# Dimension (size system)

In a size system , the dimension of a physical size expresses its qualitative properties. In the associated system of units , each dimension corresponds to a coherent unit . This serves to express the quantitative properties of all sizes of the associated dimension. The base units therefore correspond to the dimensions of base quantities . Since there is an associated coherent unit for each dimension, one could consider a dimension as a unit type or class.

## Dimension of a base quantity

Physical
size
dimension Coherent
unity
Length l , path s Length L Meter (m)

Each base quantity is assigned a dimension with the same name. For example, in the international size system (ISQ), the dimension of the base size length is also called length . A size is symbolized with a letter written in italics - in the case of length with " l ". The symbol of a dimension is an upright, sans serif capital letter - in the case of the length " L ". The corresponding coherent unit of the dimension length is the meter .

The following table shows the dimensions of the seven basic sizes of the international size system and the corresponding basic units of the associated international system of units (SI) according to the 9th edition of the so-called SI brochure

Base size and
dimension name
Size
symbol
Dimension
symbol
Base unit unit-
sign
length l L. meter m
Dimensions m M. kilogram kg
time t T second s
electr. Amperage I. I. amp A.
Thermodynamic
temperature
T Θ Kelvin K
Amount of substance
( amount of substance)
n N Mole mol
Light intensity I V J Candela CD

The selection of the basic sizes is a matter of convention. So was z. B. in the technical system of measurement (no longer permitted in Germany since 1978) instead of mass the force is used as a dimension.

The number of basic sizes determines the degree of the size system and the dimensionality of the unit system. The ISQ is therefore a size system of the seventh degree and the associated SI is a seven-dimensional system of units .

## Dimension of a derived quantity

 dim Q = X α * Y β * Z γ Specification of the dimension of any size Q in a size system of the third degree (with three basic sizes of the dimensions X, Y and Z).

The dimension of a derived quantity expresses the relation of its coherent unit to the base units as a product of powers (power product). Each power consists of a base and an exponent. The base is the dimension of a base quantity. The exponent is called the dimension exponent of this base quantity. For example, the dimension of a speed (distance per time interval) is put together as L 1 · T −1 from those of the basic quantities length and time. The dimension exponents designated as α , β , γ etc. can each assume zero, as well as a positive or negative number of a small amount (generally ≤ 4). In addition to integer exponents, non-integer fractions - often in steps of 12  - are also common in some size systems.

In the international system of sizes, the dimension of any size Q is given by the following dimensional equation:

dim QL α · M β · T γ · I δ · .theta .epsilon. · N ζ · J η

Correspondingly, the coherent unit of the same quantity Q in the international system of units can be given by the following unit equation:

[ Q ] = m α · kg β · s γ · A δ · K ε · mol ζ · cd η

Different sizes of the same coherent unit also have the same dimension. Sometimes different types of sizes can be distinguished among these sizes . For example, the variables diameter , wavelength and amount of precipitation all have the same coherent SI unit - namely the meter - the base unit of length. Therefore they also have the same dimension, namely the length, with the symbol " L ". In general, diameter and wavelength are considered to be of the same magnitude, but not the amount of precipitation. However, there are no clear definitions to delimit different types of sizes. From this point of view, it follows that sizes of the same dimension do not necessarily have to belong to the same type of size. Conversely, sizes of the same size type always have the same dimension. Sizes of different dimensions can therefore never be counted as belonging to the same type of size.

Derived quantities can also have the dimension of a base quantity.

Other quantities whose dimension exponents are all zero are called quantities of the dimension number . Such quantities can be specified as pure numbers without a unit, but so-called auxiliary units are often used here for the sake of clarity . In the interests of clarity, it is sometimes advisable to carry special units with you instead of unit 1, such as rad / s (radians per second) instead of s −1 for an angular velocity .