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.^{[1]}^{[2]} It is a system in which every quantity has a unique unit, or one that does not use conversion factors.^{[3]}

A **coherent derived unit** is a derived unit 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.^{[1]}

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 and James Clerk Maxwell and promoted by the British Science Association. 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 (SI) was designed in 1960 around the principle of coherence.

In the SI system, the derived unit m/s is a coherent derived unit for speed or velocity^{[4]}^{: 139 } but km/h 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 m/s uses the base units of the SI system.^{[1]} The derived unit km/h requires numerical factors to relate to the SI base units: 1000 m/km and 3600 s/h.

In the cgs system, m/s is not a coherent derived unit. The numerical factor of 100 cm/m is needed to express m/s in the cgs system.

The earliest units of measure devised by humanity bore no relationship to each other.^{[citation needed]} As both humanity's understanding of philosophical concepts and the organisation of society developed, so units of measurement were standardised – first particular units of measure had the same value across a community, then different units of the same quantity (for example feet and inches) were given a fixed relationship. Apart from Ancient China 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.^{[5]}

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, 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.^{[6]} 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* (barleycorns) in a *shu-si* (finger) and 30 shu-si in a *kush* (cubit).^{[7]}

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 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 (defined as kg⋅m^{−1}⋅s^{−2}), but the bar (defined as 100000 kg⋅m^{−1}⋅s^{−2}) 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 100000, 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.

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 was 0.001 m^{3} and the are (from which we get the hectare) was 100 m^{2}. 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.^{[8]}

The CGS system had two units of energy, the erg that was related to mechanics and the calorie that was related to thermal energy, so only one of them (the erg, equivalent to the g⋅cm^{2}/s^{2}) 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.^{[9]}

This list catalogues coherent relationships in various systems of units.

Main article: SI coherent derived unit |

The following is a list of quantities with corresponding coherent SI units:

- frequency (hertz) = reciprocal of time (inverse seconds)
- force (newtons) = mass (kilograms) × acceleration (m/s
^{2}) - pressure (pascals) = force (newtons) ÷ area (m
^{2}) - energy (joules) = force (newtons) × distance (metres)
- power (watts) = energy (joules) ÷ time (seconds)
- potential difference (volts) = power (watts) ÷ electric current (amps)
- electric charge (coulombs) = electric current (amps) × time (seconds)
- equivalent radiation dose (sieverts) = energy (joules) ÷ mass (kilograms)
- absorbed radiation dose (grays) = energy (joules) ÷ mass (kilograms)
- radioactive activity (becquerels) = reciprocal of time (s
^{−1}) - capacitance (farads) = electric charge (coulombs) ÷ potential difference (volts)
- electrical resistance (ohms) = potential difference (volts) ÷ electric current (amperes)
- electrical conductance (siemens) = electric current (amperes) ÷ potential difference (volts)
- magnetic flux (weber) = potential difference (volts) × time (seconds)
- magnetic flux density (tesla) = magnetic flux (webers) ÷ area (square metres)

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

- acceleration (gals) = distance (centimetres) ÷ time
^{2}(s^{2}) - force (dynes) = mass (grams) × acceleration (cm/s
^{2}) - energy (ergs) = force (dynes) × distance (centimetres)
- pressure (barye) = force (dynes) ÷ area (cm
^{2}) - dynamic viscosity (poise) = mass (grams) ÷ (distance (centimetres) × time (seconds))
- kinematic viscosity (stokes) = area (cm
^{2}) ÷ time (seconds)

The following is a list of coherent foot–pound–second (FPS) system of units:

- force (poundal) = mass (pounds) × acceleration (ft/s
^{2})