A coherent system of units is a system of

units of measurement A unit of measurement, or unit of measure, is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other qua ...

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. It is a system in which every quantity has a unique unit, or one that does not use conversion factors. A coherent derived unit is a

derived unit A base unit of measurement (also referred to as a base unit or fundamental unit) is a unit of measurement adopted for a '' base quantity''. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the ...

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 only one 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 The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...

and

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...

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 Chief ...

. The concept was initially applied to the centimetre–gram–second (CGS) in 1873 and the foot–pound–second systems (FPS) of units in 1875. The

International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...

(SI) was designed in 1960 to incorporate the principle of coherence.

Examples

In the SI, the derived unit is a coherent derived unit for

speed In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

but is not a coherent derived unit. Speed or velocity is defined by the change in distance divided by a change in time. The derived unit uses the base units of the SI system. The derived unit requires numerical factors to relate to the SI base units: and . In the cgs system, is not a coherent derived unit. The numerical factor of is needed to express in the cgs system.

History

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. ...

developed, so units of measurement were standardized—first particular units of measure had the same value across a

community A community is a social unit (a group of people) with a shared socially-significant characteristic, such as place, set of norms, culture, religion, values, customs, or identity. Communities may share a sense of place situated in a given g ...

, 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 u ...

(for example feet and inches) were given a fixed relationship. Apart from

Ancient China The history of China spans several millennia across a wide geographical area. Each region now considered part of the Chinese world has experienced periods of unity, fracture, prosperity, and strife. Chinese civilization first emerged in the Y ...

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 Enlightenment or enlighten may refer to: Age of Enlightenment * Age of Enlightenment, period in Western intellectual history from the late 17th to late 18th century, centered in France but also encompassing (alphabetically by country or culture): ...

Relating quantities of the same kind

The history of the measurement of length dates back to the early civilization of the

Middle East The Middle East (term originally coined in English language) is a geopolitical region encompassing the Arabian Peninsula, the Levant, Turkey, Egypt, Iran, and Iraq. The term came into widespread usage by the United Kingdom and western Eur ...

(10000 BC – 8000 BC). Archaeologists have been able to reconstruct the units of measure in use in

Mesopotamia Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...

India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...

, the Jewish culture and many others. Archaeological and other evidence shows that in many civilizations, 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 Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...

, multiples with prime factors aside from 2, 3 and 5 were sometimes 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 "han ...

. In 2150 BC, the Akkadian emperor Naram-Sin rationalized the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of which the only prime factors were 2, 3 and 5; for example there were 6 ''she'' (

barley Barley (), a member of the grass family, is a major cereal grain grown in temperate climates globally. It was one of the first cultivated grains; it was domesticated in the Fertile Crescent around 9000 BC, giving it nonshattering spikele ...

corns) in a ''shu-si'' (

finger A finger is a prominent digit (anatomy), digit on the forelimbs of most tetrapod vertebrate animals, especially those with prehensile extremities (i.e. hands) such as humans and other primates. Most tetrapods have five digits (dactyly, pentadact ...

) 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 Noah ...

Relating quantities of different kinds

Non- commensurable quantities have different

physical dimension Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or volume. Length can be generalized to other linear dimensio ...

s, which means that adding or subtracting them is not meaningful. For instance, adding the

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

of an object to its

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

has no physical meaning. However, new quantities (and, as such, units) can be derived via multiplication and

exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

of other units. As an example, the

SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...

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 is a coherent unit of

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...

(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

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 ( Commonwealth spelling) or liter ( American 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 metres (m3). A ...

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), that is, square metres (), and is primarily used in the measurement of land. There are 100 hectares in one square kilometre. ...

) 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 10−7joules (100Nano-, nJ). It is not an SI unit, instead originating from the centimetre–gram–second system of units (CGS). Its name is derived from (), a Greek language, Greek word meaning 'work' or ' ...

that was related to

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

and the

calorie The calorie is a unit of energy that originated from the caloric theory of heat. The large calorie, food calorie, dietary calorie, kilocalorie, or kilogram calorie is defined as the amount of heat needed to raise the temperature of one liter o ...

that was related to

thermal energy The term "thermal energy" is often used ambiguously in physics and engineering. It can denote several different physical concepts, including: * Internal energy: The energy contained within a body of matter or radiation, excluding the potential en ...

, 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 The joule ( , or ; symbol: J) is the unit of energy in the International System of Units (SI). In terms of SI base units, one joule corresponds to one kilogram- metre squared per second squared One joule is equal to the amount of work d ...

Dimension-related coherence

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 cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

and

power Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Math ...

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 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

's mass–energy equation, , does not require extraneous constants when expressed in coherent units.

Isaac Asimov Isaac Asimov ( ; – April 6, 1992) was an Russian-born American writer and professor of biochemistry at Boston University. During his lifetime, Asimov was considered one of the "Big Three" science fiction writers, along with Robert A. H ...

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. Definition The English E ...

), 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 that uses the pound and the pound-force, one of which is a base unit and the other, a non-coherent 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 , 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., , 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 kilometre (km) and the hour (h) are non-coherent derived units. The metre per second (mps) is defined as the velocity of an object that travels one metre in one second, and the kilometre 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 non-coherent derived unit. Suppose that we choose to use the kilometre per hour as the unit of velocity in the system. Then the system becomes non-coherent, 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 non-coherent derived unit, with 1 m/s = 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.

List of coherent units

This list catalogues coherent relationships in various systems of units.

SI

The following is a list of quantities, each with its corresponding coherent SI unit: :

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

(

hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...

) = reciprocal of time (

inverse second The inverse second or reciprocal second (s−1), also called ''per second'', is a unit of measurement, unit defined as the multiplicative inverse of the second (a unit of time). It is applicable for physical quantities of dimension (physics), dime ...

) :

( newton) = mass (kilogram) × acceleration (m/s²) :

( pascal) = force (newton) ÷

area Area is the measure of a region's size on a 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 surface or the boundary of a three-di ...

(m²) :

(

) = force (newton) × distance (metre) :

(

watt The watt (symbol: W) is the unit of Power (physics), power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantification (science), quantify the rate of Work ...

) = energy (joule) ÷ time (second) :

potential difference Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge ...

(

volt The volt (symbol: V) is the unit of electric potential, Voltage#Galvani potential vs. electrochemical potential, electric potential difference (voltage), and electromotive force in the International System of Units, International System of Uni ...

) = power (watt) ÷ electric current (ampere) :

electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...

(

coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...

) = electric current (ampere) × time (second) : equivalent radiation dose (

sievert The sievert (symbol: SvPlease note there are two non-SI units that use the same Sv abbreviation: the sverdrup and svedberg.) is a derived unit in the International System of Units (SI) intended to represent the stochastic health risk of ionizin ...

) = energy (joule) ÷ mass (kilogram) : absorbed radiation dose (

gray Grey (more frequent in British English) or gray (more frequent in American English) is an intermediate color between black and white. It is a neutral or achromatic color, meaning that it has no chroma. It is the color of a cloud-covered s ...

) = energy (joule) ÷ mass (kilogram) : radioactive activity (

becquerel The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as an activity of one per second, on average, for aperiodic activity events referred to a radionuclide. For applicatio ...

) = reciprocal of time (s⁻¹) :

capacitance Capacitance is the ability of an object 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 are two closely related ...

(

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, International System of Units (SI), equivalent to 1 coulomb per volt (C/V). It is named afte ...

) = electric charge (coulomb) ÷ potential difference (volt) :

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 paral ...

(

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 (1 ...

) = potential difference (volt) ÷ electric current (ampere) :

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 paral ...

(

siemens Siemens AG ( ) is a German multinational technology conglomerate. It is focused on industrial automation, building automation, rail transport and health technology. Siemens is the largest engineering company in Europe, and holds the positi ...

) = electric current (ampere) ÷ potential difference (volt) :

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 we ...

( weber) = potential difference (

) × time (second) :

magnetic flux density A magnetic field (sometimes called B-field) is a physical 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 ...

( tesla) = magnetic flux (weber) ÷ area (square metre)

CGS

The following is a list of coherent centimetre–gram–second (CGS) system of units: :

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

( gals) = distance (centimetre) ÷ time (s²) : force (

dyne The dyne (symbol: dyn; ) is a derived units of measurement, unit of force (physics), force specified in the centimetre–gram–second system of units, centimetre–gram–second (CGS) system of units, a predecessor of the modern International S ...

) = mass (gram) × acceleration (cm/s²) : energy (

) = force (dyne) × distance (centimetre) : 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 (dyne) ÷

(cm²) : dynamic

viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

( poise) = mass (gram) ÷ (distance (centimetre) × time (second)) : kinematic

( stokes) = area (cm²) ÷ time (second)

FPS

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

References

{{DEFAULTSORT:Coherent units of measurement Systems of units Dimensional analysis