\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)
Appendix A: Miscellany
6.

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 $6.3\mathbf{\times 10^3}\u{m}$.

Complete List
yotta Y 1024
zetta Z 1021
exa E 1018
peta P 1015
tera T 1012
giga G 10 9
mega M 10 6
kilo k 10 3
hecto h 10 2
deka dk 10 1
deci d 10-1
centi c 10-2
milli m 10-3
micro μ 10-6
nano n 10-9
pico p 10-12
femto f 10-15
atto a 10-18
zepto z 10-21
yocto y 10-24

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

NameSymbolStands ForNotes
megaM106
E6		can also be read as "million"
kilok103
E3		can also be read as "thousand"
centic10-2
E-2		usually only seen in "centimeters"
millim10-3
E-3
microμ10-6
E-6		This is the Greek letter "mu": a u with a ponytail
nanon10-9
E-9		I remember this by saying "nano nine-o"
picop10-12
E-12
femtof10-15
E-15
If you are given a value with a metric prefix, you don't have to bother converting it first. Instead, replace the prefix with the appropriate power of ten when plugging the number into your calculator. For example, to calculate \(5\mathrm{km}/3\mathrm{cs}\), you type into your calculator 5E3/3E-2. There are two things you have to look out for, however:
  1. If the unit is squared or cubed, then the metric prefix is also squared or cubed. For example, the expression $5\u{cm^2}$ is equivalent to

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

    2. 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 $$35.6\u{g}=35.6\times 10^{-3}\u{kg}$$

    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, $5.2\times 10^{-9}\u{m}$ 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 $$4\times 10^{-4}\u{m} \implies 0.4\times 10^{-3}\u{m} \implies 0.4\u{mm}$$ or $$4\times 10^{-4}\u{m} \implies 400\times 10^{-6}\u{m} \implies 400\u{\mu m}$$

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