Information about all additional units of the metric system.

 

These units are not, strictly speaking, part of the modern metric system, but the International Committee for Weights and Measures (CIPM) has decided to define these units officially in terms of SI basic units and classified these units as “to be maintained”. All the units below are accepted for use with the metric system, although they lack the coherency of the other units and do require conversion factors to be converted into base units.

Some units like liter or tonne have been an important part of the metric system when it was created, but have fallen short of the consistency required by the modern metric system. However, the units listed below were kept and can be used just like any other metric measure.

 

Go to: bel, day, degree of arc, hour, liter, minute, minute of arc, neper, second of arc, tonne.

 

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Bel is a measure for logarithmic quantities and is used to describe a ratio. Strictly speaking, the bel is not a measure, because it does not measure a physical quantity. Instead it describes the ratio of a physical measurement (which can be anything, watts, pascals, watts per square meter, etc.) to a set starting point of the bel scale. Bels are most frequently used to measure sound or noise and attenuation (the weakening or reduction of a signal) in radio engineering.

What is a logarithmic quantity?

Normally, in a series of numbers, a doubling of numbers indicates that the quantity has doubled (4 is twice as much as 2).

This is not the case in logarithmic tables. A logarithmic table has a defined base number, for example 10. Numbers in this logarithmic scale are then used as exponents of that base number. An exponent is a number that indicates what power of a base number is to be taken. Exponents are written in superscript to the left of the base number (such as 10¹⁷) and indicate how often the base number should be multiplied by itself. For example: 2³ = 2 × 2 × 2 = 8. 10⁴ = 10 × 10 × 10 × 10 = 1 000.

Exponents are used by mathematicians instead of writing very large numbers. To write one septillion, we usually would write down a one, followed by 24 zeros: 1 000 000 000 000 000 000 000 000. This number is equal to multiplying the number 10 with itself twenty-four times. We can thus write 10²⁴. The larger or the smaller numbers get, the more frequently exponents are used. Nobody wants to write out a number containing one hundred zeros.

The following is a sample of the table for the common logarithm (base 10):

   Exponent (base 10)  Calculation     Resulting number

   0          10⁰      1

   1          10¹      10

   2          10²      100

   3          10³      1 000

   4          10⁴      10 000

   5          10⁵      100 000

   6          10⁶      1 000 000

   7          10⁷      10 000 000

As the table shows, an increase of the exponent number by 1 makes the result ten times bigger. Changing the exponent from 2 to 4 does not double the resulting number, but increases it 100-fold! Logarithmic scales can be deceiving, as small changes can quickly lead to enormous changes in the resulting number.

As the bel is based on the common logarithm (base 10), a measure of 2 bel is 10 times 1 bel. The difference between 10 bel and 0 bel is ten billion. It is easiest to remember that +1 bel means a tenfold increase. Bels are used with measures that cover such a large range of numbers or increase so quickly that a logarithmic scale is used to keep the numbers small and manageable.

How is the bel used?

A bel itself does not measure anything; it is only a logarithmic quantity. The physical quantity actually measured must always be stated when bel is used. In addition, the starting point of the bel scale (which is 0 bel) can be set to different magnitudes. In the logarithmic scale, 0 as an exponent gives the number 1 as result. This “1” can be defined to equal any measurement. As mentioned before, bel describes a ratio. This is the ratio between the starting point of the scale (0 B) and the actual measurement.

Example: If bels are used to measure watts, the starting point of the scale (0 B) could be set to 100 watts. A measurement of 10 000 watts is a hundred times larger and therefore equals 2 bels on this scale. Such a measurement would be written as: 2 B (re 100 W). This tells the reader what physical quantity was measured and how the starting point of the scale is defined.

A measurement in bels without this additional information is almost useless.

 

Decibels are more common

By custom, the measure used is usually not bels, but decibels. One decibel is 1/10 of a bel, 10 decibels are 1 bel. This usually allows one to work with whole numbers without decimals.

Using decibels instead of bels means that an increase of 10, not 1, increases the measure ten times. Example: 10 decibels = 10; 20 decibels = 100; 30 decibels = 1 000; 40 decibels = 10 000; 50 decibels = 100 000; etc.). The difference between 0 decibel and 100 decibels is 10 billion.

 

Measuring sound with bels

Bel is most often used to measure sound. Sound is basically air waves. If the sound waves follow each other in quick succession, the noise sounds high (high frequency), if they follow each other more slowly, the noise sounds low (low frequency). The higher the power or the pressure of the sound waves, the louder the noise sounds.

Sound intensity, which is the power of sound per area, is measured in watt per square meter (W/m²). The higher this measure, the louder the sound appears to the human ear. However, human sound perception is not linear. Humans can hear an amazing range of sound intensities. If a sound appears twice as loud to a human, the sound intensity in watt per square meter has usually increased ten times. This is why bels are used to measure sound.

Other physical quantities of sound that can be measured, are sound pressure (the pressure exerted by the air waves that make up sound) and sound power. Sound pressure is measured in pascals; sound power is measured in watts.

What measure of sound is used for what purpose?

Sound power (in watts) is used to measure the total sound emissions of a sound source (for example a dishwasher or vacuum cleaner). Sound intensity (in watts per square meter) measures how powerful a sound is in a given region and best describes the loudness of a sound. When a sound intensity originating from a single source is measured, the distance from that source should be stated together with the sound intensity measurement.

How to define the starting point of the bel scale when measuring sound

The beginning of the scale (0 decibels) can be set to different points. Usually, when sound intensity is measured, the scale starts at one picowatt per square meter, since this is approximately the faintest sound humans can hear (0 dB = 1 pW/m²). This means that 10 dB correspond to 10 pW/m², 20 dB = 100 pW/m²; and 100 dB = 10 mW/m².

The starting point of the decibel scale for measuring sound pressure is usually 20 micropascals (20 μPa) and for sound power 1 picowatt (1 pW).

As mentioned before, the quantity measured and the starting point of the scale have to be stated when using bels. A decibel measurement starting at 1 pW is usually written as: number dB (re 1pW).

Taking the human factor into account when measuring sound

The human ear does not hear all frequencies with the same sensitivity. Some frequencies sound louder, some frequencies sound quieter – even when the sound power level or sound pressure is identical.

To accommodate for this, so-called sound filters are often used when measuring sound. They take the mix of different frequencies of a sound into account and, using the correct sensitivities of the human ear, calculate the resulting loudness a human would hear. Three such filters, called A, B and C, are in use. The most widely used filter is the A weighting scale. The dB (A) scale is easy to measure, but it still does not respond in quite the same way the human ear does, meaning that the measured loudness and the perceived loudness might still differ to a very small degree. The less often used C scale is suitable for measuring very loud sound levels only; the B scale is intermediate between A and C. When such a weighting scale is used, it is customary to quote the scale in brackets after the dB symbol: dB (A).

 

Bel and neper: Common and natural logarithms

The bel is closely related to the other metric unit for logarithmic quantities, the neper (Np). The neper is based on exactly the same principle, but instead of using the common logarithm (base 10), utilizes the natural logarithm. The natural logarithm is based on the so-called Euler number (symbol e; ≈ 2.718 281 828).

This Euler number is important in many areas of mathematics. It is a so-called surd, which is the name for an infinitely long number that cannot be expressed in finite terms of ordinary numbers or as a fraction.

When used with the unit neper, a number describes the exponent of the Euler number (that is how often the Euler number must be multiplied by itself), just as a number used with the unit bel specifies how often the number 10 should be multiplied by itself.

The formula for converting bels into nepers uses the natural logarithm (symbol ln) and is:

1 B = 0.5 × (ln 10) Np or 1 B ≈ 1.151 282 Np

To convert nepers into bels:

1 Np = 2 / (ln 10) B or 1 Np ≈ 0.868 588 B

The bel is far more common than the neper.

 

 

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Being the oldest unit of time, the day used to be the fundamental unit of time, from which all other units of time were derived. Naturally, the day used to be defined as a mean solar day – the time period the sun takes to appear at the same position in the sky each day. Since the duration of the solar day varies, this definition of the day was not precise enough for modern needs. The standard day was then defined as the length of 1 January 1900, but this definition also proved to be problematic.

The most modern definition uses the second as the base unit for time instead of the day. The second is now defined as the period of time it takes for a certain atom to fluctuate a certain number of times. Other units of time, like minute, hour and day are then defined as multiples of the second.

 

The day is therefore defined as exactly 86 400 seconds (= 60 seconds × 60 minutes × 24 hours). This definition means that our units of time do not change in length anymore; each day has exactly the same duration. However, since the length of the solar day fluctuates, it sometimes becomes necessary to change the length of a particular calendar day by inserting or deducting a leap second. This practice ensures that our calendar stays aligned with the movements of the earth and the sun. The International Earth Rotation Service in Paris measures the exact length of the solar days and is responsible for calculating leap seconds. When necessary, a leap second is inserted into the Coordinated Universal Time (UTC), which is maintained by the BIPM (International Bureau of Weights and Measures). The UTC, which replaced the older Greenwich Mean Time (GMT), is the basis for worldwide time keeping. The official UTC time is displayed on the BIPM website.

 

Due the practice of inserting leap seconds, a particular calendar day can therefore have 86 399, 86 400 or 86 401 seconds. However, the insertion of leap seconds is very rare.

 

 

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One full circle is divided into 360 degrees of arc. One half circle is therefore 180°, one quarter circle is 90°.

This convention of dividing the circle into 360 degrees probably comes from the ancient Egyptians, who based it on even older knowledge from the Babylonians (2000 BC) and Sumerians (3000 BC). Their numbering system was mainly based on the number 60 (sexagesimal system), not on the number 10 (decimal), like most numbering systems since then. However, the reason for the choice of 360 is not exactly known. Perhaps it was chosen to because 360 was a convenient multiple of both 60 and 12, the number of the signs in the zodiac or because it comes close to 365, the number of days in a year.

The small circle “°” we use to designate degrees is probably an Egyptian hieroglyphic, perhaps representing the sun, meaning “day”.

It was probably the Greek astronomer and mathematician Ptolemy who divided the degrees of a circle into 60 minutes of arc and the minutes further into 60 seconds. In doing so, he was paying tribute to the old Babylonian and Sumerian system.

The official unit for measuring plane angles is the radian, and degrees of arc are defined in terms of the radian. One degree of arc equals (2 × p / 360) radian (about 0.017 453 rad). One radian equals 180 / π° (about 57.296°). When using the radian as a measurement for angles, the number π is eliminated from calculations.

In the metric system, one degree of arc can be subdivided into 60 minutes of arc or 3 600 seconds of arc. Degrees of arc are also increasingly subdivided decimally, following a recommendation of the International Standards Organization (ISO).

Some scientists have suggested dividing a circle into 400 degrees instead, but this practice has not caught on and was never implemented into the metric system.

 

The correct symbol for degree of arc is the degree sign, not a superscript zero or letter o. In the Unicode character encoding table, the degree sign is number U+00B0. Please note that, as an exception to the rule, the symbol for degree of arc follows the preceding number without space (50°, not 50 °). For more information about special characters please look at the page of rules.

 

 

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Today, like all units of time, the hour is defined in terms of the base unit of time, the second. One hour is 3 600 seconds (60 seconds × 60 minutes), 60 minutes or 1/24 of day.

The unit day, which is probably the oldest unit of time, comes naturally with the earth rotating and the change between night and day resulting from it. The hour was then the oldest division of the day, giving smaller units of time.

The day used to be divided into one or two 12 hour-periods. The division of the day into 24 equal hours likely stems from the invention of the mechanical clock in the Middle Ages, when more precision for time keeping was needed. Another theory is that the ancient Greeks already used 24 equal hours.

Ancient Egyptians divided the day into two twelve hour periods. This system was probably inherited from the Mesopotamians who adopted it from the Babylonians (2000 BC) and Sumerians (3000 BC). While the 12 daylight hours were of equal lengths and the 12 night hours were of equal lengths, night and day hours differed in duration. The relation between night and day hours changed with the relation of the duration of daylight and night through the seasons. Daylight and night hours were only equal twice a year, at the equinoxes in March and September.

It is often assumed that the reason for having 12 or two times 12 hours in a day is that ancient cultures used a numbering system based on the number 12 (duodecimal system). However, this is not correct.

The Sumerian and Babylonian numbering system appears strange to our society, since it was based on the number 60 (called sexagesimal system), as opposed to our system based on 10 (decimal system). 60, of course, is a very large base number for a numbering system, and it would be difficult to create 60 different and unique symbols for numbers. That is why the Sumerians and Babylonians divided their number symbols into groups of ten, giving them in effect a mixture of a decimal and a true sexagesimal system. In their system, the number 12 had no special significance.

The roots for the division of the day in twelve hours are therefore not clear. One explanation is that the number 60 was divided by the number of then known planets (5), which gives the result 12. Another, simpler explanation is that the number 12 was chosen because of its easy division into parts of 6, 4, 3, and 2. The twelve signs of the zodiac could also have been a factor in choosing 12 hours.

The hour was further divided into smaller parts, the minute and the second, when the need for accuracy in timekeeping increased.

 

 

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The liter is used to measure volume or capacity. Another way of measuring volume is to use cubed forms of the meter, like cubic meter [m³] or cubic centimeter [cm³]. The liter was originally used for measuring volumes of liquid substances, and the cubic meter for dry substances. This difference has long gone, and both the cubic meter and the liter are used for measuring any kind of volume, no matter what the measured substance is.

The liter is defined as one cubic decimeter [dm³]. A cubic decimeter is the volume that a perfect cube with side lengths of one decimeter or ten centimeters occupies. One liter can therefore be easily visualised as 10 cm by 10 cm by 10 cm. This makes one milliliter equal to one cubic centimeter, which can again be easily visualised. One thousand liters are equal to one cubic meter.

Because of its definition, the liter is not a separate unit; it is just special name for the cubic decimeter.

Usually, volumes larger than 1 000 liters are measured in cubic metres, cubic kilometres etc., and volumes of 1 000 liters and less are measured in liters, milliliters, microliters etc. This is not a general rule, just a widely observed custom.

 

Definition and size of the liter changed in 1964

An earlier definition of the liter was the volume that one kilogram of water occupies (at its maximum density and standard atmospheric pressure). Technical difficulties arising from this earlier definition resulted in the liter being later defined as one cubic decimeter. After the prototype of the kilogram was made in the late 18^th century, it turned out to be about 0.002 8 % heavier than the mass of one cubic decimeter of water (either the kilogram was too heavy or the liter was too small). Since the liter was linked by definition to the kilogram prototype, it was then equal to 1.000 028 cubic decimeters.

This problem was finally solved by the 12^th CGPM in 1964 that redefined the liter as a special name for the cubic decimeter. Consequently, the volume of one liter changed from 1.000 028 dm³ to 1 dm³. Understandably, the base units kilogram and meter had to be kept constant at all times. That is why the liter is the only metric unit that has actually changed its magnitude since its first official definition. All other metric units have been kept as close to their original standards and definitions as possible. However, the change in the volume of the liter was only 0.002 8 % - much too small to have any impact on most measurements.

When the definition of the liter was changed in 1964, the CGPM recommended not using the liter for high-accuracy measurements. However, this change of the liter took place such a long time ago that the liter can be used just like any other measurement unit – today’s accuracy of the unit ‘liter’ is equal to the accuracy of all other metric units.

 

The liter makes calculations involving volumes and water easy

The earlier definition, linking the liter to the weight of water, results in a very practical effect: One liter of water weighs about one kilogram. This means that the weight of water and of many other substances, which have similar density, can be estimated very easily: A ten liter bucket of water weighs ten kilograms. A crate of wine with 12 bottles of 0.75 liters weighs a minimum of 9 kg. A 10 cm thick sheet of ice on a 6 meter by 3 meter garage roof weighs 1 800 kg.

Since the density of water varies depending on the temperature, the estimates are never absolutely correct. More information about the weight of water can be found in the table below.

 

Defining the liter as being equal to one cubic decimeter helps to calculate the volume of containers etc. If a box is about 10 cm by 10 cm by 30 cm, it should hold about 3 liters.

In meteorology, rainfall is usually measured in millimetres – the height the rain would reach on an even surface, if it would not drain off. One millimeter of rainfall is equal to one liter per square meter (or one million liters per square kilometer). Conveniently, the density of fresh snow is about 10 % of the density of water, so that one centimeter of snowfall equals one millimeter of rainfall or one liter of water per square meter.

 

The liter has two symbols – l and L

The symbol for the liter is a lowercase ‘l’. This conforms to the rule that symbols of units that have been named after people start with an uppercase letter (such as ‘A’ for ampere) and that symbols of generic units start with a lowercase letter (such as ‘m’ for meter).

However, some countries have argued that a lowercase ‘l’ could be easily confused with the number ‘1’. This problem arises with some computer fonts as well as with the custom of handwriting in many English-speaking countries where the number one lacks an upstroke and is written just as a vertical line.

That’s why several countries, including the USA, have adopted the uppercase ‘L’ as the legal symbol for the litre. Consequently, the 16^th CGPM in 1979 permitted the uppercase ‘L’ as an alternative symbol.

The liter is therefore the only measurement unit that has two official symbols, ‘l’ and ‘L’. However, the ISO prefers the lowercase ‘l’.

Please note that sometimes a script l (such as ℓ) is used to denote the unit liter. This is neither good practice nor part of the rules of the metric system. Naturally, the font and the appearance of the letters in a text are chosen by the author or typesetter – but there is no need to set metric unit symbols in a different font from the rest of a document. A script or handwritten ‘l’ should thus be only used when the rest the text is written in the same style.

Both the SI standard and the ISO recommend setting symbols for names and prefixes in upright or roman case only.

The page of rules has more information about the correct use of the name and symbols for the liter in the sections general rules, capitalization, spelling and special characters.

 

 

 

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The minute is defined in terms of the base unit of time, the second. One minute equals 60 seconds, 1/60 of an hour and 1/1440 of a day (24 hours × 60 minutes).

As explained in the entry under “hour”, the oldest unit of time, the day, was first subdivided into hours. Much later, when the need for more accuracy in timekeeping arose, the hour was further subdivided.

Perhaps in the 13th century, with the invention of accurate mechanical clocks, the hour was subdivided into minutes (from the Latin pars minuta prima, or "first small part") and seconds (partes minutae secundae).

The division of the hour into 60 minutes was a tribute to the old Babylonian sexagesimal system. The ancient Babylonians (around 2000 BC) and Sumerians (around 3000 BC) used a numbering system based on the number 60 (sexagesimal system) as opposed to our decimal system (based on the number 10). Later, it is most likely the Egyptians who invented the convention of dividing the circle in 360 degrees. The Greek mathematician and astronomer Ptolemy worked with these degrees of arc and further divided the degrees into 60 minutes of arc, with each minute containing 60 seconds of arc.

Medieval chronologists then divided the hour into 60 minutes and the minutes further into 60 seconds, following the ancient division of the degrees of arc.

 

 

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The minute of arc is the 1/60^th part of one degree of arc (1 degree of arc = 60 minutes of arc); one full circle is divided into 360 × 60 = 21 600 minutes of arc. The minute of arc can be further subdivided into seconds of arc.

This division of one degree into 60 parts comes from the ancient Babylonians, who used a numerical system based on the numbers 60 and 10.

Increasingly, decimal fractions of degrees of arc are being used (e.g. 60.4° instead of 60° 24′) and the minute of arc is becoming obsolete. This follows a recommendation by the International Standards Organization (ISO) to subdivide the degree decimally rather than using the minute of arc and second of arc.

 

The correct symbol for minute of arc is the prime, not the apostrophe or single quotation mark. In the Unicode character encoding table, the prime symbol is number U+2032. Please note that, as an exception to the rule, the symbol for minute of arc follows the preceding number without space (50′, not 50 ′). For more information about special characters please look at the page of rules.

 

 

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Neper is a measure for logarithmic quantities and is used to describe a ratio. Strictly speaking, the neper is not a measure, because it does not measure a physical quantity. Instead it describes the ratio of a physical measurement (which can be anything, watts, pascals, watts per square meter, etc.) to a set starting point of the neper scale.

The neper is closely related to the other measure for logarithmic quantities, the bel. A more detailed explanation can be found in this dictionary under ‘bel’. The bel is far more common than the neper.

 

Neper is based on exactly the same principle as the bel, but instead of using the common logarithm (base 10), utilizes the natural logarithm. The natural logarithm is based on the so-called Euler number (symbol e). e equals about 2.718 281 828.

This Euler number is important in many areas of mathematics. It is a so-called surd, which is the name for an infinitely long number that cannot be expressed in finite terms of ordinary numbers or as a fraction.

When used with the unit neper, a number describes the exponent of the Euler number (that is how often the Euler number must be multiplied by itself), just as a number used with the unit bel specifies how often the number 10 should be multiplied by itself.

The formula for converting bels into nepers uses the natural logarithm (symbol ln) and is:

1 B = 0.5 × (ln 10) Np or 1 B ≈ 1.151 282 Np

To convert nepers into bels:

1 Np = 2 / (ln 10) B or 1 Np ≈ 0.868 588 B

The natural logarithm (ln) is also sometimes called napieran logarithm, named after John Napier, who published fundamental works on logarithms.

 

The neper is coherent with the metric system and has been published in the official SI brochure already. However, the CGPM has not yet officially adopted it as a metric unit.

 

 

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The second of arc is the 1/60^th part of one minute of arc, which is the 1/60^th part of one degree of arc (1 minute of arc = 60 seconds of arc; 1 degree of arc = 60 × 60 = 3 600 seconds of arc); one full circle is divided into 60 × 60 × 360 = 1 296 000 seconds of arc.

This division of one minute of arc into 60 parts comes from the ancient Babylonians, who used a numerical system based on the numbers 60 and 10.

Increasingly, decimal fractions of degrees of arc are being used (e.g. 60.412 5° instead of 60° 24′ 45″) and the second of arc is becoming obsolete. This follows a recommendation by the International Standards Organization (ISO) to subdivide the degree decimally rather than using the minute of arc and second of arc.

 

The correct symbol for second of arc is the double prime, not the quotation mark. In the Unicode character encoding table, the double prime symbol is number U+2033. Please note that, as an exception to the rule, the symbol for second of arc follows the preceding number without space (50″, not 50 ″). For more information about special characters please look at the page of rules.

 

 

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The tonne is an additional unit of measurement for mass. One tonne is defined as 1 000 kilograms.

Usually, masses larger than 1 000 kilograms are measured in tonnes, kilotonnes, megatonnes, etc., and masses of 1 000 kilograms and less are measured in kilograms, grams, milligrams, micrograms, etc. This is not a general rule, just a widely observed custom. Instead of tonne, the term megagram could be used, but this has never caught on.

One possible explanation as to why the tonne has remained an official unit of measurement for so long, without any attempts to replace it with megagrams etc. is that the kilogram is the only base measurement unit that has a prefix already in place.

 

In English, the word “ton” is used to refer to an old, non-metric measurement of roughly the same size as the tonne. In English-speaking countries that have only recently changed to the metric system, the tonne is sometimes unofficially still called “metric ton”.

 

 

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