# How metric units fit together

After celebrating the arrival of decimal currency in the UK 50 years ago, Metric Views now goes back to basics on decimal measurement. No, we shall not be looking at nautical distances or Gunter’s chain!

UKMA often describes the metric system as a single, rational, consistent, simple and coherent measurement system. This article examines the rationale for these descriptions and demonstrates how the metric system fits together. It will also look at common base units and prefixes and their common uses.

At its core, the metric system is made up of basic units and prefixes. The modern metric system, known as the International System of Units or SI for short, is based on the principle that all measurable phenomena covered by the system – pretty well everything in the known natural world! – have one basic named unit. However, there are also some metric units outside SI that are approved for use with SI and they are all compatible with SI. All the multiples and subdivisions then follow the same logical structure using prefixes. All units and prefixes in the metric system have common language-independent symbols. These symbols are case-sensitive and are not abbreviations. In the metric system, each prefix always means the same multiple or subdivision of a unit wherever it is used. They are all simple multiples of or divisions by ten (and powers of ten) and they all apply to every basic named unit. Unlike the imperial system, there are no measurement tables to learn.

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The USMA poster above shows the links between various metric units.

The metre (symbol: m), the metric unit of length, comes from “metron”, the Greek word for measure. It was originally defined as the ten millionth part of the distance from the North Pole to the equator but is now defined as the distance travelled by light in a vacuum in a given fraction of a second.

The litre (symbol: l or L), the metric unit of volume, comes from “litra”, a Sicilian monetary unit. It is equal in volume to the tenth part of a metre cubed. There are 1000 litres in one cubic metre. Litres are widely used on drink labels and, for example, to describe dustbin and fridge capacity.

The kilogram (symbol: kg), the metric unit of mass, is the only base unit in SI that contains a prefix. The gram (symbol: g), from which the kilogram is derived, comes from late Latin and Greek “gramma”, meaning a small weight. The kilogram was originally defined in 1795 as the mass of a litre of water. However, several problems were subsequently discovered with this definition so the kilogram was eventually redefined as the mass of the international prototype of the kilogram. For all practical purposes, a litre of distilled water at a certain temperature has a mass of approximately 1 kilogram. The tonne (symbol: t) is equal to 1000 kilograms.

The hectare (symbol: ha), the metric unit of area, is made up of “hecto-” plus “are” (pronounced “air”). The “hecto” part of hectare comes from the Greek “hekaton”, meaning hundred. The “are” part of hectare comes from the Latin “area”, which literally means “vacant piece of level ground”. The hectare is equal to 10 000 square metres, which is equivalent to a square that is 100 metres on each side.  The hectare is used for land registration across the European Union and widely used for referring to large land areas.

The newton (symbol: N), named after Sir Isaac Newton, is the metric unit of force. The newton is defined as the force which will accelerate a mass of 1 kilogram at a rate of 1 metre per second squared. The newton is not seen very much outside scientific contexts but is mentioned here because several common metric units are derived from the newton.

The joule (symbol: J), named after English physicist James Prescott Joule, is the metric unit of energy. The joule is defined as the energy required to accelerate a mass of 1 kilogram at a rate of 1 metre per second squared over a distance of a metre (= 1 N m). The joule is widely seen in the form of kilojoules (1000 joules) on food nutrition labels.

The watt (symbol: W), named after Scottish engineer James Watt, is the metric unit of power. The watt is defined as the rate of expenditure of energy equal to 1 joule per second. The watt is commonly used for smaller electrical appliances.

The ampere (symbol: A), or amp for short, the metric unit for electric current, is named after the French physicist André-Marie Ampère. Its definition is based on a constant current of a pair of parallel wires placed exactly one metre apart in a vacuum that would produce a force equal to a fixed number of newtons per metre of length. This is the familiar ‘amp’ we associate with things like fuses and electric cable. It is the fundamental base unit for electricity and magnetism.

The volt (symbol: V), named after Italian physicist Alessandro Volta, is the metric unit of electric potential difference. The volt is defined as one watt per ampere. The volt is the familiar unit used for batteries, electricity supplies, power lines, and electrical devices.

The pascal (symbol: Pa), named after French mathematician and physicist Blaise Pascal, is the metric unit of pressure. The pascal is defined as one newton per square metre. The bar (symbol: bar), is equal to 100 000 pascals and is widely used in weather mapping.

The hertz (symbol: Hz), named after German physicist Heinrich Rudolf Hertz, is the metric unit of frequency. The hertz is defined as one cycle per second and is used for such things as radio channels, sound (pitch) and computer processors (clock speed where 1 ‘tick’ is a cycle).

Here are the diagrams of the links between metric units:

Simplified Diagram Full Diagram
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Here are the most common prefixes used in the metric system:

Prefix Symbol Multiple/Fraction Origin
giga G billion Greek “gigas”, meaning giant
mega M million Greek “megas”, meaning great
kilo k thousand Greek “khilioi”, meaning thousand
deci d tenth Latin “decimus”, meaning tenth
centi c hundredth Latin “centum”, meaning hundred
milli m thousandth Latin “mille”, meaning thousand
micro μ millionth Greek “micros”, meaning small
nano n billionth Greek “nanos”, meaning dwarf

Prefixes are combined with units to form multiples and fractions of units. The prefix tells you the multiple or fraction of the unit that the name represents. It is best to demonstrate this fact with common examples. Here are some common combinations of prefixes and units:

Name Symbol Meaning Common Usage
gigawatt GW a billion watts Power station output
gigahertz GHz a billion hertz Computer processor speed
megawatt MW a million watts Power station output
megahertz MHz a million hertz Computer processor speed
kilometre km a thousand metres Marathon distances, European road speeds and distances, etc.
kilogram kg a thousand grams Goods sold by weight, heavy objects, etc.
kilojoule kJ a thousand joules Food nutrition labels
kilowatt kW a thousand watts Electrical heaters, consumer and business power consumption, boilers, car engines, etc.
decilitre dl or dL a tenth of a litre Blood sugar levels
centimetre cm a hundredth of a metre DIY products, furniture, etc.
centilitre cl or cL a hundredth of a litre Wine bottles
millimetre mm a thousandth of a metre Construction industry, dimensions of small objects, etc.
millilitre ml or mL a thousandth of a litre Drinks
milligram mg a thousandth of a gram Drug doses, drink driving tests, blood sugar levels, etc.
millibar mbar a thousandth of a bar Weather mapping reports
micrometre μm a millionth of a metre Precision engineering, paper thickness
microgram μg a millionth of a gram Very small drug doses
nanometre nm a billionth of a metre Widths of transistors on computer chips

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(1) Reproduced with kind permission of the US Metric Association (USMA). Website: usma.org

(2) Reproduced from the NIST website: www.nist.gov

Sources for etymologies of units: Oxford English Dictionary, Wikipedia

This article uses the term “billion” in the sense of 1000 million. This is the normal meaning of the word as used in the media.

## 13 thoughts on “How metric units fit together”

1. enigmascience says:

Ronnie’s review of the metric system was excellent and comprehensive. However, I think it is worth pointing out that the modern SI is more philosophically based which to my mind makes the SI one of the most remarkable of human cultural achievements. The modern SI is based on “The seven constants”. These are only well understood with an advanced understanding of Modern Physics. It does mean for example that the Ampere is no longer defined in terms of the force between two wires (which was never practical anyway). Also, you didn’t mention the definition of the second, but with the 7 constants, this is actually a definition of the frequency of an electronic transition, so is actually now in Hz.

This whole new infrastructure is based on the constants being universal. In other words we could (in principle!) instruct an alien in another part of the universe how many electrons per second constitute an Ampere and the alien would then know exactly how big our Ampere was. Similarly the prototype kilogram no longer exists as it is replaced by combinations of the seven constants. On a similar note, the USMA poster is not quite correct in that it suggests that Celsius is part of the SI (although the poster is a bit vague on that point). However you look at it, the old idea of using “fixed points” like the melting of ice and the boiling of water is no longer strictly valid. Rather temperature is a measure of microscopic energy via the Kelvin scale, with Celsius as a derived scale. I apologise for being a bit of a nit picker, but without the 7 constants the SI units could in principle drift over the centuries as indeed has happened it seems to the former prototype kilogram since 1900.

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2. m says:

I like the choice of the term “nit picker”. The nit is a non-SI name for the SI unit candela per square metre.
It’s worth remembering that the ampere, like all SI unit names, should be written without a capital letter, apart from when it occurs at the beginning of a sentence. The exception being the degree Celsius.
The degree Celsius is certainly part of the SI. It is one of 22 derived units that have special names in the SI. (see Table 4 in the 9th edition of the SI Brochure).
When used as a difference between two temperatures, the degree Celsius is equal in value to the kelvin. When it is used as a value on a temperature scale, its value is calculated by subtracting 273.15 from the thermodynamic temperature measured in kelvins.
On the subject of non-SI units, it’s worth pointing out that the bar (and millibar mentioned in the article) is not an SI unit. The SI unit for pressure is the pascal. Weather forecasts should be using the hectopascal, symbol hPa, instead of the millibar (which is now obsolete).

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3. Daniel says:

enigmascience,
With the changes that took place in 2019, would the seven base units be the same as before? Technically, is the kilogram (mass) a base unit, since the kilogram is defined from energy/power?
Base units to me would be units that are directly defined from natural constants and not other units, even if the natural constants involve another unit. For example, the second is defined from the transitions of the caesium atom. The meter, also a base unit since it is defined from the speed of light, but the speed of light relies on the definition of the second.
The ampere can no longer be a base unit since it is not defined from a natural constant but from the coulomb and the second, both being defined from natural constants.
The kilogram, being defined from the watt-balance equates electrical energy to mechanical energy such that 1 N.m/s = 1 V.A = 1 V.C/s. The newton becomes a derived unit defined from the coulomb, the metre, the second and the volt. The kilogram is then defined from the newton in the relation that 1 N = 1 kg.m/s^2 or 1 kg = 1 N.s^2/m. Thus the kilogram is a derived unit, not a base unit.
The volt becomes a base unit defined from the Josephson voltage standard, which depend on fixed constants for the elemental charge e and Planck’s constant h.
Thus the new seven base units become the candela, the coulomb, the kelvin, the metre, the mole, the second, and the volt.

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4. Daniel says:

m,
Weather forecasts should be expressed in kilopascals, not hectopascals. Hectopascals are a lazy way out of not getting rid of the millibar where the millibar and the hectopascal are exactly the same. Metrication isn’t just about converting from imperial/USC to SI, it is also about converting from old metric to SI. Millibars are old metric units that need to be deprecated. Renaming them to hectopascals is not moving to SI where in technical areas the prefix hecto is not encouraged.

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5. m says:

@Daniel
You are right that, following the new definitions, the kilogram is now a derived unit, in the sense that all base units are now derived from the seven SI defining constants. However, you can’t just unilaterally decide that the kilogram is no longer a base unit, and that there are now seven new base units.
The SI Brochure has this to say about the current status of base units:
“2.3 Definitions of the SI units
Prior to the definitions adopted in 2018, the SI was defined through seven base units from which the derived units were constructed as products of powers of the base units. Defining the SI by fixing the numerical values of seven defining constants has the effect that this
distinction is, in principle, not needed, since all units, base as well as derived units, may be constructed directly from the defining constants. Nevertheless, the concept of base and derived units is maintained because it is useful and historically well established”.

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6. enigmascience says:

m,
You are right about my incorrect capitalisation of the A in ampere. I should know the rule by now, but I seem to keep forgetting. Regarding hPa in weather forecasting, I was working in the Met. office a few decades ago when the transition occurred between mbar (or the incorrect symbol mb was preferred). I remember groaning at the time because kPa was not the preferred unit. I don’t know how the transition occurred but you must appreciate that weather forecasting is possibly one of the best examples of international cooperation in science. Decisions would have been needed to have been made at World Meteorological Organisation level, and the UK Meteorological Office would have been just one vote. If the symbols mb or mbar still exist today that is not the fault of the Met. Office!

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7. enigmascience says:

Daniel,
“You raise interesting points. The base units of the SI remain unchanged. In other words, the kilogram is still a base unit even though it is determined using indirect methods. Note that in principle both the metre and the ampere are similar in the sense that both are defined by a constant multiplied or divided by a time. So, if you admit that the metre is a base unit, then so still is the ampere. Respondent “m” has in fact clarified the matter. I don’t think the world is ready to reclassify the volt as a base unit instead of the kilogram. In Britain most people haven’t yet given up on stones! The point of the seven constants is to put the SI on a rigorous mathematical level. The constants have been chosen so that nobody can measure the difference between the old and new definitions. However, I do believe the second is ripe for redefinition!”

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8. Martin Vlietstra says:

It appears that the bar and mm Hg were phased out in the 9th edition of the SI Brochure. In the 8th edition, they were listed as units that could be used alongside SI.

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9. Ezra Steinberg says:

Martin,
When did the SI ditch the “micron”?

Sadly, I still see it often used in scientific newsletters (like Physics World) and in more general publications on science topics instead of “micrometer”. Very frustrating (on a par with “Centigrade” being used instead of “Celsius” in my view).

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10. Martin Vlietstra says:

@Ezra

The micron, with symbol μ, was approved by the CIPM in 1879. The GCPM revoked the decision in 1967. This is detailed in the 1st edition of the SI Brochure.

If anybody is interested to looking at back copies of the SI brochure, I suggest visiting the the article “Brochure_sur_le_SI” in the French version of Wikipedia (https://fr.wikipedia.org/wiki/Brochure_sur_le_SI). That article catalogs the original versions of the SI Brochure (in French) and also the NIST translations into English.

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