The International System of Units (SI)

The International System of Units (SI)

Officially known as the International System of Units (SI), the metric system is the international standard system of measurement units. It is based on the standard decimal number system, and is designed to be easy to learn, and simple to use.

In everyday use, it is used to measure road distances and speeds, floor areas, storage volumes, energy consumption, and the mass and volumes of food and drink.

It is also used to measure temperature, electricity and the brightness of light bulbs. It is the standard system of measurement for international trade.

However, its application is not restricted to everyday use. Since its original inception, the metric system has evolved to become a single coherent system used for measurement in all fields of human endeavour, including science, medicine, technology, industry, commerce and sport.

Units

Units of measurement in the metric system relate to each other in a logical and coherent manner. Each quantity has one unit to measure it. This, together with the use of subunits based on multiples and submultiples of 10, make all calculations using metric units as straight forward as any other calculation using decimal numbers.

Base units

The metric system consists of a set of seven base units, for quantities including length, time and mass. All seven base units are defined from a set of seven constants which, in the SI, have fixed numerical values. Each defining constant is either an unvarying property of nature, or a technical constant.

Base units
Name Symbol Quantity Base unit
kilogram kg mass
The kilogram was originally defined as the mass of one cubic decimetre, or one litre, of water.
metre m length
The metre was originally defined as 1⁄10 000 000 of the distance from the North pole to the Equator.
second s time
The second was originally defined as 1⁄86 400 of a day.
ampere A electric current
kelvin K temperature
A temperature difference of one kelvin is equal to one degree Celsius. The theoretically coldest temperature is zero kelvins, or -273.15°C
mole mol amount of substance
The mole was originally defined as the number of atoms in 12 grams of carbon-12.
candela cd light intensity

Derived units

Units for all other quantities are called derived units, and are defined as products of powers of one or more of the base units.

Examples of derived units
Name Symbol Quantity Base units
square metre m2 area
The square metre is the SI coherent derived unit of area.
cubic metre m3 volume
The cubic metre is the SI coherent derived unit of volume.
metre per second m/s speed
The metre per second is the SI coherent derived unit of speed.

Derived units with special names and symbols

To simplify their expression, some derived units have been given special names and symbols.

Examples of derived units with special names and symbols
Name Symbol Quantity Base units
newton N force kg m s-2
The newton is the special name for the kilogram metre per second squared, symbol kg m s-2.
joule J energy kg m2 s-2
The joule is the special name for the kilogram metre squared per second squared, symbol kg m2 s-2.
watt W power kg m2 s-3
The watt is the special name for the kilogram metre squared per second cubed, symbol kg m2 s-3.
degree Celsius °C Celsius temperature K
The degree Celsius is the special name for the kelvin, when using the Celsius temperature scale.

Prefixes

The use of prefixes to denote decimal multiples and sub-multiples means that only one unit is required for any given quantity.

Common prefixes
Prefix Symbol Value
giga G 109 1 000 000 000
mega M 106 1 000 000
kilo k 103 1000
hecto  h 102 100
deca da 101 10
100 1
deci d 10-1   0.1
centi c 10-2   0.01
milli m 10-3   0.001
micro µ 10-6   0.000 001
nano n 10-9   0.000 000 001

Subunits

Subunits can be constructed for any unit by combining any one of the standard set of prefixes with the unit name.

Examples of subunits
Prefix + Unit = Subunit Value
milli gram milligram 0.001 grams
milli metre millimetre 0.001 metres
kilo metre kilometre 1000 metres
mega watt megawatt 1 000 000 watts

e.g. 10 000 metres can be expressed as 10 kilometres by combining the prefix ‘kilo’ (meaning 1000) with the unit name ‘metre’.

Symbols

In addition to names, each unit and prefix in the metric system has a unique symbol. Unlike abbreviations, symbols are independent of language and alphabet, and can be universally understood. Symbols are case sensitive.

Symbols for prefixes and units are combined in the same way as their names.

Examples of subunit symbols
Prefix + Unit = Subunit Value
m g mg 0.001 g
m m mm 0.001 m
k m km 1000 m
M W MW 1 000 000 W

e.g. 10 000 m can be expressed as 10 km by combining the prefix symbol k (meaning 1000) with the unit symbol m.

Properties

The International System of Units (SI) is designed to be easy to use and widely applicable. It includes a number of desirable properties:

Units based on nature

Units in the SI are based on unchanging properties of the natural world. Originally the metre was based on the distance from the Earth’s North Pole to its equator, and the kilogram was based on the mass of water contained by a volume of 1⁄1000 of a cubic metre. Modern definitions of all SI units are more precise, but each unit is still defined in terms of one or more unchanging properties of nature, such as the speed of light in vacuum, or the charge on an electron.

Decimal system

The SI is a decimal measurement system. Calculations involving SI units are as straight forward as any other calculation using the standard decimal number system.

Standard symbols

All SI units have universally understood standard symbols. Any quantity expressed using SI units can be understood independently of language.

Rational system

The SI is a rational measurement system. Each physical quantity has only one SI unit. For example, length is measured using the metre, and time is measured using the second.

Prefixes and subunits

The SI defines a set of standard prefixes to denote decimal multiples and sub-multiples. Any SI prefix can be combined with any SI unit to form a subunit. The use of subunits provides flexibility and convenience when expressing quantities of different orders of magnitude in everyday use. For example; the use of the kilometre for road distances, and the millimetre for architectural plans.

Coherence

The SI is a coherent measurement system. A coherent system of units is a system of units based on a system of quantities in such a way that the equations between the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations between the quantities. 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.

Length, area and volume

Everyday units

The simplicity and logic of the metric system has led it to become the universal measurement system for everyday use all over the world. Factors contributing to its success include:

  • It is based on the decimal number system.
  • Only one unit is needed for each quantity such as length.
  • When dealing with small quantities and large quantities, there is no need for different units or difficult-to-learn conversion factors.
  • Large quantities and small quantities are handled simply by shifting the decimal place and adding a corresponding prefix to the unit name or symbol.

All metric measurements scale seemlessly from the very small, to the very large. Using metric units, it is easy to see the relative sizes of things, which in turn enhances our understanding of our surroundings. For instance, it is easy to see that 6 kilometres is ten times as far as 600 metres, whereas it is not immediately apparent how many miles a distance ten times as far as 600 yards would be.

Thinking in metric

When encountering metric units in everyday situations for the first time, those of us that have grown up in one of the few places on Earth that have yet to fully adopt the metric system can have a tendency to want to convert them into non-metric units that may be more familiar. This involves the use of mental arithmetic and conversion factors. Faced with learning a host of new conversion factors, newcomers to everyday metric units can easily be put off metric completely, which is unfortunate because, when used exclusively, the metric system completely removes the need for all conversion factors.

Learning to think in metric is actually much simpler than converting to non-metric units, and is ultimately far more rewarding. In place of conversion factors, the metric system only requires the learning of a handful of prefix names, and the multiples of ten that they represent. The full benefits of the metric system can only really be appreciated after learning to think exclusively in metric.

One can start thinking in metric, by learning the sizes and weights of various familiar objects as reference points:

  • For example, a compact disc has a diameter of 12 cm, and a thickness of 1.2 mm, and one lap of an Olympic athletics track is 400 m.
  • Similarly, a litre-carton of fruit juice has a mass of about 1 kilogram, most new born babies weigh between 2.5 kg and 4 kg, an average man’s weight is about 80 kg, and the mass of a small car is about 1000 kg, or 1 tonne.
  • Investing in a metric-only tape measure, to measure personal height, and the size of familiar rooms, together with using metric scales to measure body weight, are also good ways to start thinking in metric.

Length

The SI unit of length is the metre, symbol m.

Commonly used subunits include the millimetre, symbol mm, centimetre, symbol cm, and kilometre, symbol km.

The metre was originally defined as being equal to one ten-millionth of the distance from the equator to the North Pole.

The modern definition of the metre is more precise. However, for practical purposes, the distance from the equator to the North Pole remains approximately 10 000 000 metres, or 10 000 kilometres.

100 centimetres = 1 metre
1000 millimetres = 1 metre
1000 metres = 1 kilometre

Orders of magnitude

 1000 nm = 1 μm = 0.000 001 m = 10-6 m 
 1000 μm = 1 mm = 0.001 m = 10-3 m 
 1000 mm = 1 m = 1 m = 100 m 
 1000 m = 1 km = 1000 m = 103 m 
 1000 km = 1 Mm = 1 000 000 m = 106 m 

 

Area

The SI unit of area is the square metre, symbol m2.

Commonly used subunits include the square centimetre, symbol cm2, and square kilometre, symbol km2.

The hectare, symbol ha, is the special name for the square hectometre, symbol hm2. 1 hectare is equal to 10 000 square metres. Prefixes must not be used with the hectare.

 10 000 square centimetres = 1 square metre
10 000 square metres = 1 hectare
100 hectares = 1 square kilometre

Orders of magnitude

 100 mm2 = 1 cm2 = 0.0001 m2 = 10-4 m2 
 100 cm2 = 1 dm2 = 0.01 m2 = 10-2 m2 
 100 dm2 = 1 m2 = 1 m2 = 100 m2 
 100 m2 = 1 dam2 = 100 m2 = 102 m2 
 100 dam2 = 1 hm2 = 10 000 m2 = 104 m2 
 100 hm2 = 1 km2 = 1 000 000 m2 = 106 m2 

 

Volume

The SI unit of volume is the cubic metre, symbol m3.

The litre, symbol L or l, is the special name for the cubic decimetre, symbol dm3. 1 litre is equal to one thousandth of a cubic metre. It follows that 1 millilitre, symbol mL or ml, is equal to 1 cubic centimetre.

In everyday use, the millilitre and litre are the most commonly used subunits to measure volume.

1000 millilitres = 1 litre
1000 litres = 1 cubic metre

Orders of magnitude

 1000 mm3 =  1 cm3 =  1000 µL =  1 mL = 10-6 m3 
 1000 cm3 =  1 dm3 =  1000 mL =  1 L = 10-3 m3 
 1000 dm3 =  1 m3 =  1000 L =  1 kL = 100 m3 
 1000 m3 =  1 dam3 =  1000 kL =  1 ML = 103 m3 
 1000 dam3 =  1 hm3 =  1000 ML =  1 GL = 106 m3 
 1000 hm3 =  1 km3 =  1000 GL =  1 TL = 109 m3 

Mass and weight

Mass

The SI unit of mass is the kilogram, symbol kg.

The kilogram was originally defined as being equal to the mass of one cubic decimetre of water.

The modern definition of the kilogram is more precise. However, for most practical purposes, one litre of water can be regarded as having a mass of one kilogram.

The tonne, symbol t, is the special name for the megagram, symbol Mg. It is equal to 1000 kilograms. One cubic metre of water has a mass of one tonne.

1000 milligrams = 1 gram
1000 grams = 1 kilogram
1000 kilograms = 1 tonne

For historical reasons, the name of the base unit of mass includes the prefix kilo. The multiples and submultiples of the kilogram are formed by attaching prefix names to the unit name “gram”, and prefix symbols to the unit symbol “g”. Thus 10-6 kg is written as milligram, symbol mg, and not as microkilogram, µkg.

Orders of magnitude

 1000 μg = 1 mg = 0.000 001 kg = 10-6 kg 
 1000 mg = 1 g = 0.001 kg = 10-3 kg 
 1000 g = 1 kg = 1 kg = 100 kg 
 1000 kg = 1 Mg = 1000 kg = 103 kg 
 1000 Mg = 1 Gg = 1 000 000 kg = 106 kg 

Weight

When the word “weight” is used, the intended meaning should be clear. In science and technology, weight is a force, for which the SI unit is the newton. However, in commerce and everyday use, weight is usually a synonym for mass, for which the SI unit is the kilogram.

Temperature

Temperature

The SI unit of temperature is the kelvin, symbol K.

The degree Celsius, symbol °C, is the special name for the kelvin. For everyday use, the degree Celsius is the standard unit used to measure temperature.

Both the kelvin and the degree Celsius can refer either to a temperature difference, or to a distinct point on a temperature scale.

  • When used to refer to a temperature difference, the kelvin and the degree Celsius are the same size; the numerical value expressed in degrees Celsius is equal to the numerical value expressed in kelvins.
     
  • When used to refer to a point on a temperature scale, the kelvin and degree Celsius values are different. The Celsius and Kelvin scales have different zero points. The Kelvin scale has absolute zero as its zero point, whereas the Celsius scale has the freezing point of water as its zero point.
 Temperature kelvins degrees Celsius 
 boiling point of water 373.15 K 100 °C 
 normal body temperature 310.15 K 37 °C 
 freezing point of water 273.15 K 0 °C 
 absolute zero 0 K -273.15 °C 

The Celsius scale

The Celsius temperature scale was originally defined by two fixed points at standard atmospheric pressure: the freezing point of water, and the boiling point of water. The size of one degree Celsius was defined as 1⁄100 of the difference in temperature between these two points.

The modern definition of the degree Celsius is more precise, but for most practical purposes 0 °C is the temperature at which water freezes, and 100 °C is the temperature at which water boils.

Celsius is the international standard temperature scale for weather reporting and weather forecasts.

desert heat 40 – 50 °C
extreme heat 35 – 40 °C
very hot 30 – 35 °C
hot 25 – 30 °C
warm 20 – 25 °C
mild 15 – 20 °C
cool 10 – 15 °C
chilly 5 – 10 °C
cold 0 – 5 °C
freezing cold -10 – 0 °C
bitter cold -20 – -10 °C
extreme cold -30 – -20 °C

The degree Celsius is used in central heating thermostats, domestic ovens, refrigeration equipment and clinical thermometers.

Usage

Note that the symbol for degree Celsius should always include the ° sign.
i.e. °C, and never just C.

Other units

Time

The SI unit of time is the second, symbol s.

1000 milliseconds = 1 second
1000 seconds = 1 kilosecond
 
Three non-SI units of time; the minute, hour and day are accepted for use with the metric system.
 
60 seconds = 1 minute
 3600 seconds = 1 hour
 86 400 seconds = 1 day

1 \ \text{kilosecond} \mspace{12mu} = 16 \ \text{min} \ 40 \ \text{s}\\ \\ 1 \ \text{megasecond} = 11 \ \text{d} \ 13 \ \text{h} \ 46 \ \text{min} \ 40 \ \text{s}

 
SI prefixes are not used with the non-SI units of time.
 

Speed

The SI unit of speed is the metre per second, symbol m/s.

The kilometre per hour, symbol km/h, is not an SI unit, but is accepted for use with the metric system.

 3.6 kilometres per hour =  1 metre per second

 

Force

The SI unit of force, or thrust, is the newton, symbol N.

One newton is equal to the force needed to accelerate a mass of one kilogram at the rate of one metre per second squared in the direction of the applied force.

1 \ \text{N} = 1 \ \text{kg} \ \text{m} \ \text{s}^{-2}

 

Pressure

The SI unit of pressure is the pascal, symbol Pa.

One pascal is equal to the pressure exerted by a perpendicular force of one newton on an area of one square metre.

1\ \text{Pa} = 1\ \text{N/m}^{2}

Atmospheric pressure is approximately 101 325 Pa, or 1013 hPa.
 

Energy

The SI unit of energy is the joule, symbol J. It is used to measure energy in all its forms; potential, kinetic, mechanical, electrical, chemical, heat, etc.

One joule is equal to the energy transferred to an object when a force of one newton acts on it in the direction of its motion through a distance of one metre.

1\ \text{J} = 1\ \text{N m}

One joule is equal to the energy required to accelerate a mass of one kilogram at a rate of one metre per second squared through a distance of one metre.

1 \ \text{J} = 1 \ \text{kg} \ \text{m}^{2} \ \text{s}^{-2}

One joule is equal to the energy required to move an electric charge of one coulomb through an electrical potential difference of one volt.

1 \ \text{J} = 1 \ \text{C} \ \text{V}

One joule is equal to the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second.

1 \ \text{J} = 1 \ \text{A}^{2} \ \Omega \ \text{s}

One joule is equal to the energy required to produce one watt of power for one second, or one watt second, symbol W s.

1 \ \text{J} = 1 \ \text{W} \ \text{s}

It follows that, at a constant power of one kilowatt (kW), the amount of energy transferred in 3600 seconds, or one hour (h), is equal to 3.6 megajoules (MJ), or one kilowatt hour (kW h).

The kilowatt hour is a non-SI unit of energy commonly used in household energy bills.
 

Power

The SI unit of power is the watt, symbol W. It is used to measure power in all its forms; mechanical, electrical, etc.

One watt is equal to the power required to transfer energy, or do work, at a rate of one joule per second.

1 \ \text{W} = 1 \ \text{J/s}

e.g. A 2 kilowatt electric fire dissipates heat at a rate of 2 kilojoules per second.

One watt is equal to the power required, or the rate at which work is done, to hold an object at a constant velocity against a constant opposing force of one newton.

One watt is equal to the power required, or the rate at which electrical work is done, when a current of one ampere flows across an electrical potential difference of one volt.

1 \ \text{W} = 1 \ \text{A} \ \text{V}

One watt is equal to the power required when energy is dissipated as heat when an electric current of one ampere passes through a resistance of one ohm.

1 \ \text{W} = 1 \ \text{A}^{2} \ \Omega

 

Electricity

The four most common units for measuring electricity are the ampere, the volt, the ohm and the watt:

Current

The SI unit of electric current is the ampere, symbol A. The ampere is one of the seven SI base units.

One ampere is equal to the electric current in a conductor when charge passes through it at a rate of one coulomb per second.

1 \ \text{A} = 1 \ \text{C/s}

1000 milliamperes = 1 ampere
1000 amperes = 1 kiloampere

Voltage

The SI unit of electric potential difference, or voltage, is the volt, symbol V.

One volt is equal to the difference in electric potential between two points of a conducting wire when an electric current of one ampere dissipates one watt of power between those two points.

1 \ \text{V} = 1 \ \text{J/C}

Resistance

The SI unit of electric resistance is the ohm, symbol Ω.

One ohm is equal to the electrical resistance between two points of a conductor when a constant potential difference of one volt, applied to these points, produces in the conductor a current of one ampere.

1 \ \Omega = 1 \ \text{V/A}

 

Light

The SI unit of luminous flux is the lumen, symbol lm.

Luminous flux is a measure of the total quantity of visible light emitted by a source per unit time.

The brightness of domestic light bulbs is measured in lumens.

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