Rules of using SI

From the previous posts we learned about the history of SI and SI units and prefixes. But how can we use SI correctly? The answers are in this post.

As we all know, SI is used all over the world. So it is significant for people to use the same rules of using SI in order that it is convenient for international communication. However, Before 1948, the writing of metric quantities was haphazard. In 1879, the CIPM published recommendations for writing the symbols for length, area, volume and mass, but it was outside its domain to publish recommendations for other quantities. Beginning in about 1900, physicists who had been using the symbol "μ" for "micrometre" (or "micron"), "λ" for "microlitre", and "γ" for "microgram" started to use the symbols "μm", "μL" and "μg", but it was only in 1935, a decade after the revision of the Metre Convention that the CIPM formally adopted this proposal and recommended that the symbol "μ" be used universally as a prefix for 1E-6.

In 1948, the ninth CGPM approved the first formal recommendation for the writing of symbols in the metric system when the basis of the rules as they are now known was laid down.  These rules were subsequently extended by International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) and now cover unit symbols and names, prefix symbols and names, how quantity symbols should be written and used and how the values of quantities should be expressed. Both ISO and the IEC have published rules for the presentation of SI units that are generally compatible with those published in the SI Brochure. As of August 2013 ISO and IEC were in the process of merging their standards for quantities and units into a single set of compatible documents identified as the ISO/IEC 80000 Standard. The rules covering printing of quantities and units are part of ISO 80000-1:2009. 

Here are the rules of using SI now.

1. Unit Names

Names of units follow the grammatical rules associated with common nouns: in English and in French they start with a lowercase letter (e.g., newton, hertz, pascal), even when the symbol for the unit begins with a capital letter. This also applies to "degrees Celsius", since "degree" is the unit. In German, however, the names of units, as with all German nouns, start with capital letters. In addition, In English the name of the units "m, l" are written as "metre, litre", and the prefix "d" is written as "deca- (not deka-)".

Likewise, the plural forms of units follow the grammar of the language concerned: in English, the normal rules of English grammar are used, e.g. "henries" is the plural of "henry". However, the units "lux, hertz" and "siemens" have irregular plurals in that they remain the same in both their singular and plural form.

In English, when unit names are combined to denote multiplication of the units concerned, they are separated with a hyphen or a space (e.g. newton-metre or newton metre). The plural is formed by converting the last unit name to the plural form (e.g. ten newton-metres).

When a unit is used as an adjective in English, a space is recommended between the number and the unit symbol, e.g. "a 25 kg sphere". The normal rules of English apply to unit names, where a hyphen is incorporated into the adjectival sense, e.g. "a 25-kilogram sphere".

2. Unit symbols and the values of quantities

Although the writing of unit names is language-specific, the writing of unit symbols and the values of quantities is consistent across all languages and therefore the SI Brochure has specific rules in respect of writing them.

General rules:

• The value of a quantity is written as a number followed by a space (representing a multiplication sign) and a unit symbol; e.g. 2.21 kg, 7.3×10² m², 22 K. This rule explicitly includes the percent sign (%) and the symbol for degrees of temperature (°C). Exceptions are the symbols for plane angular degrees, minutes, and seconds (°, ', and ''), which are placed immediately after the number with no intervening space.
• Symbols are mathematical entities, not abbreviations, and as such do not have an appended period/full stop (.), unless the rules of grammar demand one for another reason, such as denoting the end of a sentence.
• A prefix is part of the unit, and its symbol is prepended to the unit symbol without a separator (e.g. k in km, M in MPa, G in GHz). Compound prefixes are not allowed.
• Symbols for derived units formed by multiplication are joined with a centre dot (·) or a non-breaking space; e.g. N·m or N m.
• Symbols for derived units formed by division are joined with a solidus (/), or given as a negative exponent. E.g., the "metre per second" can be written m/s, m s^−1, m·s^-1.
• Only one solidus should be used; e.g. kg/(m·s²) and kg·m^−1·s^−2 are acceptable, but kg/m/s² is ambiguous and unacceptable.
• The first letter of symbols for units derived from the name of a person is written in upper case; otherwise, they are written in lower case. E.g., the unit of pressure is named after Blaise Pascal, so its symbol is written "Pa", but the symbol for mole is written "mol". Thus, "T" is the symbol for tesla, a measure of magnetic field strength, and "t" the symbol for tonne, a measure of mass. Since 1979, the litre may exceptionally be written using either an uppercase "L" or a lowercase "l", a decision prompted by the similarity of the lowercase letter "l" to the numeral "1", especially with certain typefaces.
• Symbols of units do not have a plural form; e.g. 25 kg, not 25 kgs.
• Uppercase and lowercase prefixes are not interchangeable. E.g., the quantities 1 mW and 1 MW represent two different quantities; the former is the typical power requirement of a hearing aid (1 milliwatt or 0.001 watts), and the latter the typical power requirement of a suburban train (1 megawatt or 1000000 watts).
• The 10th resolution of CGPM in 2003 declared that "the symbol for the decimal marker shall be either the point on the line or the comma on the line." In practice, the decimal point is used in English-speaking countries and most of Asia, and the comma in most of Latin America and in continental European languages.
• Spaces should be used as a thousands separator (1000000) in contrast to commas or periods (1,000,000 or 1.000.000) to reduce confusion resulting from the variation between these forms in different countries.
• Any line-break inside a number, inside a compound unit, or between number and unit should be avoided. Where this is not possible, line breaks should coincide with thousands separators.
• Since the value of "billion" and "trillion" can vary from language to language, the dimensionless terms "ppb" (parts per billion) and "ppt" (parts per trillion) should be avoided. However, no alternative is suggested in the SI Brochure.

 

Here is a template of using SI: Acceleration due to gravity.

Note the lowercase letters (neither "metres" nor "seconds" were named after people), the space between the value and the units, and the superscript "2" to denote "squared".

Printing SI symbols

Further rules are specified in respect of production of text using printing presses, word processors, typewriters and the like.

• Symbols are written in upright (Roman) type (m for metres, s for seconds), so as to differentiate from the italic type used for quantities (m for mass, s for displacement). By consensus of international standards bodies, this rule is applied independent of the font used for surrounding text.
• In print, the space used as a thousands separator (commonly called a thin space) is typically narrower than that used between words.

In this post we learned about the rules of using SI. After reading this post, Please comment your height, weight and the temperature outside in SI units. Mind your writing!

Introducing SI - Non-SI units accepted for use with SI

In the last 2 posts we talked about SI units and prefixes thoroughly. Although, in theory, SI can be used for any physical measurement, the CIPM has recognized that some non-SI units still appear in the scientific, technical, and commercial literature, and will continue to be used for many years to come. In addition, certain other units are so deeply embedded in the history and culture of the human race that they will continue to be used for the foreseeable future. The CIPM has catalogued several such units and published them in the SI Brochure so that their use may be consistent around the world. These units have been grouped as follows:

1. Non-SI units accepted for use with the SI (Table 6)

Certain units of time, angle, and legacy non-SI metric units have a long history of consistent use. Most of mankind has used the day and its non-decimal subdivisions as a basis of time and, unlike the foot or the pound, these were the same regardless of where they were being measured. The radian, being 1/2π of a revolution, has mathematical advantages but it is cumbersome for navigation, and, as with time, the units used in navigation have a large degree of consistency around the world. The tonne, litre, and hectare were adopted by the CGPM in 1879 and have been retained as units that may be used alongside SI units, having been given unique symbols. The catalogued units are:

minute (min) [1 min = 60 s]
hour (h) [1 h = 60 min = 3600 s]
day (d) [1 d = 24 h = 1440 min = 86400 s]
degree of arc (°) [1° = (π / 180) rad]
minute of arc (') [1' = (1 / 60)° = (π / 10800) rad]
second of arc ('') [1'' = (1 / 60)' = (1 / 3600)° = (π / 648000) rad]
hectare (ha) [1 ha = 100 a = 10000 m²= 1 hm²]
litre (l/L) [1 L = 1 dm³ = 0.001 m³ = 1000 cc = 1000 cm³]
tonne (t) [1 t = 10³ kg = 1 Mg]
astronomical unit (au) [1 au = 1.49597870691(6)E+11 m]

2. Non-SI units whose values in SI units must be obtained experimentally (Table 7)

Physicists often use units of measure that are based on natural phenomena, particularly when the quantities associated with these phenomena are many orders of magnitude greater than or less than the equivalent SI unit. The most common ones have been catalogued in the SI Brochure together with consistent symbols and accepted values, but with the caveat that their physical values need to be measured.

electronvolt (eV) [1 eV = 6981160217652999999♠1.60217653(14)E-19 J]
dalton/unified atomic mass unit (Da/u) [1 u = 1 Da = 6973166053886000000♠1.66053886(28)E-27 kg]

3. Other non-SI units (Table 8)

A number of non-SI units that had never been formally sanctioned by the CGPM have continued to be used across the globe in many spheres including health care and navigation. As with the units of measure in Tables 6 and 7, these have been catalogued by the CIPM in the SI Brochure to ensure consistent usage, but with the recommendation that authors who use them should define them wherever they are used.

bar (bar) [1 bar = 1E+5 Pa]
millimetre of mercury (mmHg) [1 mmHg = 7002133322387415000♠133.322387415 Pa at 0 °C = 7002133322387415000♠133.322387415 N/m² at 0 °C = 7002133322387415000♠133.322387415 kg/(cm·s²) at 0 °C]
torr (Torr) [1 Torr = 1/760 atm = 133.322368421... Pa = 133.322368421... N/m² = 133.322368421... kg/(cm·s²)]
atmosphere (atm) [1 atm = 1013.25 mbar = 1013.25 hPa = 7005101325000000000♠101325 Pa]
ångström (Å) [1 Å = 1E-10 m]
nautical mile (nmi) [1 nmi = 1852 m ]
are (a) [1 a = 1 dam² = 100 m²]
barn (b) [1 b = 1E-28 m²]
knot (kn) [1 kn = 1.852 km/h]
neper (Np) [1 Np = 1]
bel/decibel  (B/dB) [1 B = ln(10)/2 Np = ln(10)/2, 1 dB = 0.1 B]

Some of these units are very common in our life and can be seen almost everywhere, while some units are for some special areas. In short, SI units as well as these accepted non-SI units are commonly used both in life and in research. In the next post we will learn about the rules of using SI.

Introducing SI - SI Prefixes

In the last post we talked about SI base units and derived units. These units contain almost everything we can measure. However, sometimes we will meet very large or small numbers when using these units (e.g. 10000000 m or 0.000000001 V). To avoid these very large or very small numbers, SI uses a set of prefixes of units to describe these numbers.

SI prefixes are the unit prefixes which precede a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand; one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.

Decimal multiplicative prefixes have been a feature of all forms of the metric system with six dating back to the system's introduction in the 1790s. Today the prefixes are standardized for use in SI by BIPM in resolutions dating from 1960 to 1991. Since 2009, they have formed part of the International System of Quantities.

By now the BIPM specifies twenty prefixes for SI.

Each prefix name has a symbol which is used in combination with the symbols for units of measure. For example, the symbol for kilo- is k, and is used to produce km, kg, and kW, which are kilometre, kilogram, and kilowatt, respectively.

Prefixes may not be used in combination. This also applies to mass, for which the SI base unit (kilogram) already contains a prefix. For example, milligram (mg) is used instead of microkilogram (µkg).

In arithmetic of measurements having prefixed units, the prefixes must be expanded to their numeric multiplier, except when adding or subtracting values with identical units. Hence, 5 mV × 5 mA =5E-3 V × 5E-3 A = 25E-6 W = 25 µW.

Prefixes corresponding to an integer power of one thousand are generally preferred. Hence 100 m is preferred over 1 hm (hectometre) or 10 dam (decametres). The prefixes hecto, deca, deci, and centi were commonly used for everyday purposes; especially the centimeter (cm) is common.

However, some modern building codes require that the millimetre be used in preference to the centimetre, because "use of centimeters leads to extensive usage of decimal points and confusion".

When units occur in exponentiation, for example, in square and cubic forms, the multiplication prefix must be considered part of the unit, and thus included in the exponentiation.

• 1 km² means one square kilometre, or the area of a square of 1000 m by 1000 m and not 1000 square metres.
• 2 Mm³ means two cubic megametres, or the volume of two cubes of 1000000 m by 1000000 m by 1000000 m or 2E+18 m³, and not 2000000 cubic metres (2E+6 m³).

Examples:

• 5 cm = 5×10−2 m = 5E-2 m = 0.05 m
• 9 km² = 9 × (km)² = 9 × (10³ m)² = 9 × (10³)² × m² = 9E+6 m² = 9 × 1000000 m² = 9000000 m²
• 3 MW = 3E+6 W = 3 × 1000000 W = 3000000 W

The use of prefixes can be traced back to the introduction of the metric system in the 1790s, long before the 1960 introduction of the SI. The prefixes, including those introduced after 1960, are used with any metric unit, whether officially included in the SI or not (e.g. millidynes and milligauss).

The choice of prefixes with a given unit is usually dictated by convenience of use. Unit prefixes for amounts that are much larger or smaller than those actually encountered are seldom used, though they remain valid combinations. In most contexts only a few most common combinations are established as standard.

Here are some special situations when using SI prefixes.

Mass:
The kilogram, gram, milligram, microgram, and smaller are common. However, megagram (and gigagram, teragram, etc.) are rarely used; tonnes (and kilotonnes, megatonnes, etc.) or scientific notation are used instead. Megagram is occasionally used to disambiguate the metric tonne from the various non-metric tons. An exception is pollution emission rates, which are typically on the order of Tg/yr. Sometimes only one element is denoted for an emission, such as Tg C/yr or Tg N/yr, so that inter-comparisons of different compounds are easier.

Volume:
The litre, millilitre (equal to a cubic centimetre), microlitre, and smaller are common. In Europe, the centilitre is often used for packaged products but the use of the decilitre is rare everywhere. (The latter two items include prefixes corresponding to an exponent that is not divisible by three.)
Larger volumes are usually denoted in kilolitres, megalitres or gigalitres, or else in cubic metres (1 cubic metre = 1 kilolitre) or cubic kilometres (1 cubic kilometre = 1 teralitre).

Length:
The kilometre, metre, centimetre, millimetre, and smaller are common. (However, the decimetre is rarely used.) The micrometre is often referred to by the non-SI term micron. In some fields such as chemistry, the angstrom (equal to 0.1 nm) historically competed with the nanometre. The femtometre, used mainly in particle physics, is usually called a fermi. For large scales, megametre, gigametre, and larger are rarely used. Often used are astronomical units, light years, and parsecs; the astronomical unit is mentioned in the SI standards as an accepted non-SI unit.

Time and angles:
The second, millisecond, microsecond, and shorter are common. The kilosecond and megasecond also have some use, though for these and longer times one usually uses either scientific notation or minutes, hours, and so on.
Official policies about the use of these prefixes vary slightly between BIPM and the NIST;  and some of the policies of both bodies are at variance with everyday practice. For instance, the NIST advises that "…to avoid confusion, prefix symbols (and prefixes) are not used with the time-related unit symbols (names) min (minute), h (hour), d (day); nor with the angle-related symbols (names) ° (degree), ′ (minute), and ″ (second)."
The BIPM’s position on the use of SI prefixes with units of time larger than the second is the same as that of the NIST but their position with regard to angles differs: they state "However astronomers use milliarcsecond, which they denote mas, and microarcsecond, µas, which they use as units for measuring very small angles."

Temperature:
Official policy also varies from common practice for the degree Celsius (°C). NIST states; "Prefix symbols may be used with the unit symbol °C and prefixes may be used with the unit name 'degree Celsius'. For example, 12 m°C (12 millidegrees Celsius) is acceptable." In practice, it is more common for prefixes to be used with the kelvin when it is desirable to denote extremely large or small absolute temperatures or temperature differences. Thus, temperatures of star interiors may be given in units of MK (megakelvins), and molecular cooling may be described in mK (millikelvins).

In this post we talked about the SI prefixes and its usage. Now we can describe the numbers with just a prefix in a unit. But sometimes it is still inconvenient with only these SI units. In the next post we will introduce some non-SI units accepted by SI.

Introducing SI - Base Units and Derived Units

In the last post, we introduced SI and its history briefly. In the following posts we will introduce SI thoroughly. In this post we will talk about the SI units.

It is known to all that SI consists of a set of base units, a set of derived units, and a set of decimal-based multipliers that are used as prefixes. Here we will talk about SI base units and derived units.

The SI base units are the building blocks of the system and all other units are derived from them. When Maxwell first introduced the concept of a coherent system, he identified three quantities that could be used as base units: mass, length and time. Giorgi later identified the need for an electrical base unit. Theoretically any one of electrical current, potential difference, electrical resistance, electrical charge or a number of other quantities could have provided the base unit, with the remaining units then being defined by the laws of physics. In the event, the unit of electric current was chosen for SI. Another three base units (for temperature, substance and luminous intensity) were added later.

There are 7 base units in SI. Here we will give the histories of the definitions of these base units.

Length - m (metre)

• Original (1793): 1/10000000 of the meridian through Paris between the North Pole and the Equator.
• Interim (1960): 1650763.73 wavelengths in a vacuum of the radiation corresponding to the transition between the 2p10 and 5d5 quantum levels of the krypton-86 atom.
• Current (1983): The distance travelled by light in vacuum in 1/299792458 second.

Mass - kg (kilogram)

• Original (1793): The grave was defined as being the weight [mass] of one cubic decimetre of pure water at its freezing point.
• Current (1889): The mass of the international prototype kilogram.

Time - s (second)

• Original (Medieval): 1/86400 of a day.
• Interim (1956): 1/31556925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.
• Current (1967): The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

Electric Current - A (ampere)

• Original (1881): A tenth of the electromagnetic CGS unit of current. The [CGS] electromagnetic unit of current is that current, flowing in an arc 1 cm long of a circle 1 cm in radius creates a field of one oersted at the centre. 
• Current (1946): The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2E-7 newtons per metre of length.

Thermodynamic Temperature - K (kelvin)

• Original (1743): The centigrade scale is obtained by assigning 0 °C to the freezing point of water and 100 °C to the boiling point of water.
• Interim (1954): The triple point of water (0.01 °C) defined to be exactly 273.16 K. 
• Current (1967): 1/273.16 of the thermodynamic temperature of the triple point of water.

Amount of Substance - mol (mole)

• Original (1900): The molecular weight of a substance in mass grams.
• Current (1967): The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. 

Luminous Intensity - cd (candela)

• Original (1946): The value of the new candle is such that the brightness of the full radiator at the temperature of solidification of platinum is 60 new candles per square centimetre.
• Current (1979): The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540E+12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Note:

1. Despite the prefix "kilo-", the kilogram is the base unit of mass. The kilogram, not the gram, is used in the definitions of derived units. Nonetheless, units of mass are named as if the gram were the base unit.
2. In 1954 the unit of thermodynamic temperature was known as the "degree Kelvin" (symbol °K; "Kelvin" spelt with an upper-case "K"). It was renamed the "kelvin" (symbol "K"; "kelvin" spelt with a lower case "k") in 1967.
3. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

The derived units in the SI are formed by powers, products or quotients of the base units and are unlimited in number. Derived units are associated with derived quantities, for example velocity is a quantity that is derived from the base quantities of time and length, so in SI the derived unit is metres per second (symbol m/s). The dimensions of derived units can be expressed in terms of the dimensions of the base units.

Coherent units are derived units that contain no numerical factor other than 1 - quantities such as standard gravity and density of water are absent from their definitions. In the example above, one newton is the force required to accelerate a mass of one kilogram by one metre per second squared. Since the SI units of mass and acceleration are kg and m/s² respectively and F is directly proportional to m*a, the units of force (and hence of newtons) is formed by multiplication to give kg/(m*s²). Since the newton is part of a coherent set of units, the constant of proportionality is 1.

For the sake of convenience, some derived units have special names and symbols. Such units may themselves be used in combination with the names and symbols for base units and for other derived units to express the units of other derived quantities. For example, the SI unit of force is the newton (N), the SI unit of pressure is the pascal (Pa)—and the pascal can be defined as "newtons per square metre" (N/m²).

There are 22 named derived units in SI:

Note:
The radian and steradian, once given special status, are now considered dimensionless derived units.

SI contains many units, including base units and derived units. However, we will meet different types of numbers, which will produce very large or small numbers in these units. So prefixes will be used to avoid this situation. In the next post we will talk about the SI prefixes.

What is SI

In the last post, we talked about the history metric system. In this post we will talk about SI - the system of units as we are using now.

SI is the short form of the French name "Système International d'Unités", in English it is called "International System of Units". SI is the modern form of the metric system and is the world's most widely used system of measurement, used in both commerce and science. It comprises a coherent system of units of measurement built on seven base units. It defines twenty-two named units, and includes many more unnamed coherent derived units. The system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units.

SI was published in 1960 as the result of an initiative that started in 1948. In 1948 an overhaul of the metric system was set in motion which resulted in the development of the "Practical system of units" which, on its publication in 1960, was given the name "The International System of Units". In 1954 the 10th CGPM identified electric current as the fourth base quantity in the practical system of units and added two more base quantities—temperature and luminous intensity-making six base quantities in all. The units associated with these quantities were the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K) and candela (cd). In 1971 a seventh base quantity, amount of substance represented by the mole (mol), was added to the definition of SI.

The motivation for the development of SI was the diversity of units that had sprung up within the CGS systems and the lack of coordination between the various disciplines that used them. The CGPM, which was established by the Metre Convention of 1875, brought together many international organizations to not only agree on the definitions and standards of the new system but also agree rules on writing and presenting measurements in a standardized manner around the world.

SI consists of a set of base units, a set of derived units with special names, and a set of decimal-based multipliers that are used as prefixes.

Base units:

The SI base units are the building blocks of the system and all other units are derived from them. There are 7 base units in SI:

Length: m
Mass: kg
Time: s
Electric current: A
Thermodynamic temperature: K
Amount of substance: mol
Luminous intensity: cd

Derived units:

The derived units in the SI are formed by powers, products or quotients of the base units and are unlimited in number.  Derived units are associated with derived quantities. The dimensions of derived units can be expressed in terms of the dimensions of the base units. For the sake of convenience, there are 22 derived units have special names and symbols.  Such units may themselves be used in combination with the names and symbols for base units and for other derived units to express the units of other derived quantities.

Prefixes:

Prefixes are added to unit names to produce multiple and sub-multiples of the original unit. All multiples are integer powers of ten and above a hundred or below a hundredth all are integer powers of a thousand. The prefixes are never combined in SI.

Here is an example: If the displacement of an object is 10 cm, and it takes 10 s to move the object, the velocity of this object is 10 cm/s. 

In this sentence, "m" is a base unit, and the prefix "centi-" stands for 0.01, so "cm" means 0.01m, "s" is a base unit as well, then "m/s" is a derived unit, derived by length and time.

In the next post, we will talk about SI units thoroughly. We will talk about the definitions of base units, 22 derived units which have special names and symbols, SI prefixes, and other non-SI units adopted by SI, etc.