ISO 80000-1:2009 is a standard describing scientific and mathematical quantities and their units. The standard, whose full name is Quantities and units Part 1: General was developed by the International Organization for Standardization (ISO), superseding ISO 31-0. It provides general information concerning quantities and units and their symbols, especially the International System of Quantities and the International System of Units, and defines these quantities and units. It is a part of a group of standards called ISO/IEC 80000.
The standard is divided into the following chapters:
ISO 80000-1 gives "general information and definitions concerning quantities, systems of quantities, units, quantity and unit symbols, and coherent unit systems, especially the International System of Quantities, ISQ, and the International System of Units, SI."
The standard includes the following definitions.
The special choice of base quantities and quantity equations, given in ISO 80000 and IEC 80000 defines the International System of Quantities, denoted “ISQ” in all languages. Derived quantities can be defined in terms of the base units by quantity equations. There are seven base quantities in the ISQ: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Section 7.3.2 states that the decimal mark is either a comma or a point on the line and further states that whichever decimal mark is used, that same mark should be consistently used. This is in agreement with prior ISO standards and revisions, notably resolution 10 of the 22nd CGPM, 2003.
Annex C introduces the concepts of power quantities and root-power quantities, and deprecates field quantity.
The unified atomic mass unit or dalton (symbol: u, or Da or AMU) is a standard unit of mass that quantifies mass on an atomic or molecular scale (atomic mass). One unified atomic mass unit is approximately the mass of one nucleon (either a single proton or neutron) and is numerically equivalent to 1 g/mol. It is defined as one twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest, and has a value of 1.660539040(20)×10−27 kg, or approximately 1.66 yoctograms. The CIPM has categorised it as a non-SI unit accepted for use with the SI, and whose value in SI units must be obtained experimentally.The atomic mass unit (amu) without the "unified" prefix is technically an obsolete unit based on oxygen, which was replaced in 1961. However, many sources still use the term amu but now define it in the same way as u (i.e., based on carbon-12). In this sense, most uses of the terms atomic mass units and amu, today, actually refer to unified atomic mass unit. For standardization, a specific atomic nucleus (carbon-12 vs. oxygen-16) had to be chosen because the average mass of a nucleon depends on the count of the nucleons in the atomic nucleus due to mass defect. This is also why the mass of a proton or neutron by itself is more than (and not equal to) 1 u.
The atomic mass unit is not the unit of mass in the atomic units system, which is rather the electron rest mass (me).
Until the 2019 redefinition of SI base units, the number of daltons in a gram is exactly the Avogadro number by definition, or equivalently, a dalton is exactly equivalent to 1 gram/mol. Thereafter, these relationships will no longer be exact, but they will still be extremely accurate approximations.Binary prefix
A binary prefix is a unit prefix for multiples of units in data processing, data transmission, and digital information, notably the bit and the byte, to indicate multiplication by a power of 2.
The computer industry has historically used the units kilobyte, megabyte, and gigabyte, and the corresponding symbols KB, MB, and GB, in at least two slightly different measurement systems. In citations of main memory (RAM) capacity, gigabyte customarily means 1073741824 bytes. As this is a power of 1024, and 1024 is a power of two (210), this usage is referred to as a binary measurement.
In most other contexts, the industry uses the multipliers kilo, mega, giga, etc., in a manner consistent with their meaning in the International System of Units (SI), namely as powers of 1000. For example, a 500 gigabyte hard disk holds 500000000000 bytes, and a 1 Gbit/s (gigabit per second) Ethernet connection transfers data at 1000000000 bit/s. In contrast with the binary prefix usage, this use is described as a decimal prefix, as 1000 is a power of 10 (103).
The use of the same unit prefixes with two different meanings has caused confusion. Starting around 1998, the International Electrotechnical Commission (IEC) and several other standards and trade organizations addressed the ambiguity by publishing standards and recommendations for a set of binary prefixes that refer exclusively to powers of 1024. Accordingly, the US National Institute of Standards and Technology (NIST) requires that SI prefixes only be used in the decimal sense: kilobyte and megabyte denote one thousand bytes and one million bytes respectively (consistent with SI), while new terms such as kibibyte, mebibyte and gibibyte, having the symbols KiB, MiB, and GiB, denote 1024 bytes, 1048576 bytes, and 1073741824 bytes, respectively. In 2008, the IEC prefixes were incorporated into the international standard system of units used alongside the International System of Quantities (see ISO/IEC 80000).Decibel
The decibel (symbol: dB) is a unit of measurement used to express the ratio of one value of a power or field quantity to another on a logarithmic scale, the logarithmic quantity being called the power level or field level, respectively. It can be used to express a change in value (e.g., +1 dB or −1 dB) or an absolute value. In the latter case, it expresses the ratio of a value to a fixed reference value; when used in this way, a suffix that indicates the reference value is often appended to the decibel symbol. For example, if the reference value is 1 volt, then the suffix is "V" (e.g., "20 dBV"), and if the reference value is one milliwatt, then the suffix is "m" (e.g., "20 dBm").Two different scales are used when expressing a ratio in decibels, depending on the nature of the quantities: power and field (root-power). When expressing a power ratio, the number of decibels is ten times its logarithm to base 10. That is, a change in power by a factor of 10 corresponds to a 10 dB change in level. When expressing field (root-power) quantities, a change in amplitude by a factor of 10 corresponds to a 20 dB change in level. The decibel scales differ by a factor of two so that the related power and field levels change by the same number of decibels in, for example, resistive loads.
The definition of the decibel is based on the measurement of power in telephony of the early 20th century in the Bell System in the United States. One decibel is one tenth (deci-) of one bel, named in honor of Alexander Graham Bell; however, the bel is seldom used. Today, the decibel is used for a wide variety of measurements in science and engineering, most prominently in acoustics, electronics, and control theory. In electronics, the gains of amplifiers, attenuation of signals, and signal-to-noise ratios are often expressed in decibels.
In the International System of Quantities, the decibel is defined as a unit of measurement for quantities of type level or level difference, which are defined as the logarithm of the ratio of power- or field-type quantities.Field, power, and root-power quantities
A power quantity is a power or a quantity directly proportional to power, e.g., energy density, acoustic intensity, and luminous intensity. Energy quantities may also be labelled as power quantities in this context.A root-power quantity is a quantity such as voltage, current, sound pressure, electric field strength, speed, or charge density, the square of which, in linear systems, is proportional to power. The term root-power quantity was introduced in the ISO 80000-1 § Annex C; it replaces and deprecates the term field quantity.
It is essential to know which category a measurement belongs to when using decibels (dB) for comparing the levels of such quantities. A change of one bel in the level corresponds to a 10× change in power, so when comparing power quantities x and y, the difference is defined to be 10×log10(y/x) decibel. With root-power quantities, however the difference is defined as 20×log10(y/x) dB. In linear systems, these definitions allow the distinction between root-power quantities and power quantities to be ignored when specifying changes as levels: an amplifier can be described as having "3 dB" of gain without needing to specify whether voltage or power are being compared; for a given linear load (e.g. an 8 Ω speaker), such an increase will result in a 3 dB increase in both the sound pressure level and the sound power level at a given location near the speaker. Conversely, when ratios cannot be identified as either power or root-power quantities, the units neper (Np) and decibel (dB) cannot be sensibly used.
In the analysis of signals and systems using sinusoids, field quantities and root-power quantities may be complex-valued.ISO/IEC 80000
ISO 80000 or IEC 80000 is an international standard promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC).
The standard introduces the International System of Quantities (ISQ). It is a style guide for the use of physical quantities and units of measurement, formulas involving them, and their corresponding units, in scientific and educational documents for worldwide use. In most countries, the notations used in mathematics and science textbooks at schools and universities follow closely the guidelines in this standard.The ISO/IEC 80000 family of standards was completed with the publication of Part 1 in November 2009.ISO 31-0
ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI. It is intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the ISO 31 standard. ISO 31-0 was withdrawn on 17 November 2009. It is superseded by ISO 80000-1. Other parts of ISO 31 have also been withdrawn and replaced by parts of ISO 80000.International System of Quantities
The International System of Quantities (ISQ) is a system based on seven base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Other quantities such as area, pressure, and electrical resistance are derived from these base quantities by clear, non-contradictory equations. The ISQ defines the quantities that are measured with the SI units and also includes many other quantities in modern science and technology. The ISQ is defined in the international standard ISO/IEC 80000, and was finalised in 2009 with the publication of ISO 80000-1.The 14 parts of ISO/IEC 80000 define quantities used in scientific disciplines such as mechanics (e.g., pressure), light, acoustics (e.g., sound pressure), electromagnetism, information technology (e.g., storage capacity), chemistry, mathematics (e.g., Fourier transform), and physiology.International System of Units
The International System of Units (SI, abbreviated from the French Système international (d'unités)) is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the ampere, kelvin, second, metre, kilogram, candela, mole, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.
The base units are derived from invariant constants of nature, such as the speed of light in vacuum and the triple point of water, which can be observed and measured with great accuracy, and one physical artefact. The artefact is the international prototype kilogram, certified in 1889, and consisting of a cylinder of platinum-iridium, which nominally has the same mass as one litre of water at the freezing point. Its stability has been a matter of significant concern, culminating in a revision of the definition of the base units entirely in terms of constants of nature, scheduled to be put into effect on 20 May 2019.Derived units may be defined in terms of base units or other derived units. They are adopted to facilitate measurement of diverse quantities. The SI is intended to be an evolving system; units and prefixes are created and unit definitions are modified through international agreement as the technology of measurement progresses and the precision of measurements improves. The most recent derived unit, the katal, was defined in 1999.
The reliability of the SI depends not only on the precise measurement of standards for the base units in terms of various physical constants of nature, but also on precise definition of those constants. The set of underlying constants is modified as more stable constants are found, or may be more precisely measured. For example, in 1983 the metre was redefined as the distance that light propagates in vacuum in a given fraction of a second, thus making the value of the speed of light in terms of the defined units exact.
The motivation for the development of the SI was the diversity of units that had sprung up within the centimetre–gram–second (CGS) systems (specifically the inconsistency between the systems of electrostatic units and electromagnetic units) and the lack of coordination between the various disciplines that used them. The General Conference on Weights and Measures (French: Conférence générale des poids et mesures – CGPM), which was established by the Metre Convention of 1875, brought together many international organisations to establish the definitions and standards of a new system and standardise the rules for writing and presenting measurements. The system was published in 1960 as a result of an initiative that began in 1948. It is based on the metre–kilogram–second system of units (MKS) rather than any variant of the CGS. Since then, the SI has been adopted by all countries except the United States, Liberia and Myanmar.Quantity calculus
Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities. (Here the term calculus should be understood in its broader sense of "a system of computation", rather than in the sense of differential calculus and integral calculus.) Its roots can be traced to Fourier's concept of dimensional analysis (1822). The basic axiom of quantity calculus is Maxwell's description of a physical quantity as the product of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a "unit of measurement"). De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full axiomatization has yet to be completed.Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7 m". Unlike algebra, the unit symbol represents a measurable quantity such as a meter, not an algebraic variable.
A careful distinction needs to be made between abstract quantities and measurable quantities. The multiplication and division rules of quantity calculus are applied to SI base units (which are measurable quantities) to define SI derived units, including dimensionless derived units, such as the radian (rad) and steradian (sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity. Johansson proposes that there are logical flaws in the application of quantity calculus, and that the so-called dimensionless quantities should be understood as "unitless quantities".How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook on Quantities, Units and Symbols in Physical Chemistry.Rounding
Rounding a number means replacing it with a different number that is approximately equal to the original, but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.
Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement or estimate; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is usually better stated as "about 123,500".
On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain ill-conditioned cases they may make the result meaningless.
Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma".
Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals.
A wavy equals sign (≈: approximately equal to) is sometimes used to indicate rounding of exact numbers, e.g., 9.98 ≈ 10. This sign was introduced by Alfred George Greenhill in 1892.Ideal characteristics of rounding methods include:
Rounding should be done by a function. This way, when the same input is rounded in different instances, the output is unchanged.
Calculations done with rounding should be close to those done without rounding.
As a result of (1) and (2), the output from rounding should be close to its input, often as close as possible by some metric.
To be considered rounding, the range will be a subset of the domain. A classical range is the integers, Z.
Rounding should preserve symmetries that already exist between the domain and range. With finite precision (or a discrete domain), this translates to removing bias.
A rounding method should have utility in computer science or human arithmetic where finite precision is used, and speed is a consideration.But, because it is not usually possible for a method to satisfy all ideal characteristics, many methods exist.
As a general rule, rounding is idempotent; i.e., once a number has been rounded, rounding it again will not change its value. Rounding functions are also monotonic; i.e., rounding a larger number results in the same or larger result than rounding the smaller number.
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