Dimensions of Physical Quantities

Derived physical quantities can be both dimensional and dimensionless. A dimension of a physical quantity is an expression in the form of a power monomial that is composed of the products of symbols for base physical quantities raised to different powers. The expression reflects the relationship between a given physical quantity and physical quantities taken for the base ones in the given system of quantities, which has a proportionality coefficient equal to 1.

The concept “dimension” also spreads to the base quantities. The dimension of a base quantity in respect to itself is equal to 1 and does not depend on other quantities; that is, the formula for the dimension of the base quantity coincides with its symbol. For example, a dimension of length is L, that of mass is M, and that of time is T, and so on.

In accordance with International Standard ISO 31/0, the dimension of quantities is

designated as “dim” (abbreviation of “dimension”).

To find a dimension of a derived physical quantity in a certain quantity system, it is

necessary that the dimensions of quantities should be substituted for their symbols to the

right side of the defining equation of this quantity. For example, in the LMT system for Eq. (2) we obtain the following:

dim v = LM0

T-1 = LT-1 (4)

The exponents to which the dimensions of the base physical quantities were raised in Eq. (4)—(+1) for L, 0 for M, (-1) for T—are called the exponents of the dimension of the physical quantity. They may take different values, that is, they may be integer or fractional, positive or negative, and may be equal to 0.

A dimension of any derived physical quantity in the LMT system of quantities may be expressed by an exponential monomial: dim L M T X α β γ = (5).

A dimension of a derived physical quantity in a system of quantities corresponding to the Système International d’Unités (SI) that is based on quantities such as length, mass, time, electric current, thermodynamic temperature, luminous intensity, and amount of substance can be expressed, in a general form, as:

dim L M T I Q J Nq X α β γδ ε ρ = (6)

In Eqs. (5) and (6), α,,,..., βγ ρ q are the exponents of a dimension of a derived physical quantity in the LMTIQJN system of quantities.

Thus, physical quantities in the dimensions of which any one of the base physical quantities is raised to a power not equal to zero refer to derived physical quantities.

Dimensionless physical quantities are those in the dimensions of which base physical quantities enter with exponents equal to zero.

Dimensionless quantities refer to ratios of two quantities of the same kind, that is, relative quantities or their functions (exponents, products of exponents, logarithms, trigonometric functions, and so on).

The dimension of a derived physical quantity is also the dimension of its unit. Mention may be also made of a number of practical applications of the concept “dimension of physical quantity.”

Firstly, it is possible, by reference to the dimension of a quantity, to determine by how much the dimension of a unit of a given derived physical quantity would vary under a change of the dimensions of the units of the quantities taken for the base ones.

Secondly, by means of the dimensions of physical quantities, one can check the correctness of equations resulting from theoretical conclusions, by using the principle of dimensional similarity of the terms of physical equations. Thus, the dimensions on the right and left hand sides of the equation linking different physical quantities should be identical.

Analysis of dimensions and the theory of physical similarity, which allow us to study complicated physical phenomena using reduced models, are also based on the principle of dimensional similarity of the terms of physical equations.

By the turn of the eighteenth century, the growth in the world’s industry and commerce generated the need to establish a unified system of measures that could serve “à tous les temps à tous les peuples.”

French scientists created the decimal metric system, based on a single unit, a unit of length, the meter, which was originally intended to be one forty millionth part of Earth’s meridian. The units of area and volume were defined as the square meter and the cubic meter, respectively, and the unit of mass, the kilogram, as the mass equal to the mass of a cubic decimeter of pure water at 4 °C.

The decimal metric system was used to create multiples and submultiples of all derived units and of the base unit, the meter, as well; factors equal to 10n were used, where n is a positive or negative integer number.

A series of the following first six prefixes were introduced, namely, deca (10¹), hecto (10²), kilo (10³), deci (10^-1), centi (10^-2) and milli (10^-3).

The decimal metric system adopted in France at the end of the eighteenth century gained recognition in the second part of the nineteenth century. In this connection, a demand arose for a standing international organization, whose principal task was to use the metric system and to contribute to its further improvement and propagation. In Paris; at the international diplomatic conference held in May 20, 1875, representatives of seventeen nations signed the Convention du Mètre, in accordance with which the Bureau International des Poids et Mésures (BIPM) was established for the following purposes:

• To maintain the international prototypes of the meter and the kilogram.

• To carry out (at regular intervals) comparisons of national prototypes of the meter

and kilogram against their international prototypes.

• To modify the metric system.

The International System of units meets the following requirements, which have been used as the basis for its establishment:

• Possibility of assuring the unification of units of physical quantities in all countries throughout world.

• Versatility, which means that the system is useful for all fields of human activity.

• Uniqueness of a unit for each physical quantity.

• Consistency (coherence), which is achieved by a choice of a small number of base units while derived units are formed with the use of defining equations with coefficients equal to 1.

• Possibility of assuring high accuracy in realization of units experimentally (base units first). Units of the Système International d’Unités (SI) can be divided into two classes: SI base units and SI derived units.

In any measurement system, information about the object to be measured is ‘transmitted’ from the object, via an instrument, to the operator.

As in any transmission system, information can be altered (measurement error) or lost (measurement uncertainty) at various stages in the passage of information through the measurement system.

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