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OR: A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. A coherent derived unit is a
derived unit SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate ...
that, for a given system of quantities and for a chosen set of base units, is a product of powers of base units, with the proportionality factor being one. If a system of quantities has equations that relate quantities and the associated system of units has corresponding base units, with one base unit for each base quantity, then it is coherent if and only if every derived unit of the system is coherent. The concept of coherence was developed in the mid-nineteenth century by, amongst others, Kelvin and
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
and promoted by the
British Science Association The British Science Association (BSA) is a charity and learned society founded in 1831 to aid in the promotion and development of science. Until 2009 it was known as the British Association for the Advancement of Science (BA). The current Chie ...
. The concept was initially applied to the centimetre–gram–second (CGS) in 1873 and the
foot–pound–second system The foot–pound–second system or FPS system is a system of units built on three fundamental units: the foot for length, the (avoirdupois) pound for either mass or force (see below), and the second for time.. Variants Collectively, the var ...
s (FPS) of units in 1875. The International System of Units (1960) was designed around the principle of coherence.

# Example

In SI, which is a coherent system, the unit of power is the watt, which is defined as one joule per second. In the US customary system of measurement, which is non-coherent, the unit of power is the
horsepower Horsepower (hp) is a unit of measurement of power, or the rate at which work is done, usually in reference to the output of engines or motors. There are many different standards and types of horsepower. Two common definitions used today are the ...
, which is defined as 550 foot-pounds per second (the pound in this context being the
pound-force The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit of force used in some systems of measurement, including English Engineering units and the foot–pound–second system. Pound-force should not be confused with pou ...
); similarly the gallon is 231 cubic inches.

# Before the metric system

The earliest units of measure devised by humanity bore no relationship to each other. As both humanity's understanding of philosophical concepts and the organisation of
society A society is a group of individuals involved in persistent social interaction, or a large social group sharing the same spatial or social territory, typically subject to the same political authority and dominant cultural expectations. Societ ...
developed, so units of measurement were standardised – first particular units of measure had the same value across a
community A community is a social unit (a group of living things) with commonality such as place, norms, religion, values, customs, or identity. Communities may share a sense of place situated in a given geographical area (e.g. a country, village, tow ...
, then different units of the same
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a un ...
(for example feet and inches) were given a fixed relationship. Apart from
Ancient China The earliest known written records of the history of China date from as early as 1250 BC, from the Shang dynasty (c. 1600–1046 BC), during the reign of king Wu Ding. Ancient historical texts such as the ''Book of Documents'' (early chapter ...
where the units of capacity and of mass were linked to red millet seed, there is little evidence of the linking of different quantities until the Enlightenment.

## Relating quantities of the same kind

The history of the measurement of length dates back to the early civilisations of the Middle East (10000 BC – 8000 BC). Archeologists have been able to reconstruct the units of measure in use in Mesopotamia,
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
, the Jewish culture and many others. Archeological and other evidence shows that in many civilisations, the ratios between different units for the same quantity of measure were adjusted so that they were integer numbers. In many early cultures such as Ancient Egypt, multiples of 2, 3 and 5 were not always used—the Egyptian royal cubit being 28 fingers or 7
hands A hand is a prehensile, multi- fingered appendage located at the end of the forearm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs. A few other vertebrates such as the koala (which has two opposable thumbs on each ...
. In 2150 BC, the Akkadian emperor Naram-Sin rationalised the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of 2, 3 or 5, for example there were 6 ''she'' (
barley Barley (''Hordeum vulgare''), a member of the grass family, is a major cereal grain grown in temperate climates globally. It was one of the first cultivated grains, particularly in Eurasia as early as 10,000 years ago. Globally 70% of barley ...
corns) in a ''shu-si'' ( finger) and 30 shu-si in a ''kush'' (
cubit The cubit is an ancient unit of length based on the distance from the elbow to the tip of the middle finger. It was primarily associated with the Sumerians, Egyptians, and Israelites. The term ''cubit'' is found in the Bible regarding ...
). ## Relating quantities of different kinds

Non- commensurable quantities have different physical dimensions, which means that adding or subtracting them is not meaningful. For instance, adding the mass of an object to its volume has no physical meaning. However, new quantities (and, as such, units) can be derived via multiplication and exponentiation of other units. As an example, the
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
for force is the newton, which is defined as kg⋅m⋅s−2. Since a coherent derived unit is one which is defined by means of multiplication and exponentiation of other units but not multiplied by any scaling factor other than 1, the
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
is a coherent unit of pressure (defined as kg⋅m−1⋅s−2), but the bar (defined as ) is not. Note that coherence of a given unit depends on the definition of the base units. Should the standard unit of length change such that it is shorter by a factor of , then the bar would be a coherent derived unit. However, a coherent unit remains coherent (and a non-coherent unit remains non-coherent) if the base units are redefined in terms of other units with the numerical factor always being unity.

# Metric system

## Rational system and use of water

The concept of coherence was only introduced into the metric system in the third quarter of the nineteenth century; in its original form the metric system was non-coherent – in particular the
litre The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre ( ...
was 0.001 m3 and the are (from which we get the
hectare The hectare (; SI symbol: ha) is a non-SI metric unit of area equal to a square with 100-metre sides (1 hm2), or 10,000 m2, and is primarily used in the measurement of land. There are 100 hectares in one square kilometre. An acre is a ...
) was 100 m2. A precursor to the concept of coherence was however present in that the units of mass and length were related to each other through the physical properties of water, the gram having been designed as being the mass of one cubic centimetre of water at its freezing point. The CGS system had two units of energy, the
erg The erg is a unit of energy equal to 10−7 joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' ...
that was related to
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
and the
calorie The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of on ...
that was related to thermal energy, so only one of them (the erg, equivalent to the g⋅cm2/s2) could bear a coherent relationship to the base units. By contrast, coherence was a design aim of the SI, resulting in only one unit of energy being defined – the joule.

## Dimension-related coherence

Work of
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
and others Each variant of the metric system has a degree of coherence – the various derived units being directly related to the base units without the need of intermediate conversion factors. An additional criterion is that, for example, in a coherent system the units of
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
, energy and
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may ...
be chosen so that the equations :' = ' × ' :' = ' × ' :' = ' / ' hold without the introduction of constant factors. Once a set of coherent units have been defined, other relationships in physics that use those units will automatically be true –
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's mass–energy equation, , does not require extraneous constants when expressed in coherent units. Isaac Asimov wrote, "In the cgs system, a unit force is described as one that will produce an acceleration of 1 cm/sec2 on a mass of 1 gm. A unit force is therefore 1 cm/sec2 multiplied by 1 gm." These are independent statements. The first is a definition; the second is not. The first implies that the constant of proportionality in the force law has a magnitude of one; the second implies that it is dimensionless. Asimov uses them both together to prove that it is the pure number one. Asimov's conclusion is not the only possible one. In a system that uses the units foot (ft) for length, second (s) for time, pound (lb) for mass, and pound-force (lbf) for force, the law relating force (''F''), mass (''m''), and acceleration (''a'') is . Since the proportionality constant here is dimensionless and the units in any equation must balance without any numerical factor other than one, it follows that 1 lbf = 1 lb⋅ft/s2. This conclusion appears paradoxical from the point of view of competing systems, according to which and . Although the pound-force is a coherent derived unit in this system according to the official definition, the system itself is not considered to be coherent because of the presence of the proportionality constant in the force law. A variant of this system applies the unit s2/ft to the proportionality constant. This has the effect of identifying the pound-force with the pound. The pound is then both a base unit of mass and a coherent derived unit of force. One may apply any unit one pleases to the proportionality constant. If one applies the unit s2/lb to it, then the foot becomes a unit of force. In a four-unit system (
English engineering units Some fields of engineering in the United States use a system of measurement of physical quantities known as the English Engineering Units. Despite its name, the system is based on United States customary units of measure; it is not used in Englan ...
), the pound and the pound-force are distinct base units, and the proportionality constant has the unit lbf⋅s2/(lb⋅ft). All these systems are coherent. One that is not is a three-unit system (also called English engineering units) in which ''F'' = ''ma'' that uses the pound and the pound-force, one of which is a base unit and the other, a noncoherent derived unit. In place of an explicit proportionality constant, this system uses conversion factors derived from the relation 1 lbf = 32.174 lb⋅ft/s2. In numerical calculations, it is indistinguishable from the four-unit system, since what is a proportionality constant in the latter is a conversion factor in the former. The relation among the numerical values of the quantities in the force law is = 0.031081 , where the braces denote the numerical values of the enclosed quantities. Unlike in this system, in a coherent system, the relations among the numerical values of quantities are the same as the relations among the quantities themselves. The following example concerns definitions of quantities and units. The (average) velocity (''v'') of an object is defined as the quantitative physical property of the object that is directly proportional to the distance (''d'') traveled by the object and inversely proportional to the time (''t'') of travel, i.e., ''v'' = ''kd''/''t'', where ''k'' is a constant that depends on the units used. Suppose that the metre (m) and the second (s) are base units; then the kilometer (km) and the hour (h) are noncoherent derived units. The metre per second (mps) is defined as the velocity of an object that travels one metre in one second, and the kilometer per hour (kmph) is defined as the velocity of an object that travels one kilometre in one hour. Substituting from the definitions of the units into the defining equation of velocity we obtain, 1 mps = ''k'' m/s and 1 kmph = ''k'' km/h = 1/3.6 ''k'' m/s = 1/3.6 mps. Now choose ''k'' = 1; then the metre per second is a coherent derived unit, and the kilometre per hour is a noncoherent derived unit. Suppose that we choose to use the kilometre per hour as the unit of velocity in the system. Then the system becomes noncoherent, and the numerical value equation for velocity becomes = 3.6 /. Coherence may be restored, without changing the units, by choosing ''k'' = 3.6; then the kilometre per hour is a coherent derived unit, with 1 kmph = 1 m/s, and the metre per second is a noncoherent derived unit, with 1 mps = 3.6 m/s. A definition of a physical quantity is a statement that determines the ratio of any two instances of the quantity. The specification of the value of any constant factor is not a part of the definition since it does not affect the ratio. The definition of velocity above satisfies this requirement since it implies that ''v''1/''v''2 = (''d''1/''d''2)/(''t''1/''t''2); thus if the ratios of distances and times are determined, then so is the ratio of velocities. A definition of a unit of a physical quantity is a statement that determines the ratio of any instance of the quantity to the unit. This ratio is the numerical value of the quantity or the number of units contained in the quantity. The definition of the metre per second above satisfies this requirement since it, together with the definition of velocity, implies that ''v''/mps = (''d''/m)/(''t''/s); thus if the ratios of distance and time to their units are determined, then so is the ratio of velocity to its unit. The definition, by itself, is inadequate since it only determines the ratio in one specific case; it may be thought of as exhibiting a specimen of the unit. A new coherent unit cannot be defined merely by expressing it algebraically in terms of already defined units. Thus the statement, "the metre per second equals one metre divided by one second", is not, by itself, a definition. It does not imply that a unit of velocity is being defined, and if that fact is added, it does not determine the magnitude of the unit, since that depends on the system of units. In order for it to become a proper definition both the quantity and the defining equation, including the value of any constant factor, must be specified. After a unit has been defined in this manner, however, it has a magnitude that is independent of any system of units.

# Catalogue of coherent relations

This list catalogues coherent relationships in various systems of units.

## SI

The following is a list of coherent SI units: : frequency ( hertz) = reciprocal of time ( inverse seconds) :
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
(
newtons The newton (symbol: N) is the unit of force in the International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in ...
) = mass (kilograms) × acceleration (m/s2) : pressure ( pascals) = force (newtons) ÷
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open ...
(m2) : energy ( joules) = force (newtons) × distance (metres) :
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may ...
( watts) = energy (joules) ÷ time (seconds) : potential difference ( volts) = power (watts) ÷ electric current (amps) : electric charge (
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary cha ...
s) = electric current (amps) × time (seconds) : equivalent radiation dose (
sievert The sievert (symbol: SvNot be confused with the sverdrup or the svedberg, two non-SI units that sometimes use the same symbol.) is a unit in the International System of Units (SI) intended to represent the stochastic health risk of ionizing rad ...
s) = energy (joules) ÷ mass (kilograms) : absorbed radiation dose (
gray Grey (more common in British English) or gray (more common in American English) is an intermediate color between black and white. It is a neutral or achromatic color, meaning literally that it is "without color", because it can be composed ...
s) = energy (joules) ÷ mass (kilograms) : radioactive activity (
becquerel The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. For applications relati ...
s) = reciprocal of time (s−1) :
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized a ...
(
farad The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
s) = electric charge (coulombs) ÷ potential difference (volts) :
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual paralle ...
(
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (bo ...
s) = potential difference (volts) ÷ electric current (amperes) :
electrical conductance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual paralle ...
(
siemens Siemens AG ( ) is a German multinational conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad. The principal divisions of the corporation are ''Industry'', ' ...
) = electric current (amperes) ÷ potential difference (volts) :
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...
( weber) = potential difference ( volts) × time (seconds) :
magnetic flux density A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
( tesla) = magnetic flux (webers) ÷ area (square metres)

## CGS

The following is a list of coherent centimetre–gram–second (CGS) system of units: :
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
( gals) = distance (centimetres) ÷ time2 (s2) :force (
dyne The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI. History The name dyne was first proposed as a CGS unit of force in 1873 by a Committee o ...
s) = mass (grams) × acceleration (cm/s2) :energy (
erg The erg is a unit of energy equal to 10−7 joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' ...
s) = force (dynes) × distance (centimetres) :pressure (
barye The barye (symbol: Ba), or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetre–gram–second (CGS) unit of pressure. It is equal to 1 dyne per square centimetre. : =  =  = =  = See also *Pasca ...
) = force (dynes) ÷
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open ...
(cm2) :dynamic viscosity ( poise) = mass (grams) ÷ (distance (centimetres) × time (seconds)) :kinematic viscosity ( stokes) = area (cm2) ÷ time (seconds)

## FPS

The following is a list of coherent foot–pound–second (FPS) system of units: :force (poundal) = mass (pounds) × acceleration (ft/s2)