When we want to know how long something is we often get a numerical answer, like 10 m, with a unit after the number showing us the type and size of the measurement. In this case it is metres so we know it is about distance, and we have an idea about how big 1 metre is. If we were to ask about the length of a river it might be 9000 m long. In this case it might be easier to talk about it as being 9 km long. Similarly , if we are measuring an insect its size might be 0.002 m in length. So we might write this as 2 mm instead.

\begin{aligned} 9000 \text{ m} &= 9 \times 10^3 \text{ m} &= &\,\,9 \text{ km} \\ 0.002 \text{ m} &=2 \times 10^{-3} \text{ m} &=&\,\,2\text{ mm} \end{aligned}

All of these units are related to the metre (m). We added prefixes ("k" and "m") to this unit ("m") to make the unit bigger or smaller by a certain amount. This makes it easier to talk about values that are very large or very small.

Prefix	Letter	Size
\text{yotta}	\text{Y}	{10}^{24}
\text{zetta}	\text{Z}	{10}^{21}
\text{exa}	\text{E}	{10}^{18}
\text{peta}	\text{P}	{10}^{15}
\text{tera}	\text{T}	{10}^{12}
\text{giga}	\text{G}	{10}^{9}
\text{mega}	\text{M}	{10}^{6}
\text{kilo}	\text{k}	{10}^{3}
\text{hecto}	\text{h}	{10}^{2}
\text{deca}	\text{da}	{10}^{1}
		{10}^{0}
\text{deci}	\text{d}	{10}^{-1}
\text{centi}	\text{c}	{10}^{-2}
\text{milli}	\text{m}	{10}^{-3}
\text{micro}	\mu	{10}^{-6}
\text{nano}	\text{n}	{10}^{-9}
\text{pico}	\text{p}	{10}^{-12}
\text{femto}	\text{f}	{10}^{-15}
\text{atto}	\text{a}	{10}^{-18}
\text{zepto}	\text{z}	{10}^{-21}
\text{yocto}	\text{y}	{10}^{-24}

Here is the table of all the currently approved metric prefixes. Many of these are rarely used.

Each row in the table contains a prefix which is how we would write it out "femto", a letter prefix is added to the unit "f", and the size says how much smaller that unit is compared to the original unit. For a metre we would have "femtometre", "fm" with 1\text{ fm} = 1 \times {10}^{-24} \text{ m}.

The units commonly used with these prefixes include grams (g), litres (L), metres (m), seconds (s), kelvin (K), joules (J) and watts (W). Note that some combinations of units and prefixes are rarely used. For example, hectograms (hg), megagrams (Mg) and gigametres (gm) are not widely used. Other combinations have other names commonly used such as a micrometre (\mu \text{m}) being referred to as a micron.

The most common prefixes used are in the middle of the table (although deca and hecto are also fairly rare). The top half of the tables shows prefixes which make the unit bigger. The bottom half shows the prefixes that make the unit smaller. We can also see that the larger prefixes (after kilo) have capital letters as their prefix.

To convert between units with a different prefix it is easiest to reduce it to the original unit.

Examples

Example 1

Which of the following is the same wattage as 1\,000\,000 W?

1 kW

1 \, \muW

1 MW

1 GW

Worked Solution

Create a strategy

Write the value in front of the unit in scientific notation, in the form 1 \times {10}^{n}. Use the table below as reference.

Prefix	Letter	Size
\text{giga}	\text{G}	{10}^{9}
\text{mega}	\text{M}	{10}^{6}
\text{kilo}	\text{k}	{10}^{3}
\text{hecto}	\text{h}	{10}^{2}
\text{deca}	\text{da}	{10}^{1}

Apply the idea

1\,000\,000 can also be written as 1 \times {10}^{6}. So 1 \,000 \,000 W is the same as 1 MW, option C.

Example 2

Write 52 ms in ns.

Worked Solution

Create a strategy

The prefix of ms is "m" which is the "milli" prefix.

The prefix of ns is "n" which is the "nano" prefix.

Use the table to compare the sizes of the two prefixes:

Prefix	Letter	Size
\text{deci}	\text{d}	{10}^{-1}
\text{centi}	\text{c}	{10}^{-2}
\text{milli}	\text{m}	{10}^{-3}
\text{micro}	\mu	{10}^{-6}
\text{nano}	\text{n}	{10}^{-9}

Apply the idea

From the table, we can see that the difference in size between a ms and a ns is {10}^{6} since 10^{-9} \times {10}^{6}=10^{-3}. So we should multiply by this to convert to ns.

\displaystyle 52 \text{ ms}	\displaystyle =	\displaystyle 52 \times {10}^{6} \text{ ns}	Multiply by {10}^{6}
	\displaystyle =	\displaystyle 52 \,000 \, 000 \text{ ns}	Evaluate
	\displaystyle =	\displaystyle 5.2 \times {10}^{7} \text {ns}	Write in scientific notation

Example 3

Write 5\times {10}^{5} dL in hL.

Worked Solution

Create a strategy

The prefix of dL is "d" which is the "deci" prefix.

The prefix of hL is "h" which is the "hecto" prefix.

Use this table to compare the sizes of the two prefixes

Prefix	Letter	Size
\text{mega}	\text{M}	{10}^{6}
\text{kilo}	\text{k}	{10}^{3}
\text{hecto}	\text{h}	{10}^{2}
\text{deca}	\text{da}	{10}^{1}
\text{deci}	\text{d}	{10}^{-1}
\text{centi}	\text{c}	{10}^{-2}
\text{milli}	\text{m}	{10}^{-3}
\text{micro}	\mu	{10}^{-6}

Apply the idea

From the table, we can say that the difference in size between a dL and a hL is 10^{-3} since 10^{2} \times {10}^{-3}=10^{-1}. So we should multiply by this to convert to hL.

\displaystyle 5 \times {10}^{5} \text{ dL}	\displaystyle =	\displaystyle 5 \times {10}^{5} \times {10}^{-3} \text{ hL}	Multiply by {10}^{-3}
	\displaystyle =	\displaystyle 5 \times {10}^{2} \text{ hL}	Evaluate

Idea summary

Units can have prefixes added to make them larger or smaller. Below is a table of the most common prefixes:

Prefix	Letter	Size
\text{giga}	\text{G}	{10}^{9}
\text{mega}	\text{M}	{10}^{6}
\text{kilo}	\text{k}	{10}^{3}
\text{hecto}	\text{h}	{10}^{2}
\text{deca}	\text{da}	{10}^{1}
		{10}^{0}
\text{deci}	\text{d}	{10}^{-1}
\text{centi}	\text{c}	{10}^{-2}
\text{milli}	\text{m}	{10}^{-3}
\text{micro}	\mu	{10}^{-6}
\text{nano}	\text{n}	{10}^{-9}

Outcomes

MA5.1-9MG

interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures

1.05 Metric prefixes

Ideas

Metric prefixes

Examples

Example 1

Example 2

Example 3

Outcomes

MA5.1-9MG

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