# Per mille

Auxiliary unit of measure
Unit name Per mille, thousandths
Unit symbol ${\ displaystyle {} ^ {0 \!} \! / \! _ {00}}$
Formula symbol ${\ displaystyle {\ mathit {p}}}$
Type quotient
definition ${\ displaystyle 1 \, {} ^ {0 \!} \! / \! _ {00} = 1 \ cdot 10 ^ {- 3} = 0 {,} 001}$
Named after Italian per mille , "from the thousand"

A per mille (composed of the Latin pro 'in proportion to' and the Latin mille 'thousand' ) or thousandth stands for the fractional number 0.001. Per mille information is usually indicated by the alcohol symbol ‰ .

## definition

The per mille or thousandth is an auxiliary unit of measurement for proportions with the meaning of 1/1000. By trailing the degree sign preceding it is permillage divided so by a thousand: ${\ displaystyle p}$

${\ displaystyle p \, {} ^ {0 \!} \! / \! _ {00} = {\ frac {p} {1 {\,} 000}}}$

## Examples

• ${\ displaystyle 1 \, {} ^ {0 \!} \! / \! _ {00} = {\ frac {1} {1 {\,} 000}} = 0 {,} 001 = \ \ \ \ 10 ^ {- 3} = 0 {,} 1 \, \%}$
• ${\ displaystyle 2 \, {} ^ {0 \!} \! / \! _ {00} = {\ frac {2} {1 {\,} 000}} = 0 {,} 002 = 2 \ cdot 10 ^ {- 3} = 0 {,} 2 \, \%}$

## calculation

The basic equation applies:

${\ displaystyle {\ frac {\ text {per mille value}} {\ text {basic value}}} = {\ frac {\ text {per mille rate}} {\ text {1000}}}}$
or.
${\ displaystyle {\ text {Basic value}} \ cdot {\ text {Per mil rate}} = {\ text {Per mil value}} \ cdot {\ text {1000}}}$

This results in:

${\ displaystyle {\ text {per mille value}} = {\ frac {{\ text {basic value}} \ cdot {\ text {per mille rate}}} {\ text {1000}}}}$
${\ displaystyle {\ text {per mille rate}} = {\ frac {{\ text {per mille value}} \ cdot {\ text {1000}}} {\ text {basic value}}}}$
${\ displaystyle {\ text {Basic value}} = {\ frac {{\ text {Per mille value}} \ cdot {\ text {1000}}} {\ text {Per mille rate}}}}$

as well as the formula that can be used as a calculation control:

${\ displaystyle {\ frac {{\ text {Basic value}} \ cdot {\ text {Per mill rate}}} {\ text {Per mill value}}} = {\ text {1000}}}$

## Applications

### Banking and insurance

Parameters given in per mille are often found in insurance and banking . An example from the insurance industry is the specification of a loss frequency per 1000 risks.

### Alcohol in traffic

The term per mil is colloquially related to the alcohol content in the blood of the respective person. The blood alcohol content is usually given in per mille and can be calculated using the Widmark formula and measured in the breath or in the blood. The unit of measurement is the mass ratio of milligrams of alcohol per gram of blood (mg / g) - see article Blood alcohol concentration . With a handy Alcotest device, which is usually used on the street during police checks or during accident recordings, and into which one blows, it is possible to determine the alcohol concentration in the breath of the person concerned. The unit of measurement is the amount of alcohol in milligrams per liter of breath (mg / l). For example, the value 1.0 mg / l breath alcohol concentration corresponds approximately to a blood alcohol concentration of 2.1 mg / g, i.e. 2.1 per mille. The exact alcohol content - should it be suspected to be too high - is then determined at the police station either with a much more complex breath alcohol measuring device or by a doctor with a blood alcohol measurement (with a small amount of blood drawn from the person concerned).

### Inclines / slopes of railway lines

20 ‰ gradient for the next 1371 m (board at ČD )

While the inclination is usually given in percent for roads , the horizontal inclination of stretches for adhesive webs ( gradients ) is given in per mille. The value corresponds to the difference in height in millimeters per meter of horizontal distance (or the difference in height in meters per kilometer of distance), i.e. the tangent of the angle of incline. The value is required to calculate the maximum possible towing weight or the trailer load on inclines, and also to determine the maximum permissible speed on slopes (depending on the braking force and towing weight).

In Switzerland, routes with inclines of up to 10 ‰ are considered flat runs . In Germany, according to the Railway Building and Operating Regulations, 12.5 ‰ is the maximum incline for main lines .

Routes with a gradient of more than 50 ‰ can usually only be traveled with the help of a gear drive. Exceptions are e.g. B. Lisbon's tram , the maximum gradient of which is 13.5%, i.e. 135 ‰, the Pöstlingbergbahn in Linz with 105 ‰, the Uetlibergbahn with 79 ‰, the Bernina train with 72 ‰ and the Erzbergbahn with 71 ‰.

## Alcohol symbol

The alcohol symbol ‰ ( Unicode : U + 2030, HTML / XML:) was created &permil;from the percent symbol  %, which in turn was shortened from the Italian per cento .

Between the numerical value and the alcohol symbol there is a space (possibly protected against breakage ) ; Example: “19 ‰”. Sometimes a small space is used instead.

In the German standard DIN 5477 of February 1983, details on the correct use of the per mille are specified. Accordingly, its use is restricted to the use of "the specification of quotients of numbers or sizes of the same dimension including money". In the international standard ISO 80000-1 ( sizes and units , Chapter 6.5.5) from August 2013, per mil is used "as part of the coherent unit one "; the symbol ‰ is used like a unit symbol. According to DIN 5008 , the substitute representation "o / oo" (with lower case letter o ) is possible in text works .

### Enter / key combinations

German standard keyboard layout E1
- 5
Windows
Alt+0137
macOS (Germany keyboard)
⌥ Alt+ ⇧ Shift+E
macOS (Swiss keyboard)
⌥ Alt+ ⇧ Shift+§
Linux, Unix
Strg+ ⇧ Shift+ U+2030