}{)}^2=5\times ({10}^{-2}\mathrm{m}{)}^2=5\times {10}^{-4}{\mathrm{m}}^2,$$ not $5\times {10}^{-2}{\mathrm{m}}^2$.

Note

If you find this frustrating, you're not alone! (I think it would be much better if we gave the kilogram a different name, like "klonk", and then grams would be "milliklonks" and everything would be consistent. Alas, though I tempted to do so, it's not likely to catch on.
2. Because we are using the SI system, the base unit of mass is the kilogram, not the gram. Thus if you have a mass like 3.5kg, you should NOT plug it into your calculator as 3.5E3, but just as 3.5. Conversely, if you are given a mass in grams, you need to convert it into kilograms, so $$35.6\mathrm{g}=35.6\times {10}^{-3}\mathrm{kg}$$

   In this book we will always give masses in kilograms and never use prefixes with mass quantities, to avoid confusion.

If the answer to a problem is written a scientific notation, then you may choose to write the answer using metric prefixes instead. If the exponent is one of those on the list, then the process is straightforward: for example, $5.2\times {10}^{-9}\mathrm{m}$ is 5.2nm, because "n" (nano) is an abbreviation for $\times {10}^{-9}$. If the exponent isn't on the table, then you will need to adjust the exponent until it matches one (like we did in Scientific Notation), which usually means moving it to the nearest multiple of three (higher or lower, your choice). So for instance, you can write $$4\times {10}^{-4}\mathrm{m}⟹0.4\times {10}^{-3}\mathrm{m}⟹0.4\mathrm{mm}$$ or $$4\times {10}^{-4}\mathrm{m}⟹400\times {10}^{-6}\mathrm{m}⟹400\mu \mathrm{m}$$

(Remember that to decrease the exponent you must increase the mantissa, and vice versa.)