Mars Polar Orbiter

Representation of numerical values and SI units in character strings for information interchanges

Version Released Terms
MIXF-08 2008-02-07 RFC


This document describes a character string encoding for numerical values and units which:


According to [NASA 1999] Arthur Stephenson, chairman of the Mars Climate Orbiter Mission Failure Investigation Board:

"The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units in a segment of ground-based, navigation-related mission software, ..."

Although the [ISO 6093] standard for automated interchange of numerical data is widely used, standardized measurement units (other than for page formating) are not routinely attached to interchange data.

Relation to Previous Work

The 1986 standard Representations for U.S. Customary, SI, and Other Units to Be Used in Systems with Limited Character Sets [ANSI X3.50] states:

This standard was not designed for ... usage by humans as input to, or output from, data systems. ... They should never be printed out for publication or for other forms of public information transfer.

[ANSI X3.50] representations of units are ambiguous. "min" is both "minute" and "milliinch"; "cd" is both "candela" and "centiday".

Apart from SI units, [ANSI X3.50] supports only U.S. local units, is not complete in that support, and has no provision for extension to other locales. But non-SI unit systems are in such disarray that using them for interchange is not practical. Unit names signify different volumes in different locales; the Canadian gallon is 4.54609 liters, while the U.S. gallon is 3.785412 liters. The CRC Handbook of Chemistry and Physics [CRC] lists no less than six distinct (incompatible) systems of wire gauges.

The character set limitations targeted by [ANSI X3.50], namely single alphabetic case, are no longer common in data interchanges. But much of its double case "Form I" SI unit representations are similar to those presented here.

The audience for metric standards has changed and grown. In the preface to Guide for the Use of the International System of Units (SI) [NIST 811], B. Taylor writes:

The International System of Units, universally abbreviated SI, is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce.

[NIST 811] details a methodology for expressing measurement units in both text and symbolic form in scientific and other documents. Its unit expressions combine over 40 metric base and derived unit symbols unambiguously. Taylor's unit symbols are the basis for this metric interchange format.

Metric Interchange Format

In the expression for the value of a quantity, the unit symbol is placed after the numerical value. A dot (PERIOD, ".") is placed between the numerical value and the unit symbol.

Within a compound unit, each of the base and derived symbols can optionally have an attached SI prefix.

Unit symbols formed from other unit symbols by multiplication are indicated by means of a dot (PERIOD, ".") placed between them.

Unit symbols formed from other unit symbols by division are indicated by means of a SOLIDUS ("/") or negative exponents. The SOLIDUS must not be repeated in the same compound unit unless contained within a parenthesized subexpression.

The grouping formed by a prefix symbol attached to a unit symbol constitutes a new inseparable symbol (forming a multiple or submultiple of the unit concerned) which can be raised to a positive or negative power and which can be combined with other unit symbols to form compound unit symbols.

The grouping formed by surrounding compound unit symbols with parentheses ("(" and ")") constitutes a new inseparable symbol which can be raised to a positive or negative power and which can be combined with other unit symbols to form compound unit symbols.

Compound prefix symbols, that is, prefix symbols formed by the juxtaposition of two or more prefix symbols, are not permitted.

Prefix symbols are not used with the time-related unit symbols min (minute), h (hour), d (day). No prefix symbol may be used with dB (decibel). Only submultiple prefix symbols may be used with the unit symbols L (liter), Np (neper), o (degree), oC (degree Celsius), rad (radian), and sr (steradian). Submultiple prefix symbols may not be used with the unit symbols t (metric ton), r (revolution), or Bd (baud).

A unit exponent follows the unit, separated by a CIRCUMFLEX ("^"). Exponents may be positive or negative. Fractional exponents must be parenthesized.

Alphabetic Case

The case of letters in unit symbols must match the symbols specified here. Unit symbols are composed of lower-case letters except that:

The prefix symbols Y (yotta), Z (zetta), E (exa), P (peta), T (tera), G (giga), and M (mega) are printed in upper-case letters while all other prefix symbols are printed in lower-case letters.

SI Prefixes

1e24 yottaY
1e21 zettaZ
1e18 exa E
1e15 peta P
1e12 tera T
1e9 giga G
1e6 mega M
1e3 kilo k
1e2 hectoh
1e1 deka da
1e-1 deci d
1e-2 centic
1e-3 millim
1e-6 microu
1e-9 nano n
1e-12pico p
1e-18atto a

Binary Prefixes

These binary prefixes are valid only with the units B (byte) and bit. However, decimal prefixes can also be used with bit; and decimal multiple (not submultiple) prefixes can also be used with B (byte).

1.073741824e9  230gibiGi
1.048576e6  220mebiMi
1.024e3  210kibiKi

Unit Symbols

Type of Quantity Name Symbol Equivalent
time second s
time minute min= 60.s
time hour h= 60.min
time day d= 24.h
frequency hertz Hz  s^-1
signaling rate baud Bd  s^-1
length meter m
volume liter L  dm^3
plane angle radian rad
solid angle steradian sr  rad^2
plane angle revolution r=*6.283185307179586.rad
plane angle degree o=*2.777777777777778e-3.r
information capacity bit bit
information capacity byte, octet B= 8.bit
mass gram g
mass ton t  Mg
mass unified atomic mass unit u=
amount of substance mole mol
catalytic activity katal kat  mol/s
thermodynamic temperature kelvin K
temperature degree CelsiusoC
luminous intensity candela cd
luminous flux lumen lm
illuminance lux lx  lm/m^2
force newton N^-2
pressure, stress pascal Pa  N/m^2
energy, work, heat joule J  N.m
energy electronvolt eV= 1.602176487e-19.J
power, radiant flux watt W  J/s
logarithm of power ratio neper Np
logarithm of power ratio decibel dB=*0.1151293.Np
electric current ampere A
electric charge coulomb C  s.A
electric potential, EMF volt V  W/A
capacitance farad F  C/V
electric resistance ohm Ohm  V/A
electric conductance siemens S  A/V
magnetic flux weber Wb  V.s
magnetic flux density tesla T  Wb/m^2
inductance henry H  Wb/A
radionuclide activity becquerel Bq  s^-1
absorbed dose energy gray Gy  m^2.s^-2
dose equivalent sievert Sv  m^2.s^-2
*The exact formulas are:
r/rad = 8 * atan(1)
o/r = 1 / 360
db/Np = ln(10) / 20

Unit Examples

Type of Quantity Name Symbol
area square meter m^2
volume cubic meter m^3
speed, velocity meter per second m/s
acceleration meter per second squared m/s^2
wave number reciprocal meter m^-1
mass density (density) kilogram per cubic meter kg/m^3
specific volume cubic meter per kilogram m^3/kg
current density ampere per square meter A/m^2
magnetic field strength ampere per meter A/m
concentration mole per cubic meter mol/m^3
luminance candela per square meter cd/m^2
angular velocity radian per second rad/s
angular acceleration radian per second squared rad/s^2
dynamic viscosity pascal second Pa.s
moment of force newton meter N.m
surface tension newton per meter N/m
heat flux density watt per square meter W/m^2
radiant intensity watt per steradian W/sr
radiance watt per square meter steradian W/(m^
heat capacity, entropy joule per kelvin J/K
specific heat or entropy joule per kilogram kelvin J/(kg.K)
specific energy joule per kilogram J/kg
thermal conductivity watt per meter kelvin W/(m.K)
energy density joule per cubic meter J/m^3
electric field strength volt per meter V/m
electric charge density coulomb per cubic meter C/m^3
electric flux density coulomb per square meter C/m^2
permittivity farad per meter F/m
permeability henry per meter H/m
molar energy joule per mole J/mol
molar entropy or heat joule per mole kelvin J/(mol.K)
exposure (x and g rays) coulomb per kilogram C/kg
rotational speed revolution per minute r/min
catalytic concentration katal per cubic meter kat/m^3
data rate mebibit per second Mib/s
noise voltage density nanovolt per root hertz nV/Hz^(1/2)

Use of Metric Units by Computer Programs

Metric units attached to individual numerical values have the format described above. An unattached unit can be used to specify the units applying to a row, column, or entire table of numerical values; or for other purposes.

Programming language support for metric interchange should be provided by a function of two unit arguments returning a conversion factor. Multiplying a numerical value expressed in the second unit by the returned conversion factor yields the numerical value expressed in the first unit. This function must return a non-positive number if either of its arguments is not a syntactically valid unit; or if the conversion factor does not exist.


    UCF("km/s", "m/s" ) --> 0.001     UCF("N"   , "m/s" ) --> 0
    UCF("moC" , "oC"  ) --> 1000      UCF("mK"  , "oC"  ) --> 0
    UCF("rad" , "o"   ) --> 0.0174533 UCF("K"   , "o"   ) --> 0
    UCF("K"   , "K"   ) --> 1         UCF("oK"  , "oK"  ) --> -3
    UCF(""    , "s/s" ) --> 1         UCF("km/h", "mph" ) --> -2

Programming Language Extension

Lexical numerical constants in the programming languages C, Pascal, and Scheme could be extended to incorporate Metric Interchange Syntax compatibly with their current syntaxes; but this is not required for supporting input and output of units.


Portability of Numbers

"Representation of numerical values in character strings for information interchanges", [ISO 6093], specifies the three machine-readable presentations in widespread use (Integer, Decimal, and Exponential notations) using only the characters:

<left-parenthesis> (
<right-parenthesis> )
<comma> ,
<plus-sign> +
<hyphen-minus> -
<period> .
<E> E
<e> e
<digit> 0 - 9

In [UTF-7] the character PLUS-SIGN ("+") is not directly encoded, requiring multi-octet encoding. But every [ISO 6093] numeric value can be expressed without the use of PLUS-SIGN. So the number syntax given here does not include PLUS-SIGN.

Locale charsets all support the digits 0 to 9. There are only 3 LC_NUMERIC attributes: decimal_point, thousands_sep, and grouping. [ISO 6093] specifies use of either "." or "," for the decimal point. [ISO 6093] does not allow grouping. There is no LC_NUMERIC attribute for exponent. Thus Latin characters ("e" or "E") must be available in all languages which support [ISO 6093].

The programming languages C, Fortran, PL/I, Pascal, and Scheme accept [ISO 6093] numbers both as lexical constants and as input data.

Portable Character Set

Of the SI symbols, the "micro" prefix (GREEK-SMALL-LETTER-MU or MICRO-SIGN), "ohm" symbol (GREEK-CAPITAL-LETTER-OMEGA), and "degree" symbol (DEGREE-SIGN) are not supported by all charset encodings. By substituting "u", "Ohm", and "o" respectively, the unit symbols remain readable while preserving the system's unambiguity.

Taylor recommends using the MIDDLE-DOT character between multiplied unit symbols. To support those charset encodings lacking MIDDLE-DOT, metric interchange format instead uses PERIOD (".").

The unit superscript exponents could be formed using SUPERSCRIPT-MINUS, SUPERSCRIPT-ONE, SUPERSCRIPT-TWO, SUPERSCRIPT-THREE, etc. But these characters are not universal. So the CIRCUMFLEX ("^") is placed between a unit and its exponent, written with a portable (HYPHEN-MINUS and) digit.

The symbol for the liter, L, was adopted by the General Conference on Weights and Measures in order to avoid the risk of confusion between the letter l and the number 1 (see [NIST 811] - Units Outside the SI).

Metric Interchange Format (including numbers) uses only the characters:

<left-parenthesis> (
<right-parenthesis> )
<comma> ,
<hyphen-minus> -
<period> .
<solidus> /
<circumflex> ^
<digit> 0 - 9
<upper> A - Z
<lower> a - z

Binary Units and Prefixes

Computer professionals sometimes use the term "kilobyte" to mean 1024 bytes. However, standards for data interchange must be unambiguous in all contexts. In December 1998 the International Electrotechnical Commission (IEC) approved as an IEC International Standard [IEC 60027-2] names and symbols for prefixes for binary multiples for use in the fields of data processing and data transmission.

As of 2000, the units bit and byte have not been accepted for use with SI, but are in widespread use. The IEC symbols are "B" for byte and "bit" for bit. To avoid conflict for "B", the bel was replaced by the decibel (dB).


Because white noise power in a bandwidth is proportional to that bandwidth, electronic noise units can have fractional exponents as in nV/Hz^(1/2) (nanovolt per root hertz).

Degree Celsius (oC) is not convertible to kelvin (K) by multiplication of a constant. Thus the formula "oC = K - 273.15" does not appear in the "Unit Symbols" table; and the conversion-factor function must return a non-positive number when called to convert between oC and K.

Programming Language Syntax Extension

Because a PERIOD (".") after a numerical lexical constant is not specified in the syntax of the programming languages C, Pascal, and Scheme, the syntax of their lexical constants could be extended to incorporate SI unit symbols. The syntax of "double" in Java could similarly be extended.


Arnold G. Reinhold helped complete and clarify ideas and presentation. Jon Krom discovered disparities between the text and syntax; and suggested clarifications.


[ANSI X3.50]
ANSI, Representations for U.S. customary, SI, and other units to be used in systems with limited character sets, ANSI X3.50, 1986.
P. Mohr and B. Taylor, CODATA Recommended Values of the Fundamental Physical Constants, National Institute of standards and Technology, 2006.
Chemical Rubber Company, CRC handbook of chemistry and physics, CRC Press, 67th edition, 1986.
[IEC 60027-2]
IEC, Amendment 2 to IEC International Standard IEC 60027-2: Letter symbols to be used in electrical technology - Part 2: Telecommunications and electronics., January 1999.
[ISO 2955]
ISO, Information processing-Representation of SI and other units in systems with limited character sets, ISO 2955:1983.
[ISO 6093]
ISO, Representation of numerical values in character strings for information interchanges, ISO 6093:1985.
[NASA 1999]
NASA, Mars Climate Orbiter Failure Board Releases Report,, November 1999.
[NIST 811]
Taylor, B., Guide for the Use of the International System of Units (SI), NIST Special Publication 811, 1995 Edition.
Portable Character Set
The Open Group Base Specifications Issue 6
IEEE Std 1003.1, 2004 Edition
ISO, Universal Multiple-Octet Coded Character Set (UCS) - Part 1: Architecture and Basic Multilingual Plane (BMP), ISO/IEC 10646-1, March 2000.
The Unicode Consortium, The Unicode Standard, Version 3.0 Addison-Wesley Pub Co, February, 2000.
D. Goldsmith, UTF-7, A Mail-Safe Transformation Format of Unicode, RFC 2152, May 1997.
F. Yergeau, UTF-8, a transformation format of ISO 10646, RFC 2279, January 1998.

Metric Interchange Syntax

Here is a YACC-like syntax for metric quantities (with [ISO 6093] numbers).
        : numerical_value
        | numerical_value '.' unit

        : unit_product
        | unit_product '/' single_unit

        : single_unit
        | unit_product '.' single_unit

        : punit
        | punit '^' uxponent
        | '(' unit ')'
        | '(' unit ')^' uxponent

        : uinteger
        | '-' uinteger
        | '(' uinteger '/' uinteger ')'
        | '(-' uinteger '/' uinteger ')'

        : decimal_multiple_prefix unit_p_symbol
        | decimal_submultiple_prefix unit_n_symbol
        | decimal_multiple_prefix unit_b_symbol
        | decimal_submultiple_prefix unit_b_symbol
        | binary_prefix 'B'
        | binary_prefix 'bit'
        | unit_p_symbol
        | unit_n_symbol
        | unit_b_symbol
        | unit___symbol

        : 'E' | 'G' | 'M' | 'P' | 'T' | 'Y' | 'Z' | 'da' | 'h' | 'k'

        : 'a' | 'c' | 'd' | 'f' | 'm' | 'n' | 'p' | 'u' | 'y' | 'z'

        : 'Ei' | 'Gi' | 'Ki' | 'Mi' | 'Pi' | 'Ti'

        : 'B' | 'Bd' | 'r' | 't'

        : 'L' | 'Np' | 'o' | 'oC' | 'rad' | 'sr'

        : 'A' | 'Bq' | 'C' | 'F' | 'Gy' | 'H' | 'Hz' | 'J' | 'K' | 'N'
        | 'Ohm' | 'Pa' | 'S' | 'Sv' | 'T' | 'V' | 'W' | 'Wb' | 'bit'
        | 'cd' | 'eV' | 'g' | 'kat' | 'lm' | 'lx' | 'm' | 'mol' | 's'

        : 'd' | 'dB' | 'h' | 'min' | 'u'

        : ureal
        | '-' ureal

        : numerical_value
        | numerical_value suffix

        : uinteger
        | dot uinteger
        | uinteger dot uinteger
        | uinteger dot

        : '.' | ','

        : digit uinteger
        | uinteger

        : exponent_marker uinteger
        | exponent_marker '-' uinteger

        : 'e' | 'E'

        : '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'

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