^{1,a)}

### Abstract

We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit systems, there exists a relation of preorder such that one unit system is transferable to the other unit system. The transfer (or conversion) is possible only when all of the quantities distinguishable in the latter system are always distinguishable in the former system. By utilizing this structure, we can systematically compare the representations in different unit systems. Especially, the equivalence class of unit systems (EUS) plays an important role because the representations of physical quantities and equations are of the same form in unit systems belonging to an EUS. The dimension of quantities is uniquely defined in each EUS. The EUS’s form a partially ordered set. Using these mathematical structures, unit systems and EUS’s are systematically classified and organized as a hierarchical tree.

The authors would like to thank S. Tanimura, Y. Nakata, and S. Tamate for their helpful discussions. The present study was supported in part by Grants-in-Aid for Scientific Research (22109004 and 22560041) and by the Global COE program “Photonics and Electronics Science and Engineering” of Kyoto University.

I. INTRODUCTION

II. BASICS OF UNIT SYSTEMS

A. Ensembles of quantities

B. Representation of quantities with base units

III. PREORDER OF UNIT SYSTEMS

A. Unit-system dependent distinguishability of quantities

B. Transferability of unit systems

IV. CONVERSION OF UNIT SYSTEMS

A. Mapping from one unit system to another

B. Transfer matrix

C. Composition of transformations

V. QUANTITIES IN EQUIVALENT UNIT SYSTEMS

A. Equivalence class of unit systems

B. Relation among equivalence classes of unit systems — partial order

C. Dimension of e-quantities

D. Dimensional analysis and the Buckingham Pi-theorem

VI. STANDARD FORM OF TRANSFORMATION

A. Decomposition of the transfer matrix

B. Quantities transferred to unity

VII. COMPARISON OF UNIT SYSTEMS WITH NORMALIZED QUANTITIES

A. Normalized quantities

B. Comparison of incomparable unit systems

VIII. PRACTICAL UNIT SYSTEMS

A. Examples of the use of normalized quantities

IX. RELATIONS BETWEEN REAL UNIT SYSTEMS

A. Example 1

B. Example 2

C. MKSA to CGS emu

D. MKSA to CGS esu

E. MKSA to a symmetric three-base unit

F. The modified Heaviside-Lorentz system

G. Examples of dimensions

X. CONCLUSION

### Key Topics

- Electric currents
- 6.0
- Maxwell equations
- 4.0
- Calculus
- 3.0
- Charged currents
- 3.0
- Constitutive relations
- 3.0

## Figures

Ordering of unit systems U and V. A quantity Q ∈ is mapped to and for the representation in the respective unit systems. In the case of U≿V, the preimage is always included in . Only then we can naturally define the mapping .

Ordering of unit systems U and V. A quantity Q ∈ is mapped to and for the representation in the respective unit systems. In the case of U≿V, the preimage is always included in . Only then we can naturally define the mapping .

Two equivalence classes of unit systems (EUS's) satisfying
, with the mapping
. Each EUS contains equivalent unit systems as
,
. There are invertible mappings between any pair of unit systems in an EUS, for example,
and
between U and U
^{′}. There is a (one-way) mapping from any unit system in
to any unit system in
. These mappings (
and
in this example) are related as
. Note that there are also mappings between U and V′ (
) or U
^{′} and V (
), which is not shown here.

Two equivalence classes of unit systems (EUS's) satisfying
, with the mapping
. Each EUS contains equivalent unit systems as
,
. There are invertible mappings between any pair of unit systems in an EUS, for example,
and
between U and U
^{′}. There is a (one-way) mapping from any unit system in
to any unit system in
. These mappings (
and
in this example) are related as
. Note that there are also mappings between U and V′ (
) or U
^{′} and V (
), which is not shown here.

A hierarchical tree of unit systems. Here, N is the number of base units. Arrows indicate transferability “≻,” and the associated quantity is considered to be unity on the transfer. Dashed boxes represent EUS's, and the four- and two-base unit systems listed are equivalent within each group, whereas the three-base unit systems are all incomparable.

A hierarchical tree of unit systems. Here, N is the number of base units. Arrows indicate transferability “≻,” and the associated quantity is considered to be unity on the transfer. Dashed boxes represent EUS's, and the four- and two-base unit systems listed are equivalent within each group, whereas the three-base unit systems are all incomparable.

## Tables

Four possible relations between two unit systems U and V: (strictly) transferable to (≻), (strictly) transferable from (≺), equivalent (∼), and incomparable (∥). The relations between and , which are the numbers of base units, are also listed.

Four possible relations between two unit systems U and V: (strictly) transferable to (≻), (strictly) transferable from (≺), equivalent (∼), and incomparable (∥). The relations between and , which are the numbers of base units, are also listed.

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