SI base units

A physical quantity is something that can be measured, such as mass or time. A measurement gives a numerical value and a unit: for example, a mass of $5,\mathrm{kg}$ has the numerical value $$$5$$$ and the unit kilogram.

SI base units are fundamental units from which other SI units are constructed. The relevant base quantities and their units are:

Mass: kilogram ( $\mathrm{kg}$ )
Length: metre ( $\mathrm{m}$ )
Time: second ( $\mathrm{s}$ )
Amount of substance: mole ( $\mathrm{mol}$ )
Temperature: kelvin ( $\mathrm{K}$ )
Electric current: ampere ( $\mathrm{A}$ )

Other SI units can be formed by combining base units. For example, speed can be expressed in metres per second, $\mathrm{m/s}$ , using the base units of length and time.

Unit symbols are case-sensitive. For example, the symbol for the kelvin is the capital letter $\mathrm{K}$ .

SI derived units

Some physical quantities are related to quantities measured in SI base units. Their units can therefore be built by combining base units through multiplication, division or powers. A unit formed in this way is called an SI derived unit.

For example, area is found by multiplying two lengths. Since length is measured in metres ( $\mathrm{m}$ ), area is measured in square metres ( $\mathrm{m^2}$ ). Speed is distance divided by time, so its SI unit is metres per second ( $\mathrm{m/s}$ ).

Some derived units have a special name and symbol. The joule, symbol $\mathrm{J}$ , is an SI unit of energy. It can also be written using the base units kilogram, metre and second:

\mathrm{J} = \mathrm{kg\,m^2\,s^{-2}}

The negative power means that $\mathrm{s^{-2}}$ is equivalent to dividing by $\mathrm{s^2}$ . The joule and $\mathrm{kg,m^2,s^{-2}}$ are therefore two ways to write the same unit.

SI prefixes

An SI prefix attached to a unit shows that the unit has been multiplied by a power of ten. For example, the prefix kilo means a factor of $$$10^3$$$ , so one kilometre is $$$10^3$$$ metres. The prefix milli means a factor of $10^{-3}$ , so one millimetre is $10^{-3}$ metres.

[Use the table shape for tabular content — GFM tables are disabled in md_block]PrefixSymbolFactortera$$\mathrm{T}$$$$10^{12}$$giga$$\mathrm{G}$$$$10^9$$mega$$\mathrm{M}$$$$10^6$$kilo$$\mathrm{k}$$$$10^3$$centi$$\mathrm{c}$$$$10^{-2}$$milli$$\mathrm{m}$$$$10^{-3}$$micro$$\mu$$$$10^{-6}$$nano$$\mathrm{n}$$$$10^{-9}$$pico$$\mathrm{p}$$$$10^{-12}$$femto$$\mathrm{f}$$$$10^{-15}$$

Positive powers give units larger than the unprefixed unit; negative powers give units smaller than it. To express a prefixed quantity in the unprefixed unit, multiply by the prefix factor. For example, $2.5\ \mathrm{km} = 2.5 \times 10^3\ \mathrm{m}$ and $3.6\ \mathrm{ns} = 3.6 \times 10^{-9}\ \mathrm{s}$ .

Prefix symbols are case-sensitive: $\mathrm{M}$ means mega, a factor of $$$10^6$$$ , whereas $\mathrm{m}$ as a prefix means milli, a factor of $10^{-3}$ . The symbol $\mathrm{m}$ on its own is also the unit symbol for the metre, so its position matters.

SI prefixes and standard form

In standard form, a nonzero number is written as $a \times 10^n$ , where $1 \leq |a| < 10$ and $$$n$$$ is an integer. Prefix factors are powers of ten: for example, $10^{-6}$ is $1 \times 10^{-6}$ in standard form.

Standard form can also show the size of a measured quantity in an unprefixed unit. For example, $0.0000045\ \mathrm{m} = 4.5 \times 10^{-6}\ \mathrm{m} = 4.5\ \mu\mathrm{m}$ . The micro prefix replaces the factor $10^{-6}$ attached to the metre.

Conversion between units

The same measurement can be expressed using different units of the same physical quantity. For example, a length of $1,\mathrm{m}$ is also $100,\mathrm{cm}$ . The length has not changed; only its numerical value and unit have changed.

A smaller unit gives a larger numerical value for the same quantity, because more of the smaller units are needed. A larger unit gives a smaller numerical value.

Using a conversion factor

A conversion factor is a ratio formed from two equivalent measurements. Since $1,\mathrm{m}=100,\mathrm{cm}$ , the ratio $\frac{1,\mathrm{m}}{100,\mathrm{cm}}$ has a value of 1. Multiplying by it changes the unit without changing the length.

To convert $250,\mathrm{cm}$ to metres, choose the ratio so that centimetres cancel:

250\,\mathrm{cm}\times\frac{1\,\mathrm{m}}{100\,\mathrm{cm}}=2.5\,\mathrm{m}

For the reverse conversion, use the ratio the other way round: $\frac{100,\mathrm{cm}}{1,\mathrm{m}}$ . The unit to be replaced should cancel, leaving the required unit.

Converting units raised to a power

When a unit is squared or cubed, the conversion factor must also be squared or cubed. Since $1,\mathrm{cm}=0.01,\mathrm{m}$ , a square centimetre is converted by squaring the whole relationship:

1\,\mathrm{cm^2}=(0.01\,\mathrm{m})^2=0.0001\,\mathrm{m^2}

Using $$$0.01$$$ rather than $$$0.01^2$$$ would convert a length factor, not an area factor. The same principle applies to cubed units.

Converting energy units

The electronvolt ( $\mathrm{eV}$ ), kilowatt-hour ( $\mathrm{kW,h}$ ) and joule ( $\mathrm{J}$ ) are units of energy. Their relationships are:

1\,\mathrm{eV}=1.602176634\times10^{-19}\,\mathrm{J}

1\,\mathrm{kW\,h}=3.6\times10^6\,\mathrm{J}

Use each relationship as a conversion factor in either direction. For example, multiplying by joules per electronvolt converts electronvolts to joules:

2\,\mathrm{eV}\times\frac{1.602176634\times10^{-19}\,\mathrm{J}}{1\,\mathrm{eV}}=3.204353268\times10^{-19}\,\mathrm{J}

To convert joules to electronvolts, divide by $1.602176634\times10^{-19}$ . To convert kilowatt-hours to joules, multiply by $3.6\times10^6$ ; to convert joules to kilowatt-hours, divide by that factor.

A kilowatt-hour is a unit of energy, not power. A kilowatt is a unit of power; a kilowatt-hour represents power multiplied by time.