it is thus not surprising that the first detected human measurements were done at least 30,000 years ago~\cite{haustein2001weltchronik}\ednote{Give a concrete example of such a measurement}.\ednote{describe the meaning of
measurement in this case}.
In former times, length, for instance, was measured in comparison to the human body, like in the case
In former times, length, for instance, was measured in comparison to parts of the human body, like in the case
of the units feet and cubit. Without a fixed value for these terms, we can regard the information that, for
instance, a house
has a height of 10 feet only as a loose estimate. The exact value depends on the length of the compared foot, which
is obviously impractical as a common system.
This problem can be solved by defining exact values for the terms.
Like this, we get fixed \textit{units} -- like the modern unit feet as a measure for length.
Now we can use \textit{quantity expressions}, say again 10 feet, to describe exact distances.
However different units with different values arose in different societies.
Today there are two predominant systems of measurement, the imperial system and the metric system.
Nowadays the latter is officially accepted in nearly all countries, but the former is still in use in many
countries of the former British Empire.
The latter is nowadays officially accepted in nearly all countries, but the former is still in use in many
countries that were part of the old British Empire.
The mixture of both systems already caused expensive errors like the crash of the
Mars Climate Orbiter~\cite{marsorbiter} in 1999. It happened because a part of the program calculated the impulse
Mars Climate Orbiter in 1999~\cite{marsorbiter}. It happened because a part of the program calculated the impulse
of the spacecraft in imperial units, namely in pound-seconds, while a different module expected the impulse
in Newton-seconds -- a metric unit -- as an input. The miscalculation lead to a wrong
position of the spacecraft that finally caused the crash.
in newton-seconds -- a metric unit -- as an input. The resulting miscalculation lead to a wrong
position of the spacecraft which finally caused the crash.
Efforts for a uniform system of measurement resulted in the International System of Units (abbreviated as SI) which
was resolved in 1960~\cite{ptbsi}. It is based on the seven basic units meter, kilogram, second, ampere,
kelvin, mol and candela. For convenience, the system also contains many units that can be derived from the
basic units. For instance Watt, a unit for power, is defined in terms of basic units as
Efforts for a uniform and modern system of measurement resulted in the International System of Units
(abbreviated as SI) which
was resolved in 1960~\cite{si2006}. It is based on the seven basic units meter, kilogram, second, ampere,
kelvin, mol and candela. Their value is derived mostly from natural constants. For instance, the second is defined as
``the duration of 9 192 631 770 periods of the radiation
corresponding to the transition between the two hyperfine levels of the
ground state of the caesium 133 atom.''~\cite[Page 113]{si2006} and the meter as
``the length of the path travelled by light in vacuum during a
time interval of 1/299 792 458 of a second.''~\cite[Page 112]{si2006}.
Table~\ref{tab:sibaseunits} shows the base units and their symbols together with the property they measure -- the physical dimension.
\begin{table}
\center
\begin{tabular}{|l|l|}
\hline
SI base unit & Physical Dimension \\
\hline
meter (m) & length \\
kilogram (kg) & mass \\
second (s) & time, duration \\
ampere (A) & electric current \\
kelvin (K) & temperature \\
mole (mol) & amount of substance \\
candela (cd) & luminous intensity \\
\hline
\end{tabular}
\caption{The SI base units, their symbols and their corresponding physical dimensions.
This table corresponds to Table 1 on Page 116 in~\cite{si2006}.}
\label{tab:sibaseunits}
\end{table}
For convenience, the system also contains many units that can be derived from the
basic units without any additional numerical factor.
For instance Watt, a unit for power, is defined in terms of basic units as
$1\;\rm W =1\;\rm kg \cdot\rm m^2\cdot\rm s^{-3}$ and Ohm, a unit for electrical resistance, is defined as
$1\;\rm\Omega=1\;\rm W \cdot\rm A^{-2}=1\;\rm kg \cdot\rm m^2\cdot\rm s^{-3}\cdot\rm A^{-2}$.
Table 2 to 4 in~\cite[Page 117ff]{si2006} list derived units and their definition in SI base units.
We omit a separate table here.
Decimal fractions and multiples of SI units can be formed with a fixed set of SI prefixes, which are shown in
Table~\ref{tab:siprefixes}. They range from $10^{-24}$ to $10^{24}$.
The kilogram already contains a prefix in its normal form, because its value is defined by the mass of the
international kilogram prototype.
\begin{table}
\center
\begin{tabular}{|c|c||c|c|}
\hline
Factor & Name (Symbol) & Factor & Name (Symbol) \\
