The unit symbol dictionary (see Part 5) contains symbols for a large number of commonly-used units. These are divided into two groups: base unit symbols and derived unit symbols. A base unit is not defined in terms of any other unit symbols. It represents the standard reference unit of a basic physical or dimensionless quantity. A derived unit symbol, on the other hand, is defined as an expression of other unit symbols or as a dimensionless scaling.
A unit symbol may be used in an API-SI unit expression only if it appears in the API-SI unit dictionary listed in Part 5 of this standard. API-SI unit symbols match corresponding SI unit symbols when such units are defined under SI. Additional unit symbols (including base units) have been added to the API-SI unit dictionary for commonly-used oilfield and (dimensionless) monetary quantities.
This model is based on the SI notion that from a small handful of basic quantities all other quantities can be derived by algebraic operations. The SI basic quantities are length, mass, time, electric current, temperature, luminous intensity, and amount of substance. Two dimensionless supplemental quantities - plane angle, and solid angle - are typically added. These have corresponding reference units m (meter), kg (kilogram), s (second), A (ampere), K (kelvin), cd (candela), mol (mole), rad (radian), and sr (steradian).
Given length and time, for example, velocity may be derived as length/time, and a corresponding unit for velocity may be expressed as m/s.
In addition to derived units that represent new quantities, there are derived units that represent scalings of other quantities. For example, the expression 0.3048 m is also a length. The expression 0.3048 m can be assigned to a customary unit symbol, say ft. The symbol ft is a derived unit symbol having definition ft = 0.3048 m. Both ft and 0.3048 m are derived units and may be used interchangeably. It happens that ft is also a unit symbol, which can be found in the unit dictionary and which may be used in the definition of other derived units.
SIUnit ::= ' ' | Multiplier | StandardForm . StandardForm ::= [Multiplier ' '] FactorExpression [',' ' ' Offset] . FactorExpression ::= UnitTerm ['/' UnitTerm] | 1 '/' UnitTerm . UnitTerm ::= '(' FactorExpression ')' | UnitFactors . UnitFactors ::= UnitFactor '.' UnitFactors | UnitFactor . UnitFactor ::= SIUnitSymbol [UnitExponent] . SIUnitSymbol ::= {UnitCharacter} . UnitCharacter ::= 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | `R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' | 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | '%' . UnitExponent ::= ['-'] SINum | '(' ['-'] (SINumber | SINum ['/' SINum]) ')' . Multiplier ::= ComplexNum Offset ::= ['-'] ComplexNum ComplexNum ::= X ['E' N]['/' Y] X ::= SINumber N ::= ['-'] SINum Y ::= SINumber SINumber ::= SINum | SINum '.' SINum | '0' '.' SINum SINum ::= '0' SINum | SINum '0' | NonZeroDigits NonZeroDigits ::= NonZeroDigit | NonZeroDigit {NonZeroDigit} NonZeroDigit ::= '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
Here are a few examples that illustrate the grammar:Note the power of the grammar to construct an arbitrary number of unit expressions from a relatively small number of predefined unit symbols. The successive construction method illustrated here also makes the task of defining new unit symbols very simple, especially with respect to computing conversion factors (see 6.1.2).
m/s meter per second s(0.5) root-second 1E-28 m2 barn (b) 0.0174532925198 rad degree (dega) 360 dega cycle (c) (c/s)/(m/s) (cycle per second) per (meter per second) kg.m/s2 newton (N) N.m joule (J) J/s watt (W) W/A volt (V) A/V siemens (S)
Letting M = Multiplier, E = FactorExpression, and O = Offset, the standard form expression `M E, O' is a new derived unit. Let U represent this derived unit, i.e., U = M E, O. Note that E is also a unit. Let X represent an amount expressed in unit U and Y represent the same amount expressed in unit E. Then the following unit conversion equations hold:
Applications typically need to convert between units when one is not directly expressed in terms of the other. The procedure for doing this is described in 6.3.1.
Dimensionless units are already in reduced form.
When there are no offsets,
reduction is accomplished by substituting
each unit symbol in E by its dimensionless or
standard form, applying appropriate
exponentiation and algebraic simplification, then
factoring all multipliers out to create a new multiplier.
If the new E reduces to 1, it is
dropped out to produce a dimensionless unit.
This is iterated until E has only base unit
symbols or until a dimensionless unit results.
Example -
mi/hr2 = (5280 ft) / (60 min) 2 = 5280/3600 ft/min2 = 4.4/3 ft/min2
= 4.4/3 (0.3048 m) / (60 s) 2= (4.4)(0.3048)/(3)(3600) m/s2
= 0.02794/225 m/s2
For the simple case in which a factor expression has no denominator,
and a single unit factor with exponent 1,
then reduction with offset is governed by (EQ 1).
Example -
From the unit symbol dictionary,
degF = 5/9 degC, 32 and
degC = K, -273.15.
If X degF, Y degC, and Z K all represent the same temperature, then from (EQ
1)
Z = Y + 273.15
= (5/9)(X - 32) + 273.15
= (5/9)(X + 459.67)
which by reverse application of (EQ 1) implies
degF = 5/9 K, -459.67.
When a unit factor is in a denominator
or has an exponent other than 1 or is not the
only unit factor in the factor expression,
then any offset in the definition of the unit factor
is dropped. That is, such a unit factor is considered to represent
a differential quantity.
Example -
The unit degF/ft represents a temperature difference per length. It may be
reduced by ignoring the offsets in the definition of degF and degC:
degF/ft = 5/9 degC/(.3048 m)
= 0.508/3 degC/m
= 0.508/3 K/m
Assume units U and V have been reduced to standard form expressions `MU EU, OU' and `MV EV, OV' respectively, where EU and EV have only base unit symbols. Then U and V belong to the same unit dimension if and only if EU/EV algebraically reduces to 1. Furthermore, if X represents an amount in unit U and Y represents the same amount in unit V, then from (EQ 1),(EQ 3) MU(X - OU) = MV(Y - OV) (EQ 4) Y = (MU/MV)(X - OU) + OVFor example, to convert X mi/hr to Y km/s, first reduce both expressions:mi/hr = 5280 ft / 3600 s = 4.4/3 (0.3048 m)/s = 0.44704 m/s [MU = 0.44704, OU = 0] km/s = 1000 m/s [MV = 1000, OV = 0]Note that m/s is the factor expression for both units, so U and V are convertible. Using (EQ 4), we getY = 0.44704/1000 X = 0.44704E-3 XThe same conversions shown in (EQ 3) and (EQ 4) also hold when two units reduce to dimensionless units, since only multipliers and offset are involved.
One could incorporate the dimensionless case into the standard form case by using a pseudo base unit symbol "nodim".