8  prefix operations

8.1 one prefix

Sometimes we need to convert between different orders of magnitude of the same base unit. For example:

How many kilograms does a piece of metal of 600 grams have?

In the solution below, the trick is to realize that we can always multiply a number by one!

\begin{align*} \text{(start) }&\quad&& 600 \text{ g} \\ \text{(multiply by one) }&&& 1\cdot 600 \text{ g} \\ \text{(in powers of 10) }&&& \underbrace{10^{-3}10^3}_{=1}\cdot 600 \text{ g} \\ \text{(kilo means 1000) }&&& 10^{-3}\underbrace{10^3}_{=\text{k}}\cdot 600 \text{ g} \\ \text{(put k next to g) }&&& 10^{-3}\cdot 600 \text{ kg} \\ \text{(end) }&&& 0.6 \text{ kg} \end{align*}

Now that we understand the logic behind this conversion, we can do it a bit shorter. I know that I need the result in kilograms, so I write a \text{k} next to the grams and multiply by a factor of 10^{-3}, to balance out the “kilo”:

\begin{align*} \text{(start) }&\quad&& 600 \text{ g} \\ \text{(write both k and }10^{-3}\text{) }&&& 10^{-3}\cdot 600 \text{ kg} \\ \text{(end) }&&& 0.6 \text{ kg} \end{align*}

8.2 two prefixes

Let’s try another one, this time with two prefixes.

How many centimeters are there in 2 kilometers?

In the solution below we use the fact that “centi” means one hundredth, \text{c}=10^{-2}.

\begin{align*} \text{(start) }&\quad&& 2 \text{ km} \\ \text{(write k as 1000) }&&& 2\cdot 10^3 \text{ m} \\ \text{(write both c and }10^{2}\text{) }&&& 2\cdot 10^3 \underbrace{10^2 \text{ c}}_{=1} \text{m}\\ \text{(end) }&&& 200000 \text{ cm} \end{align*}

8.3 two prefixes, but different

A solution has 5 micrograms of salt per milliliter. What is the concentration in grams per liter?

\begin{align*} \text{(start) }&\quad&& 5 \,\frac{\mu\text{g}}{\text{mL}}\\ \text{(spell out }\mu\text{ and m)}&&& 5 \,\frac{10^{-6}\text{g}}{10^{-3}\text{L}}\\ \text{(consolidate) }&&& 5\cdot 10^{-3} \,\frac{\text{g}}{\text{L}}\\ \text{(end) }&&& 0.005 \,\frac{\text{g}}{\text{L}} \end{align*}

If we are beeing loose with significant figures, then I guess that the last line counts as correct, but I would recommend stopping at the line before last.

8.4 areas and volumes

When converting units of area or volume, don’t forget to square or cube the prefixes.

How many square meters are in a square kilometer?

1\, \mathrm{km}^2 = 1\, (\mathrm{km})^2 = 1 (\underbrace{10^3}_\text{k} \mathrm{m})^2 = 1 \left( 10^3 \right)^2 \left(\mathrm{m} \right)^2 = 10^6\, \mathrm{m}^2

There are a million square meters in a square kilometers.

When we write km^2, it’s understood that the square acts on both the “kilo” and the “meter”. Nobody writes this with explicit parentheses as we did above, but that is the intended meaning.

Let’s see another example, now with volume.

How many liters are in a cubic meter? One liter is the volume of a cube of side 10 cm.

1\, \mathrm{m}^3 = (100\, \mathrm{cm})^3 = \left(10 \cdot 10\, \mathrm{cm}\right)^3 = 10^3\left(10\, \mathrm{cm}\right)^3 = 10^3\, \mathrm{L}

There are a thousand liters in a cubic meter.