# Size of dimension number

A quantity in the dimension number is a physical quantity that can be specified by a pure number. According to EN ISO 80000 , the coherent derived unit for these quantities is the number one. The unit symbol is 1, but it is almost always left out. For the sake of clarity, auxiliary units of measure can be used for many of these quantities .

The term dimensionless quantity is referred to as "out of date" in EN ISO 80000, it should no longer be used.

The term meant here by dimension is to be understood in the sense of dimension (size system) such as "length", not in the sense of dimension (mathematics) such as in "three-dimensional space".

## Examples

Examples of sizes of the dimension number are:

A natural constant with the dimension number is the Sommerfeld fine structure constant , which is composed of the electrical elementary charge, Planck's quantum of action and the speed of light. Their value is about 1/137.

Even when natural units are used in some areas of theoretical physics, it is customary to treat the relevant quantities formally as quantities of the dimension number.

## designation

According to DIN 5485 naming principles for physical quantities; Word compositions with adjectives and basic words that contain rules for renaming physical quantities for which a name is not yet available are provided for quantities of the dimension number:

• -proportion of
• coefficient
• -factor
• -Degree
• quota
• -relationship
• -number

Historical names of such quantities also contain the endings -module or -index .

The dimension cannot be recognized by the ending - coefficient . The coefficient of friction has the dimension number, but the coefficient of thermal expansion has the dimension “per temperature”.

## Theoretical background

In the international size system ISQ with its seven basic sizes and seven dimensions with the dimension symbols , every size has the dimension ${\ displaystyle {\ mathsf {T, L, M, I, \ Theta, N, J}}}$${\ displaystyle Q}$

${\ displaystyle \ dim Q = {\ mathsf {T}} ^ {\ alpha} \ {\ mathsf {L}} ^ {\ beta} \ {\ mathsf {M}} ^ {\ gamma} \ {\ mathsf { I}} ^ {\ delta} \ {\ mathsf {\ Theta}} ^ {\ varepsilon} \ {\ mathsf {N}} ^ {\ zeta} \ {\ mathsf {J}} ^ {\ eta} \. }$

A quantity where each dimension exponent is zero, that is, with , is called the dimension of the dimension number, and its value is indicated by a number. These quantities include those that are defined as the quotient of two quantities of the same dimension, and those as a number. ${\ displaystyle \ dim Q = {\ mathsf {1}}}$

In principle, it depends on the basis chosen for a size system which derived sizes have which dimensions, and thus also which sizes (apart from the quotients of identical sizes) have the dimension number. In electrostatic cgs systems, electrical capacitance and length are of the same dimension. Every quotient of these quantities therefore has the dimension number.

## Individual evidence

1. DIN EN ISO 80000-1: 2013, sizes and units - general , chap. 3.8 and 5.
2. DIN EN ISO 80000-11: 2013, sizes and units - parameters of the dimension number , chap. 0.3.
3. ^ Bureau International des Poids et Mesures : The International System of Units. SI brochure (9th ed.) , 2019, chapter 2.3.3. ( weblink PDF / online version)