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# Dimensionless number

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In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters one's system of units of measurement, for example from English units to metric units. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.

For example: "one out of every 10 apples I gather is rotten." The rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1, which is a dimensionless quantity.

Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Buckingham π-theorem of dimensional analysis, the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = nk independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.

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## Properties

• A dimensionless number has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured.
• A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the metric measurement system or the imperial measurement system.
• However, a number may be dimensionless in one system of units (e.g., in a nonrationalized cgs system of units with the electric constant ε0 = 1), and not dimensionless in another system of units (e.g., the rationalized SI system, with ε0 = 8.85419×10-12 F/m).

## An example

The power-consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:

• Length L [m]
• Time T [s]
• Mass M [kg]

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = nk = 5 − 3 = 2 independent dimensionless numbers which are in case of the stirrer

• Reynolds number (This is the most important dimensionless number; it describes the fluid flow regime)
• Power number (describes the stirrer and also involves the density of the fluid)

## List of dimensionless numbers

There are infinitely many dimensionless numbers. Some of those that are used most often have been given names, as in the following list of examples (in alphabetical order, indicating their field of use):

## Dimensionless physical constants

The system of natural units chooses its base units in such a way as to eliminate a few physical constants such as the speed of light by choosing units that express these physical constants as 1 in terms of the natural units. However, the dimensionless physical constants cannot be eliminated in any system of units, and are measured experimentally. These are often called fundamental physical constants.

These include: