# Size system

A size system is used to systematically classify physical sizes . From a practical point of view, it is defined by stipulating one or more basic quantities, from which further derived quantity types of the quantity system can be created according to agreed calculation rules .

In the VIM size system definition , 3rd edition of 2007, the existence of basic sizes is not required, but it is required that the sizes of the size system are related to one another with a "set of consistent equations"; In the definition of "base size" and of "derived size", the term system of sizes is used.

One example is the International Size System (ISQ) .

A size is inserted into the size system by its appropriate assignment to a size type. Since the basic sizes of a size system do not have to be independent of one another, it is possible that the same size type can be represented by more than one combination of basic sizes. There may also be sizes that are not classified in a size system, e.g. B. because there is no meaningful possibility.

Note : Contrary to this representation, basic quantities are viewed as independent of one another in the above-mentioned VIM edition. The base quantity is defined therein as "a quantity in a subset of a quantity system selected by agreement, whereby none of these quantities can be expressed by the other quantities of the subset". The subset mentioned is called the "set of base quantities".

The system of a size system is based on the types and dimensions of the sizes. Several different sizes can belong to the same size type, and several different sizes can belong to the same dimension. The number of dimensions of the system of sizes determines its degree .

Relationships between a system of sizes and a system of units must be defined by definition. A system of units does not automatically follow from a system of sizes, and vice versa. Different systems of units can be assigned to a size system.

## Examples

( Size types fat, sizes italics)

Definitions:

1. A size system will with the two basic sizes length L and mass M defined.
2. Further types of sizes of the system should be derived coherently, i. This means that all base quantities are given their own exponents and then multiplied with one another.

Properties:

• Practical point of view: The choice of the basic quantities makes physical sense and enables simple calculations.
• A size type G of the size system can be represented as G = L x  · M y  .
• Trivial: From the basic parameters, the derived length as L 1  · M 0  =  L and the mass as L 0  * M 1  =  M from.

Classification of sizes:

• The size diameter of a circle is initially not defined within the size system, i. that is, it is not assigned to any type of size; i.e. it is not part of the sizing system; i.e. it cannot be represented in this.
• Both the diameter and the circumference of a circle can be assigned to the size type L , i.e. i.e. defined as length and thus included in the size system.

Derived quantity types:

• The size type volume is initially not defined within the size system, i. i.e. it is not part of the sizing system; i.e. it cannot be represented in this.
• The volume V could be defined as a quantity type “length in three independent spatial dimensions” by V  = L 3  · M 0  =  L 3 . The density D , as “mass through volume”, could then be of the magnitude D  = L −3  · M 1  =  L −3  · M 1  .
• The speed S could be defined as a quantity type “length of the braking distance in water”, i. H. as S  = L 1  · M 0  =  L . The speed would then be a length . (“Length of the braking distance in water” means “The length of the path that a no (no longer) accelerated object travels to a standstill after immersion in water under certain conditions”, ie the object is decelerated according to a certain method so that the braking distance is a function of the speed.)
• The speed S , but possibly after the introduction of a third base variable time T can be derived. The size type of the speed would be, for example, S  = L 1  · M 0  · T  -1  =  L / T .
• After the introduction of a fourth basic variable speed S  = L 0  · M 0  · T 0  · S 1  , the speed would no longer be treated as a derived type of variable . So that no ambiguities arise, must be defined for corresponding variables, whether they L / T or S are to be displayed. The assignment can e.g. B. depend on the importance of size. From a computational point of view, it is not critical to mix the two modes of representation, since the relationship S = L / T is known.

## literature

• Peter Kiehl, Norbert Breutmann, Wolfgang Goethe: Small introduction to the DIN standards . BG Teubner Verlag, 2001, ISBN 978-3-519-26301-2 ( full text in the Google book search - page 1077ff).