Unit Systems

Measurements Need a Common Reference

Imagine reading the quantity value $12$ without a unit. Is it $12 \text{ cm}$ , $12 \text{ m}$ , $12 \text{ kg}$ , or $12 \text{ s}$ ? The number is the same, but the meaning can change completely.

A unit is an agreed reference size for a physical quantity. Measurements from different people, tools, and places can be compared only when their units are stated and converted when necessary.

A quantity value is the numerical value multiplied by the unit.

The numerical value answers how much. The unit answers which reference it is compared with.

From Many Systems to SI

Before one standard became widely used, several unit systems existed side by side. Three names often appear in physics books:

System	Example base units	How to read them
FPS	$\text{ft},\ \text{lb},\ \text{s}$	foot, pound-mass, and second
CGS	$\text{cm},\ \text{g},\ \text{s}$	centimeter, gram, and second
MKS	$\text{m},\ \text{kg},\ \text{s}$	meter, kilogram, and second

In this FPS row, $\text{lb}$ denotes pound-mass. It must not be confused with pound-force, commonly written $\text{lbf}$ .

These differences make measurements hard to compare when the unit system is unclear. That is why modern science uses the International System of Units or SI. SI comes from the French name Système international d'unités and is known in English as the International System of Units. BIPM, or the Bureau International des Poids et Mesures, is the international organization that maintains the SI standard.

When you need to check the official definition of SI and its base units, open the BIPM overview of the International System of Units.

SI is an agreement that $1 \text{ m}$ , $1 \text{ kg}$ , and $1 \text{ s}$ mean the same thing in different schools, laboratories, factories, and countries.

The Seven SI Base Units

Base quantity	Common symbol	SI unit	Unit symbol	Dimension
Length	$\ell,\ x,\ r$	meter	$\text{m}$	$[\mathrm{L}]$
Mass	$m$	kilogram	$\text{kg}$	$[\mathrm{M}]$
Time	$t$	second	$\text{s}$	$[\mathrm{T}]$
Electric current	$I$	ampere	$\text{A}$	$[\mathrm{I}]$
Thermodynamic temperature	$T$	kelvin	$\text{K}$	$[\Theta]$
Amount of substance	$n$	mole	$\text{mol}$	$[\mathrm{N}]$
Luminous intensity	$I_v$	candela	$\text{cd}$	$[\mathrm{J}]$

One detail often causes confusion: the SI base unit for mass is $\text{kg}$ . Convert grams to kilograms when you need the full SI base unit form.

\begin{aligned} 1 \text{ g} &= 10^{-3} \text{ kg} \\ 250 \text{ g} &= 250 \times 10^{-3} \text{ kg} = 0.25 \text{ kg} \end{aligned}

Derived Units Come from Operations on Base Units

A derived unit appears when a derived quantity is formed from base quantities. If the formula multiplies or divides quantities, the units are multiplied or divided too.

Derived quantity	Quantity formula	SI unit	Dimension
Area	$A = l \times w$	$\text{m}^2$	$[\mathrm{L}]^2$
Volume	$V = l \times w \times h$	$\text{m}^3$	$[\mathrm{L}]^3$
Density	$\rho = \frac{m}{V}$	$\text{kg/m}^3$	$[\mathrm{M}][\mathrm{L}]^{-3}$
Speed	$v = \frac{\Delta s}{\Delta t}$	$\text{m/s}$	$[\mathrm{L}][\mathrm{T}]^{-1}$
Acceleration	$a = \frac{\Delta v}{\Delta t}$	$\text{m/s}^2$	$[\mathrm{L}][\mathrm{T}]^{-2}$
Force	$F = m \times a$	$\text{N}$	$[\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2}$
Work	$W = F \times \Delta s$	$\text{J}$	$[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2}$
Power	$P = \frac{W}{t}$	$\text{W}$	$[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}$

A derived unit can be broken down into base units. For example, force uses the unit newton, and a newton is built from kilogram, meter, and second.

\begin{aligned} F &= m \times a \\ 1 \text{ N} &= 1 \text{ kg} \cdot \text{m/s}^2 \\ &= 1 \text{ kg m s}^{-2} \end{aligned}

Once force is clear, work and power can be read step by step.

\begin{aligned} 1 \text{ J} &= 1 \text{ N} \cdot \text{m} = 1 \text{ kg m}^2\text{ s}^{-2} \\ 1 \text{ W} &= \frac{1 \text{ J}}{1 \text{ s}} = 1 \text{ kg m}^2\text{ s}^{-3} \end{aligned}

Writing Large and Small Values Compactly

When a measured value is very large or very small, SI prefixes can replace long rows of zeros with powers of $10$ .

Compact Notation with SI Prefixes

Common prefixes include:

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

The table above covers prefixes from yocto through yotta. BIPM also lists ronna and ronto for $10^{27}$ and $10^{-27}$ , plus quetta and quecto for $10^{30}$ and $10^{-30}$ . You can check the complete current list in the BIPM table of SI prefixes.

Reading Prefixes in Calculations

A prefix is attached to a base unit. For example, $\text{km}$ means one thousand meters, so it remains a unit of length based on the meter.

\begin{aligned} 7.5 \text{ km} &= 7.5 \times 10^3 \text{ m} \\ &= 7500 \text{ m} \end{aligned}

For small sizes, a prefix replaces a string of decimal zeros with a named power of $10$ . A diameter of $0.1 \ \mu\text{m}$ is equal to:

\begin{aligned} 0.1 \ \mu\text{m} &= 0.1 \times 10^{-6} \text{ m} \\ &= 1.0 \times 10^{-7} \text{ m} \end{aligned}

Scientific notation separates the electron mass into a coefficient and its scale:

9.11 \times 10^{-31} \text{ kg}

For a unit such as $\text{cm}$ , $\text{mg}$ , or $\mu\text{m}$ , first separate the prefix from the base unit. Then replace the prefix with its power of $10$ and continue the calculation.