(Calvin of Farina, IL.
2000-11-05)
Current and Deprecated Prefixes
What are all of the metric prefixes?
(Only 20 are official ones.)
Official SI metric prefixes (largest to smallest)
and deprecated metric prefixes (obsolete or bogus)
SI
Value
Remarks
Obsolete
Bogus
10^{33}
una, vendeka (V)
10^{30}
dea, weka (W)
10^{27}
nea, xenna (X)
yotta-
Y
10^{24}
Adopted in 1991.
otta
zetta-
Z
10^{21}
Adopted in 1991.
hepa
exa-
E
10^{18}
Adopted in 1975.
peta-
P
10^{15}
Adopted in 1975.
tera-
T
10^{12}
Adopted in 1960.
megamega (MM)
giga-
G
10^{9}
Adopted in 1960.
kilomega (kM)
mega-
M
1000 000
CGS system since 1874. Legal in France since 1919.
100 000
hectokilo (hk)
10 000
myria (ma, my) 1795
kilo-
k
1000
Since 1793.
hecto-
h
100
Since 1793.
deca-
da
10
Since 1793. Also deka.
dk
1
Unprefixed.
deci-
d
1/10
Since 1793.
centi-
c
1/100
Since 1793.
milli-
m
1/1000
Since 1793.
1/10 000
decimilli,
dimi (dm)
myrio (mo)
1/100 000
centimilli (cm)
micro-
m,u
1/1000 000
Within CGS system since 1874 (BAAS).
nano-
n
10^{-9}
Adopted in 1960.
millimicro (mm)
pico-
p
10^{-12}
Adopted in 1960.
micromicro (mm)
femto-
f
10^{-15}
Adopted in 1964.
atto-
a
10^{-18}
Adopted in 1964.
zepto-
z
10^{-21}
Adopted in 1991.
ento
yocto-
y
10^{-24}
Adopted in 1991.
fito
10^{-27}
syto, xenno (x)
10^{-30}
tredo, weko (w)
10^{-33}
revo, vendeko (v)
The use of metric prefixes dates back to the inception of the French metric system, in 1793.
It was originally decided that the submultiples of all basic units would be prefixed
with a Latin root, corresponding to the decimal divisor
(deci for 10, centi for 100, milli for 1000), whereas the decimal
multiples would be prefixed with a Greek root, corresponding to the decimal
multiplier (deca for 10, hecto for 100, kilo for 1000).
In 1795, the Greek root myria for 10000 was added to the latter list
(it's now officially obsolete, see below).
There was soon an obvious need to extend the system beyond its original limited range.
The prefix micro (from the Greek mikros, small) was introduced to denote
one millionth of the basic unit. The prefix mega
(from the Greek megas, great) appeared around 1870
to denote a million times the basic unit.
It used to be acceptable to combine two prefixes (see above "obsolete" column).
In 1960 however, it was decided to name only powers of 1000,
not intermediary powers of 10, except for the original 1793 prefixes
(the popular myria prefix was thus deprecated in the process).
Four additional prefixes were introduced at that time:
pico (Spanish pico beak, small quantity),
nano (Greek nanos, little old man, dwarf),
giga (Greek gigas, giant),
tera (Greek teras, monster).
It was then decided that the names of future prefixes should serve as reminders of
the relevant power of 10.
This started in 1964, with the introduction of femto and atto
(Danish or Norwegian: femten for 15, atten for 18).
The former prefix was particularly convenient, because it made the widespread
abbreviation fm (for "fermi") correspond correctly to the
the officially endorsed femtometer.
After that, however, it became clear (!?) that since only powers of 1000
were to be named, the prefixes should reflect the ranks of the powers of 1000 involved.
This is why, in 1975, the prefix exa (Greek hex, 6) was chosen for
10^{18}=1000^{6},
whereas peta (Greek pente, 5) was picked to represent
10^{15}=1000^{5}.
The four latest prefixes, which were made official in 1991, are also supposed to remind
an international audience of the relevant powers of 1000:
yocto (1000^{-8}),
zepto (1000^{-7}),
zetta (1000^{7}), and
yotta (1000^{8});
the trend being that the ending "a" is used for large powers,
while "o" is used for small ones.
The 5 exceptions to this modern rule are all the 1793 prefixes, except deca
(for these 6 "low" prefixes,
the long forgotten Greek/Latin distinction applies, as mentioned above).
The last column of the above table lists as bogus 10 extreme prefixes
(revo, tredo, syto, fito, ento, hepa, otta, nea, dea, una).
The larger of these follow the etymological pattern described above,
and 4 of them "compete" with the latest official SI prefixes.
These bogus prefixes have apparently not been used by anyone and
are not endorsed by anybody, but they show up
in tables which have been floating around in Cyberspace...
This is probably the result of a minor hoax perpetrated sometime
around 1996.
[2003-06-22 update:]
Other dubious prefixes are also shown
(vendeka, xenna, xenno, vendeko)
which we discuss elsewhere.
Please, tell us
whatever you know about the issue...
Note (2002-05-01) : Usenet
Archives show
Alejandro López-Ortiz
posting 3 times, between 1998 and 2000,
a bogus list of prefixes ["7.5" dated 1998-02-20]
whose previous version ["7.1", dated 1995-12-31, last posted 1996-10-09]
didn't include any bogus information...
In a 2002-01-14 post,
Robi Buecheler plagiarized the above
text...
On 2004-12-14, Robi Buecheler
apologized: [edited summary]
I should have given you credit [and/or posted a] link.
Sorry.
Bogus prefixes are no longer spreading out of control.
However, at least one (careless) science-fiction writer has been fooled:
In his 2003 novel entitled Schild's Ladder, Greg Egan uses
the two bogus prefixes xenno and vendeka as if they
were legitimate.
(Thanks to Tom Alcorn for pointing that out, 2007-12-05.)
(J. B. of New Lenox, IL.
2001-02-09)
How many kilobytes [kB or "K"] in 2 "megs" [megabytes, MB]?
For units of information that are multiples of the bit (and only these),
the multiplicative prefixes kilo- mega- giga- tera- etc. do not have their usual
meaning as powers of 1000.
They're powers of 1024 (2 to the power of 10).
Thus, a kilobyte (kB) is 1024 bytes and a megabyte (MB) is 1024 times that
(namely 1048576 bytes). Therefore, 2 "megs" is 2048 kilobytes.
A gigabyte (GB) is 1024 MB (1073741824 bytes) and a terabyte (TB)
is 1024 GB (1099511627776 bytes). A petabyte (PB) would be
1024 times as large, namely 1125899906842624 bytes
(9007199254740992 bits or 86.2 nJ/K).
The situation may be quite confusing for several reasons.
In particular, a few commercial designations have wrongly
ignored the above binary-based convention (powers of 1024)
and used the standard decimal one (powers of 1000) in some cases.
Even worse, the two have been mixed to create a special type of digital macaronic
terms like the "megabyte of storage" which turns out to be worth 1024000 bytes,
but is only used commercially for some removable storage media.
This came about (sadly) when the capacity of
the so-called 3½" IBM microfloppies
doubled from 720 kB to 1440 kB
and the larger capacity was widely advertised as "1.44 MB"
(instead of "1.40625 MB" or "1.4 MB").
In December 1998, the International Electrotechnical Commission (IEC)
attempted to clear things up by introducing a
kilobinary system,
in which we would no longer use kilobyte to designate 1024 bytes,
but kibibyte (KiB).
The IEC proposal is slowly
gaining some ground.
However, it should only be a way to disambiguate the customary exception which has been
universally used for multiples of the bit (b) and the 8-bit byte (B),
as far as addressable computer memory is concerned.
Ideally, acceptance of the IEC
proposal should only replace "kB" or "K" by "KiB" to mean 1024 bytes.
It should never be construed as the permission to use "kB"
concurrently to mean 1000 bytes.
(Current usage does not allow unrestricted use
of metric prefixes anyway:
It's not permissible to use "kiloinch" for 25.4 m, is it?)
Otherwise, ambiguity and confusion would be increased, not decreased.
Arguably, manufacturers of digital storage who use the abbreviation "GB"
for 1000 000 000 bytes would still be shortchanging their customers by
7.4%, even if the unambiguous IEC binary prefixes gain wider acceptance.
Warning: 1 kb/s = 1 Kib/s = 1.024 kbps
Be aware that the binary exception only
applies to multiples of the bit, not
to derived units like the "bps" (bit per second),
so that 56 kbps is exactly 56000 bps.
This may not look so bad until you realize that a transfer speed of
"1 kilobit per second" is actually equal to 1.024 kbps.
The latter should only be pronounced "kilo-bee-pee-ess"
to avoid confusion with the former!
Likewise, the related baud rates
have always used decimal prefixes:
A kilobaud (kBd) is 1000 Bd.
A megabaud (MBd) is 1000000 Bd.
That's the current mess we've built for ourselves.
Careless standardization efforts could make the situation even worse
before it gets better.
"Brontobyte" [ hoax alert ]
This unit is just a joke
(2004)
nothing more! Unfortunately, the word caught the
fancy of
many unsuspecting webmasters
and is now often listed
among "serious" units of information
(even more dubious is the geobyte
of 1024 brontobytes).
In terms of entropy,
this huge amount of information is only 1.7668 J/K.
( L. K. of Owen, WI.
2000-10-10)
What has a density of 1 ?
Proper units (g/cc, lb/ft^{ 3}, etc.)
are used to express an absolute density.
A relative density is the ratio
of an absolute density to the absolute density of "water".
For the utmost precision,
it's important to specify what kind of "water" is meant.
For example, SMOW ("Standard Mean Ocean Water")
at its densest (around 3.98°C)
has an absolute density of about 0.999975 g/cc.
However, the universally accepted conversion factor
between "absolute" and "relative" density is 0.999972 g/cc !
This is one number which has acquired the
unofficial status of a defined
exact conversion factor, which has ultimately
little to do with actual water or SMOW.
In other words, the short answer to this question is:
"Water."
A more precise (somewhat cynical) long answer is:
"Anything with an absolute density of exactly 0.999972 g/cc."
(Michael of United Kingdom.
2001-02-12)
What's the difference between normal [1N] and molar [1M] solutions in acid chemistry?
Particularly for sulfuric acid.
Each liter of a molar solution (1M or 1000mM) contains a mole of a given compound
(a mole of H_{2}SO_{4}
is about 98.08 grams of it).
A normal acid (1N), on the other hand, contains the solute(s)
that could produce a mole of H^{+} ions.
In the case of sulfuric acid, you'd have 2 H^{+}
ions per molecule,
so that a normal (1N) solution of sulfuric acid is actually a 1/2 molar solution
(0.5M or 500mM).
A mole of "objects" [atoms, molecules, ions, electrons]
is defined to be as many of these as there are atoms in 12 grams of carbon-12.
The "number of things per mole of stuff" is a universal constant known as
Avogadro's Constant :
6.02214129(27)´10^{23}per mole.
[Here, the parenthesized 27 indicates an uncertainty whose standard deviation is
27 times the weight of the last decimal position shown.]
Free protons (H^{+} ions) in water are mostly a convenient
fiction, since such ions would quickly combine with nearby molecules of water to form
hydronium ions.
The dissociation of water molecules into ions
is thus best described by the following reversible chemical reaction:
2 H_{2}O
«
H_{3}O^{+} +
OH^{-}
(J. M. of College Station, TX.
2001-02-11)
Definition of the Calorie
How much energy is required to raise the temperature
of one kilogram of water [by] one degree Celsius?
If the calorie was still defined as the energy required to raise
a gram of water by 1°C,
the answer would be "1000 calories" (or 1 kcal).
However, that definition of the calorie was dependent on the starting temperature
and wasn't good enough for metrological purposes.
Several more precise definitions have been given (see table below)
including the fifteen degree calorie
which is still defined as the energy that raises a gram of water from
14.5°C to 15.5°C.
This type of calorie must be measured
to be equal to 4.1855 J at a fairly modest level of precision
(an uncertainty of about 0.0003 J). It's only good for casual use...
Since 1935, the current (thermochemical)
calorie has been defined as
exactly equivalent to 4.184 J.
No other conversion factor should be used in Science.
The recommendation is to use joules primarily.
The energy which raises a kg of water by 1°C
(under 1 atm = 101325 Pa)
is a function of temperature which features a minimum of about 4178 J
around 34.5°C.
It's about the same at the ice point (4218 J at 0°C)
and the steam point (4216 J at 100°C). All values
between 4178 J and 4216 J are
correct for two temperatures
(one below 34.5°C, one above that).
Various "calories" competing with the thermochemical calorie of 4.184 J
The dubious "IST calorie" (or "steam tables calorie") was created merely for compatibility
with the official definition of the Btu (described below).
The Btu itself is never used in Science
and it seems to be utterly unknown outside of the US and UK.
So is the obscure unit dubbed therm
which is now taken to be 100000 Btu,
although the name had been used to denote
the 4° calorie between 1888 and 1899
(as indicated in the above table).
The 5^{th }International Conference on the Properties of Steam
(London, July 1956)
gave the (final) modern definition of the Btu
(British thermal unit) based on their nominal heat capacity for water:
2326 J/kg/°F (which is equivalent to 4186.8 J/kg/°C).
This authoritative definition of the Btu had the unfortunate side-effect
of creating a bogus "IT calorie" of 4.1868 J
(IT or IST stands for "International [Steam] Tables")
which has infected many computers and most handheld calculators.
It is off by 669 ppm and shouldn't be used in connection with the precise thermochemical
data which has been lovingly compiled by our elders since 1935 (using
an exact equivalent of 4.184 J to the calorie).
The Btu had an historical definition similar to that of the calorie:
Before 1820, it had been introduced by Michael Faraday
(1791-1867)
as the quantity of heat which would raise a pound of water from
63°F to 64°F
(presumably under normal atmospheric pressure)...
In 1876, it was re-defined as the energy required to raise the temperature
of one pound (lb) of water by 1°F,
around the point of maximum density [about 3.98°C].
All told, it's best to use the IST definitions (1956) for the Btu
(1 Btu is 1055.05585262 J, namely the ratio of the pound to the kilogram
multiplied by 2326 J) and to use the standard 1935 thermochemical definition
for the calorie (1 cal is 4.184 J).
Unfortunately, you may also encounter a "thermochemical Btu"
(» 1054.35 J)
and an "IST calorie" (4.1868 J = 2326 * 0.0018 J).
cdw239
(2001-08-23)
What is the equation for converting horsepowers to watts?
The horsepower and the watt are both units of power;
there's just a conversion factor between them.
The way power is delivered (voltage, etc.) is completely irrelevant.
A horsepower (hp) is about 745.7 watts (W),
but many metric countries use another
closely related unit [best abbreviated "ch"]
of nearly 735.5 W.
The horsepower unit (hp) was originally defined by James Watt (1736-1819)
as exactly equal to 550 ft-lbf per second (lbf = "pound-force") .
Since January 1, 1959,
the foot and the pound have been defined in metric terms
(1 ft = 0.3048 m and 1 lb = 0.45359237 kg, both exactly).
Furthermore, since the third CGPM of 1901, the standard
(or conventional)
acceleration of gravity has been defined as exactly
equal to 9.80665 m/s^{2}.
That's the proper "conversion factor" to use to transform
(not "convert", please) a unit of mass
like the pound (lb) into the
unit of force best called pound-force (lbf).
The resulting exact conversion factor has 12 digits:
1 lbf = (0.45359237 kg) (9.80665 N/kg) =
4.4482216152605 N
Multiply this by the length corresponding to 550 ft
(exactly 167.64 m) and you have the equivalence of a
horsepower in watts
(as a watt "W" is simply a meter-newton per second).
This gives an exact modern conversion factor
which requires no fewer than 17 digits:
1 hp = 745.69987158227022 W
Needless to say that everybody
usually rounds this up in the most obvious way
(which is appropriate except in computerized conversion tables):
1 hp » 745.7 W
In countries where the metric system has been around for a while,
the horsepower (ch) is a 1.37% smaller unit,
called Pferdestärke (PS) in German,
paardekracht (pk) in Dutch,
hästkraft (hk) in Swedish,
caballo de vapor (CV) in Spanish,
cavalo-vapor in Portuguese and
cavalli vapore in Italian.
The French call it cheval-vapeur (ch)
or cheval (plural is chevaux).
This "metric" horsepower (ch) is defined as
75 kgf-m/s, which
engineers used to abbreviate as 75 kgm/s, using the obsolete symbol kgm
for a "technical" unit of energy called kilogrammetre or
kilogram-meter and worth 9.80665 J
(that same unit of energy was also called kilopond-meter
and abbreviated kpm ).
A metric horsepower (ch) is thus exactly :
1 ch =
(75 kg) (9.80665 N/kg) (1 m/s) = 735.49875 W
French readers should not confuse this cheval-vapeur (ch) unit
with the French cheval fiscal (CV) which is
a nonlinear rating of a motor vehicle for tax purposes
(registration cost is about $30 per CV, as of this writing).
The CV rating, or fiscal power [sic], is
(P/40)^{1.6}+ U/45,
where P is the maximum DIN power (in kW)
and U is the amount of CO_{2} emitted per unit of distance (in g/km).
From 1909 to 1947, British car taxation
was based on another unphysical rating called
RAC
horsepower (introduced by
the Royal Automobile Club ) numerically
equal to 0.4 times the square of the bore
(the diameter of each cylinder in inches) multiplied by the number of
cylinders. Several generations of British engines had
artificially long strokes which produced a larger
displacement (in cubic inches) for the same tax rating.
The assumption behind the RAC formula was that short stroke
engines would work at higher rpm: The maximum product of stroke and rpm
was taken to be a constant (6000 in/min) equal to
half the largest acceptable mean piston speed,
(estimated to be 1000 ft/min).
Paying no attention to British
tax law, American automakers designed
short stroke engines.
Adding to the confusion, a so-called electric horsepower
is defined as exactly equal to 746 W
(it's clearly a rounded-up version of the "hp").
Finally, there's an unrelated unit of power called the
boiler horsepower,
defined in 1884 as the power it takes to boil 34.5 lb of water per hour
(under 1 atm, when water is already at 100°C = 212°F).
So defined, the boiler horsepower is approximately
9809.91 W, or about 13.155 hp.
However, this is so close to 1000 kgf-m/s
(which is 9806.65 W) that I suspect
such a "metric" definition of the boiler horsepower may have been given...
(The quotes around "metric" are a reminder that "technical" units of force,
named after units of mass, are not official SI units.)
I'd be grateful if anyone could
tell me
if this is so...
(2001-05-04)
Why is 9.80665 m/s^{2} [1 G] the standard acceleration of gravity?
To an actual measurement of 9.80991 m/s^{2} in Paris,
a theoretical correction factor of 1.0003322 was applied which gives
a sea-level equivalent at 45° of latitude.
The result (9.80665223...) was rounded to five decimals
to obtain the value officially enacted by the third CGPM, in 1901.
The
above includes a centrifugal
component due to the rotation of the Earth,
whereas the gravitational field
at altitude zero has a slightly larger value,
used when tracking satellites outside
the atmosphere in nonrotating coordinates
(9.82025048(2) m/s^{2 })
which is the ratio of the Earth's gravitational constant(3.986004415(8) 10^{14 }m^{3}/s^{2 })
to the square of the conventional Earth radius
(R = 6371000 m).
The centripetal acceleration
of a satellite orbiting
the Earth at a distance R+z from its center is thus:
(9.82025 m/s^{2 }) / (1+z/R)^{ 2}
Time
(Bob J.of Clarksville, TN. 2000-09-28)
What is the term for 1/1000 of an attosecond? (This would be 10^{-21 }s.)
That's one zeptosecond (zs).
One thousandth of that is a yoctosecond :
1 ys = 10^{-24 }s Both terms were officially adopted by the CGPM in 1991.
Fred Berman,
Ph.D., P.E. (2002-11-29; e-mail)
What's a jiffy ?
Is a jiffy really the time for light to travel one centimeter in a vacuum?
A formal definition of the jiffy as a light-centimeter
(roughly equal to 33.3564 picoseconds) was first proposed, in physical chemistry, by
Gilbert
Newton Lewis (1875-1946), the American chemist who isolated heavy water and
defined a Lewis acid as an
acceptor of electron pairs (1916).
In 1926, the same Gilbert N. Lewis also coined the term photon which is now
used to denote the quantum of electromagnetic
radiation introduced by Albert Einstein
in 1905, under the German name of Lichtquant.
(Ironically, Lewis originally intended the word "photon" to mean something else entirely.)
Informally, a jiffy can be any short period of time, though.
The word was commonly used before
1785.
Jiffy meant "lightning" in
thieves' cant
(possibly as early as 1530)
but its early etymology is otherwise unknown. The jiffy
has been given several definitions in various contexts:
In the quaint context of computer engineering, a jiffy
may denote the period of the system's main clock (e.g., 10 ns for a 100 MHz clock)
but it can also be the interval between two regular
timer interrupts,
which is usually something between 1 ms and 20 ms
(most commonly 4 ms).
In electrical engineering, a jiffy
used to be the period of the electrical power grid, namely:
20 ms in Europe (50 Hz) or about 16.6667 ms in the US (60 Hz).
Nowadays, this flavor of jiffy has all but disappeared;
a modern jiffy is usually equal to 10 ms
(the resolution of an ordinary stopwatch).
On 2008-12-30, Dr.
Robin Whitty wrote: [edited summary]
I love it! Where but Numericana could you find something so minute given such
Johnsonian treatment?
A much smaller obsolete unit
[about 9.3996392(13) 10^{-24} s] is
related to the above jiffy of physical chemists:
The tempon is defined as the time required for light to travel
a distance of one classical electron radius.
The smallest recognized unit of time is called chronon,
or Planck time:
(B. D. of Australia.
2000-05-01) How long is one second?
(J. F. of Memphis, TN.
2000-10-20) Who determined the length of a second?
The "SI second" (formerly called "atomic second") is now defined as equal to
9192631770 periods of the radiation
corresponding to the transition between the two hyperfine levels of Cesium-133.
(Until recently, surprisingly, nobody seemed to care about
general relativistic effects,
which are becoming relevant:
Are we talking about cesium atoms in free fall or not?)
In 1967, this replaced officially the "Ephemeris Second", which was based on the
orbital motion of the Earth around the Sun.
An earlier definition was based on the mean solar day instead,
and was thus tied to the Earth's rotation around its own polar axis,
although fluctuations in this rotation make it a poor basis for the definition
of a precise unit of time (as was first shown by Simon Newcomb).
The Full Story:
Originally, the second was defined as 1/86400 of the mean solar day.
In other words, there are 24 hours of 3600 seconds in a day.
It is necessary to specify "mean" solar day because the length of the day varies throughout
the year,
as the angular speed of the Earth varies in its elliptical motion around the Sun.
(It is this angular speed which determines how soon the Sun will be seen again at the same
longitude in the sky, after roughly one revolution of the Earth on its axis.)
This mean solar second came under international scrutiny by the CGPM in 1954,
and the BIPM proposed (in 1956) a new official definition of the second:
The definition of the so-called ephemeris second is based entirely on the
orbital period of the Earth, which is steadier than its spin.
It is specified, as explained below, by equating
to 31 556 925.9747 ephemeris seconds the instantaneous value
at epoch 1900.0 of the tropical year.
This definition was ratified by the CGPM in 1960, but it originated in the 19th century:
The American astronomer Simon Newcomb (1835-1909) discovered that there are significant
irregularities in the rotation of the Earth on its own axis
(this was apparent to him when he analyzed the ephemerides of the Moon published by Hansen
in 1857).
Newcomb came up with a famous equation giving L,
the so-called "mean geometrical longitude of the Sun",
as a function of the time T expressed in the number of centuries
[of exactly 31 557 600 000 seconds each]
elapsed since "January 0.5 1900"
[this is either 24:00 GMT on 1899-12-31 or 0:00 GMT on 1900-01-01].
That "longitude" is measured against the vernal point [which means it integrates
the wobbling of the Earth's spin which influences the length of the
tropical year and causes the precession of equinoxes].
The qualifier "geometrical" is a reminder that the equation gives the immediate
position of the Sun, not its apparent location,
as perceived from solar light emitted about 499 seconds before.
Finally, the qualifier "mean" is a reminder of the averaging made necessary
by the variable angular speed of the Earth, in its elliptical orbit around the Sun.
L = 279° 41' 48.04" + 129602768.13" T
+ 1.089" T^{ 2}
A tropical year is the time it takes for L to increase by a full turn
(360° or 1296000"),
we may thus state that the instantaneous tropical year
at time T is a full turn divided by dL/dT.
To obtain the duration Y of this year expressed in seconds
(rather than Julian centuries),
we simply multiply by 3155760000.
This boils down to:
Y(T)
= 227214720000000000 / ( 7200153785 + 121 T )
= 31556925.9747415242... - (0.5303203455...) T + O( T^{ 2 })
It turns out that Newcomb's equation can be used backwards
to define the unit of time with far
greater precision than anything based on the rotation of the Earth.
By specifying the value of Y(0) in some unit of time,
that unit is very precisely defined
in terms of the orbital motion of the Earth around the Sun,
rather than on the less precise rotation of the Earth about its own axis.
This is precisely how the so-called ephemeris second was defined,
by making exact de jure Newcomb's value of Y(0) rounded [down]
at the fourth position after the decimal point:
Y(0) º 31556925.9747 ephemeris seconds
This definition makes the ephemeris second very slightly longer than whatever
we may call the "second" used by Newcomb himself to establish his equation.
The above rounding corresponds to a relative precision of
1.31585 10^{-12}
(roughly 50 microseconds per year).
This is lower than the combined precision of the observations used by Newcomb,
which were made between 1750 and 1892.
The solar second and the ephemeris second were identical around 1820 or 1826.
Since then, the mean solar day has been slightly longer than
86 400 ephemeris seconds, as the rotation of the Earth is slowing
down under the braking effect of the tides.
It may be amusing to record that, according to Newcomb's original equation,
the instantaneous tropical year was exactly 31556925.9747 "seconds"
about 247097 seconds after T=0: January 3, 1900, at 20:38:17 GMT.
The ephemeris second was the official definition of the second from 1960 to 1967.
During that period, the credit for determining the "length of a second"
would clearly have gone to Simon Newcomb...
Since 1967, the official definition of the second has been in "absolute"
atomic terms rather than astronomical ones. It was decided to define the second in terms
of a number of standard transitions of the Cesium atom.
In 1958, it had been determined that there were
9192631770 such transitions (give or take 20) in an ephemeris second.
This was the result of a three-year collaboration between William Markowitz at
USNO
(U.S. Naval Observatory, in Washington, DC)
and Louis Essen (1908-1997)
at NPL
(National Physical Laboratory, in Teddington, England).
USNO contributed accurate astronomical time measurements,
using a dual-rate Moon camera (invented by Markowitz in 1951)
which was compensating simultaneously for sidereal and lunar motions.
Occultations of stars by the Moon provided the best estimate of Ephemeris Time.
On the other hand, NPL provided the
World's first caesium clock standard,
which had been perfected by Louis Essen and Jack Parry since 1953.
(The two clocks were compared using synchronizing radio transmissions from the
WWV station
operated by the National Bureau of Standards, now called
NIST.)
This value of 9192631770 Cesium transitions per second was ultimately accepted as the
de jure value. Therefore, the guys who really determined
"the length of a second" are the authors of that particular measurement.
It was a team effort, by Markowitz, Hall, Essen and Parry
[See "Frequency of Cesium in Terms of Ephemeris Time"
by W. Markowitz, R. Glenn Hall, L. Essen, and J.V.L. Parry in
Physical Review Letters, Volume 1, pp. 105-106 (1958)].
That's our final answer, as long as the "Cesium standard" remains the basis
for the official definition of the second.
The international body which is responsible for making such definitions official
is the CGPM.
However, the CGPM should not be credited for the work on which its decisions are based.
Instead, we ought to remember the accomplishments of great scholars like
Simon Newcomb, Louis Essen, or William Markowitz...
(C. V. of Indianapolis, IN.
2000-10-23)
How many seconds in a day?
The short answer is 86400 (24 hours of 3600 seconds).
At a higher level of accuracy,
it may be useful to point out that there are 3 kinds of standard days,
but we may still say that there are exactly 86400 "solar" seconds
in a "mean solar day" and 86400 "ephemeris" seconds in an ephemeris day.
The "day" used in modern science is also defined as exactly equal to
86400 SI seconds (officially defined in terms of the cesium atomic standard).
When the "day" of one system is expressed in terms of the "second" of another,
the numbers are slightly off.
For example, the mean solar day "at epoch 2000.0" is
about 86400.002 SI seconds.
Now, the so-called "sidereal day" is another matter entirely because it is significantly
different from the above 3 "standard" days and has never been used as a standard unit
of time. A sidereal day is about 86164.09 SI seconds.
It is interesting to notice a weird point of etymology about "sidereal"
(which is often misspelled "sideral", as would be correct in French and/or a few other
languages).
"Sidereal" should mean that a "sidereal day" refers to the rotation of the Earth
with respect to the fixed stars (as is the case with other "sidereal" motions, by the way).
This is the definition most dictionaries will give you.
However, that's not quite so. Historically, astronomers have most often used
the term "sidereal day" to refer to the rotation with respect to the slowly moving
"vernal point" (which rotates a full turn in about 25772 years, the period of "precession
of the equinoxes").
When motion with respect to the fixed stars is meant, the unambiguous term "Galilean day"
should be used. In other words, the Earth rotates on its axis once per Galilean day
(i.e., once in each period of 86164.1 s).
The Galilean day is longer than the sidereal day by 0.0084 s. The Galilean day increases
by about 0.00164 s per century because of the braking effects of tides.
Both the sidereal day and the mean solar day also increase at almost exactly the same rate
(so the differences between these three remain roughly constant).
The drift rates are almost exactly the same because all the other relevant astronomical
motions are far more stable than the spin of the Earth on its axis.
The SI "atomic" day, on the other hand, is absolutely stable in principle
(assuming only that the laws of physics themselves do not change over time).
(2000-11-03) The scientific year:
1 a = 365.25 D = 31557600 s
(P. H. of Concord, CA.
2000-11-03
and I. I. of Canada.
2001-02-05)
How many seconds in a year?
(John of Springville, AL.
2000-10-08) What is a "scientific year" ?
jwill123 (2002-05-05)
A light-year is the distance that light travels in one year.
How many seconds in [such] a "year"?
" The recognised symbol for a year is the letter a [annum]
rather than yr, which is often used in papers in English. [...]
Although there are several different kinds of year (as there are
several different kinds of day), it is best to regard a year as a julian year
of 365.25 days (31.5576 Ms) unless otherwise specified. "
The only recognized "year" unit in scientific practice is a year of
exactly 365.25 days, based on a day of exactly 86400 seconds
(these are standard SI seconds, formerly known as "atomic seconds"). Therefore:
The number of seconds in a year is exactly 31557600.
This is the number you should use, for instance, to compute precisely the number of meters
in a light-year (which is exactly 9460730472580800).
Some scientists like to memorize the duration of a year in seconds as approximately equal
to "p times ten to the seventh".
This scientific year is longer than the average calendar year,
the Gregorian year of 365.2425 mean solar days,
and it's extremely close to the Julian year of 365.25 mean solar days.
As the mean solar day slowly drifts in duration, so do both the Gregorian year
and the Julian year.
The related tropical year is more stable than either of these calendar years,
because it is based on the orbital motion of the Earth, which is steadier than its spin.
The wobbling period of the Earth's axis
(responsible for the precession of equinoxes) affects this tropical year but not
the sidereal year
which is measured with respect to the "fixed stars"
(more precisely, the background of galactical nebulae).
However, even this sidereal year is not absolutely stable,
since the orbit of the Earth does decay...
By contrast, the scientific year of 31557600 seconds is rock stable
[more stable than any rock will ever be, actually];
it's a true unit of time.
It will never change, unless the laws of physics themselves change.
Finally, it is properly based on a local [atomic] definition, as any unit of time
should be: According to Special and General Relativity,
there is not such thing as an absolute time which would "flow" the same for all
observers, irrespective of their motions and/or surrounding gravitational fields.
Length
nara
(2000-04-11)
How long is a meter?
I know it is not the same system, but how many inches are in a meter?
There are (very) sligthly more than 39.37 inches in a meter (a more precise number
is 39.37007874).
Since January 1, 1959,
the International inch has been defined to be exactly equal to 25.4 mm
(0.0254 meter).
Now, the inch and the meter are thus almost part of the same system (well, kinda)...
Since 1866, the US Coast and Geodetic Survey has been using
another metric definition of the inch,
equating a meter to 39.37 inches.
This "US Survey" inch (of about 25.4000508 mm)
was confirmed for general use by the Mendenhall ordinance of April 5, 1893,
but it's been restricted to US surveying since 1959.
There's a noteworthy numerical coincidence
concerning the ratio of these two different
"types" of inches, since (254/10000)/(100/3937) turns out to be exactly 999998/1000000,
so that it can be stated that the modern International inch is exactly 2 ppm less
than the 1893 "US Survey" inch, whose value in mm has the following expansion:
25.400050800101600203200406400812801625603...
You may notice a pattern in the above decimal expansion which allows you to write
dozens of decimals very quickly. It comes from the fact that 1000000/999998
is the sum of a geometric progression of ratio 0.000002 and is thus equal to
1.000002000004000008000016...
The 1824 Imperial inch was based on the actual British standard yard,
which kept shrinking
(the 1760 brass artifact was lost in an 1834 fire;
new ones were made of Baily's Metal, after 1841).
This obsolete inch was "calibrated"
to be:
25.399978 mm in 1895.
25.399956 mm in 1922.
25.399950 mm in 1932.
25.399931 mm in 1947.
The 1895 and 1922 calibrations are still quoted
today in an historical context, whereas the others are all but forgotten.
The preliminary 1819 equivalence of 39.3694" to the meter
describes a larger inch (of about 25.400438 mm)
which may best match the yard made by Bird in 1760
(after an old Tower standard).
(2007-05-22) The Typographical Point
The typographer's point is exactly 0.013837" =
0.3514598 mm.
Typographers use several specific traditional units of length.
The ATA point and its spinoffs
The typographer's point was defined in term of the inch in 1886.
Thus, when the inch evolved and was redefined in terms of the meter in 1959,
so was the typographer's point.
The point is exactly 0.013837 inches.
That's precisely 0.01 ppm
below what is implied by the conversion factor of 72.27 points
to the inch,
which has been advocated by
Donald Knuth in connection
with his "TeX"
computerized typesetting system.
The difference between the genuine typographical point and the "TeX point"
is so minute that the two are interchangeable, even in the
most exacting typographical work.
(There are 72.2700007227... points to the inch.)
Not so with the coarse equivalence of 72 points to the inch,
which was part of the original specification of PostScript, the page
description language championed by the Adobe Corporation, which was
instrumental in launching the "DeskTop Publishing" (DTP) industry in the mid
1980's. This rough equivalence gave birth to a new set of "DTP" units for
computerized typography: There are, for example, exactly 6 DTP picas to the
inch, and 72 DTP points to the inch.
Other points. 155520 Didot points to the arpent...
Although the above now dominates computerized publishing worldwide,
some typographic systems are based on other unrelated "point" units.
Most notably, the Didot point was introduced by
François Ambroise Didot (1730-1804)
with the system of font measurements that we still use today
(regardless of a slight difference in scale).
Didot's father (also named François, 1689-1757)
was the founder of the printing
and publishing enterprise which survives to this day (in Paris, France)
under the name of Firmin-Didot & Cie.
The original Didot point was defined as the 72^{nd} part of the
French Royal inch (pouce).
The foot corresponding to 12 pouces
was the pied de roi.
Two (very close) metric equivalences can be given for the pied de roi.
The earlier one goes back to the very inception of the metric system itself,
since the French scientists who conceived the new system were actually
using the toise
of 6 pieds de roi in their preliminary work.
The metrological equivalence they gave now stands as a metric definition
of the old unit: 0.513074 toise to the meter.
(That would make a toise approximately equal to
1.9490366 m.)
However, the Canadians can be considered to be the rightful heirs to the
ancient French system, as they still use the arpent
of 30 toises. The modern Canadian definition is thus
just as relevant as the current definition of the International
inch (of exactly 25.4 mm)
regardless of previous definitions...
The Canadian arpent is now defined to be 191.835 ft
or 50.471308 m. This makes the toise
exactly equal to 1.9490436 m. The pied de roi is
0.3248406 m and the pouce is exactly 27.07005 mm,
The proleptic value of the Didot point is 72 times smaller than that, namely:
1 Didot point = 0.3759729166666... mm
1 Cicéro (12 Didot points) = 4.511675 mm
( exactly )
The French Imprimerie nationale (IN)
now uses a metric point of exactly
0.4 mm.
The obsolete Truchet point was exactly half of a
Didot point.
Other traditional units of length
pertaining to newsprint include the line
(the agate line of 1/14 in, called a "ruby" in the UK)
and the "SAU column width" of 36/16 in (i.e., 2-1/16
inches of print and a 1/8" gutter space between columns).
"SAU" stands for "Standard Advertising Unit" and is also called "column-inch".
The SAU is actually a unit of surface area
equal to 36/16 square inches, namely the surface of one inch of
a standard column, as described above.
That unit is used to charge advertisers for commercial
space in printed media.
cheftell
(Wilmington. 2001-02-11)
How far in miles is 20000 leagues? One league equals how many miles?
A (land) league used to be defined as an hour's walk.
It's now defined as exactly 3 statute miles (4828.032 m).
However, a nautical league is 3 nautical miles
(5556 m, or about 3.452 miles),
and that's the league Jules Vernes refers to in the title of his book
"20000 Leagues under the Sea".
So, if you are a fan of Jules Vernes and Captain Nemo,
20000 nautical leagues is 60000 nautical miles.
That's about 69047 statute miles, 111120 km or almost
3 times around the Globe [at the Equator].
(D.W. of Orangevale, CA.
2000-10-07)
What is the circumference of the Earth at the equator?
(D.N. of Grass Valley, CA.
2000-10-09)
What is the radius of the Earth?
The irregularities of the Earth are charted with respect to a perfect ellipsoid whose
dimensions were precisely defined (not measured) once and for all in 1980, by the
IUGG (International Union of Geodesy and Geophysics).
The equatorial radius of that ellipsoid is exactly 6378137 meters,
which makes the circumference at the equator equal to 40075016.685578486...m
down to the nearest (ludicrous) nanometer. That's about 24901.46 statute miles
(these are "land miles" of 1609.344 m; the circumference may also be expressed as
21638.7779 "nautical miles", the modern
nautical mile being exactly 1852 m).
The conventional "radius of the Earth" is a unit
defined to be 6371000 m.
This is almost the radius of a sphere having the
same volume as the reference ellipsoid (6371000.79 m)
or the radius of a sphere with the same area as the ellipsoid
(6371007.181 m).
(Michael of Nashville, TN.
2000-10-03) [2012 update]
Are there any units longer than a lightyear, or shorter than an ångström?
A list of extreme units of length
that have actually been used, largest first:
Big ones :
gigaparsec (Gpc). Over 3 thousand million light-years.
hubble. A thousand million light-years, by definition,
megaparsec (Mpc).
kiloparsec (kpc).
parsec (pc) = 3.261563378 light-years.
light-year (exactly 9460730472580800 m).
The parsec
(pc) is actually a very specific (irrational) multiple of the
astronomical unit (au) since it's defined as the radius of a circle
for which an arc of one second has a
length of one astronomical unit (au). In other words, a parsec is exactly
648000/p au (about 206265 au).
The astronomical unit (au)
was once defined as the radius of the circular trajectory of a tiny
mass orbiting an hypothetically isolated Sun
with a period equal to a Gaussian year.
Since August 2012, the astronomical unit is simply defined to be
149597870700 m (exactly).
The mean distance between the Earth and the Sun
was about 1.00000003641 au at J2000.0.
A light-year is about 63241.07708 au...
In 1911, the Swedish astronomer
Carl Ludwig Charlier (1862-1934)
proposed a unit of stellar distances worth one million au
(about 15.8125 light-years)
which is nearly twice the distance to Sirius.
This never caught on.
The radius of the observable Universe itself is about 4 Gpc,
so there is no need for units larger than the gigaparsec.
Small ones :
ångström (Å). 10^{-10} m = 0.1 nm.
picometer (pm). 1/100 of an ångström.
Formerly known as a micromicron, bicron, or stigma.
femtometer or fermi (fm). 1/100000 ångström.
microångström. 1/1000000 ångström.
Only the picometer (pm) and femtometer (fm) are official SI units.
The ångström and its submultiples aren't.
The prevalent US spelling has been used in the above, but the British spelling seems to be
gaining ground for the "metre" and the various standard multiples of the "metre": "picometre",
"femtometre", etc.
This happens to be closer to the original French spelling:
"mètre", "kilomètre", "centimètre", "millimètre", etc.
Well below all of these is a truly minuscule "unit", the Planck length,
which is about 1.616 10^{-35 }m
and describes a scale at which space itself is thought to lack any kind of smoothness.
At the Planck scale, the very concept of length measurement becomes meaningless.
Surface Area
robster
(2001-04-15)
10 sq. chains to the acre. How many square inches in one acre?
An acre [Greek agros, field] is precisely 1/10 of a square furlong.
A furlong being 660 feet,
a square furlong is 660^{2} = 435600 square feet
and an acre is 43560 square feet.
There are 12^{2} = 144 square inches in a square foot,
so an acre is 43560 times 144 square inches, or exactly 6272640 square inches.
A (Gunter) chain is 66 ft (1/10 of a furlong).
The chain is itself divided into 100 Gunter links (each of those is
0.001 fur, 7.92 in or 20.1168 cm).
An acre is thus the area of a rectangle whose length is one furlong
and whose width is one chain.
Historically, the relation is reversed:
The furlong ["furrow-long"] was a basic unit so strongly favored by the Tudors
that they redefined the mile so that it would be exactly 8 furlongs.
This statute mile of 8 furlongs or 5280 ft thus displaced the previous
London mile of 5000 ft, which had a definition similar to that of the
Roman mile of 1000 strides (double-paces)
of 5 Roman feet each.
The acre was thus defined to be 1/10 of a square furlong well before Edmund Gunter
introduced the chain (in 1620) as the "width" of an acre.
Gunter's invention of the chain
[divided into 100 links of exactly 7.92 inches]
actually made it much easier to work out land areas expressed in acres.
A Gunter chain is also 4 poles.
Nowadays, a pole is an odd unit of exactly 16½ feet.
However, it is a much older unit which was defined as exactly 15
Saxon feet (also called "Drusus feet").
A furlong was exactly 600 Saxon feet or 40 poles
(a Saxon foot was thus exactly 11/10 of a modern foot),
which made a lot more sense in the old days.
So, the original furlong was the Saxon equivalent of the ancient Greek
stadion, which was similarly divided into 600 feet
(the length of a Greek foot varied from one city-state to the next).
On the other hand, the related Roman stadium was 1/8 of a Roman mile,
which may explain why the Tudors wanted 8 furlongs to their [statute] mile...
A lot of the bizarre conversion factors which are now floating around
were once perfectly sensible.
The way to (numerical) hell is paved with good intentions...
Volume, Capacity
( Lacy of Fort Walton Beach, FL.
2000-12-03)
Why is the abbreviation for liter "L" instead of [a lowercase] "l" ?
This is the only metric symbol which you may choose to capitalize or not.
You also have a choice between the US spelling and the
British one: "Litre" is becoming acceptable in US English the same way
"metre" is gaining ground as a favored (not "favoured")
spelling for the SI unit of length. Not so with the original French spellings of
mass units like "gramme" and "kilogramme", which remain confined to British English.
For all other metric units, the symbol is capitalized if, and only if, it has been named
in honor of a person,
whereas the unit name is never capitalized:
V for volt, Hz for hertz, A for ampere,
E for erlang..., but
m for meter and g for gram because these two were not named
after anybody!
Up until a few years ago, the recommendation was indeed a lowercase "l"
for liter, according to the common rule, but cursive script became commonly
used to make a clear distinction between a lowercase
""
and the numeral "1". When typing, a cursive
""
may not be an option and a capital "L" became acceptable.
The lowercase symbol was the only symbol adopted by the CIPM in 1879 and this was
confirmed in 1948, by the 9th CGPM. In 1979 however,
Resolution 6 of the 16th CGPM
recognized that both "L" and"l" should be accepted until actual practice
could be monitored by the CIPM, so the 18th CGPM could rule further...
In 1990, on the recommendation of the CIPM, the 18th CGPM declined to do so.
(Joan of Norwell, MA.
2000-11-05)
What are the formulas for changing ounces or teaspoons into drops?
(T.S. of Clarksboro, NJ.
2001-01-25)
How many drops are in a milliliter?
Note : Due to the two "li" syllables, the incorrect spelling "mililiter"
is more common than the wrong spelling "milimeter".
The standard SI prefix is "milli",
so it's "milliliter", not "mililiter"...
This note should make search engines deliver this page to anybody with a "mililiter"
query who may be surprised to have so few pages to choose from!
We apologize for quoting the wrong spelling "mililiter" 4 times.
In either the US (Winchester) or the UK (Imperial) system of liquid measures, a drop
is another name for a minim and there are 480 of these in a fluid ounce.
Thus, if you have a volume in ounces, multiply by 480 to have the
number of drops in it.
However, since the US and UK ounces are slightly different, a drop is
about 0.0616 cc in the US and only 0.0592 cc in the UK.
The metric drop is exactly 0.05 cc.
Nowadays, this is the conventional value worldwide:
20 metric drops
to a cubic centimeter (or milliliter).
Similar distinctions hold for teaspoons :
A teaspoon is 1/6 of a fl oz (about 4.929 cc in the US and 4.7355 cc in the UK).
So, there are exactly 80 drops in a teaspoon
(in either the Imperial or the Winchester system).
The metric teaspoon is slightly larger (5 cc)
and the metric drop slightly smaller (0.05 cc) than the nonmetric counterparts,
so there are exactly 100 metric drops in a metric teaspoon.
In a cubic centimeter or milliliter (cc, ml, or mL), there are exactly 20 metric drops
and about 16 Winchester drops or 17 Imperial drops (more precise values being
16.23 and 16.89 respectively).
Note that all of the above are conventional values,
which are only loosely related to the results you would actually get by using a
thin dropper. So, don't be disappointed at the lack of "accuracy" if you do.
dbsafe
(2001-06-21)
How do I convert milliliters into ounces?
Roughly speaking, divide a number of milliliter by about 30 to express that
volume in fluid ounces (fl oz). For example,
300 mL is about 10 fl oz.
Actually, the fluid ounce has different
values in the Winchester (US) system and in the Imperial (UK) system.
The US ounce is about 4% larger than the British ounce
(the ratio is 1.04084273078623608419542947895884...);
about 29.6 mL (29.6 cc) to the US fl oz
and 28.4 mL to the UK fl oz.
More precisely:
There are exactly29.5735295625 milliliters in a US ounce.
In the US, the Winchester system is used and the basic unit of capacity for fluids
is the US gallon, defined to be exactly 231 cubic inches.
Since 1959, the inch has been defined to be exactly 2.54 cm, and the number of
milliters in a cubic inch is thus 2.54^{3}=16.387064.
Now, there are 128 US ounces in a US gallon,
so the number of milliliters in a US ounce is exactly
(i.e., legally) 231/128 multiplied by 16.387064, which is the number advertised above.
There are exactly28.4130625 milliliters in a UK ounce (Imperial fl oz).
The British Imperial gallon was
first introduced in 1824
as the volume occupied by 10 pounds of water at 62°F.
Unlike the US gallon, it is divided into 160 fluid ounces.
(Since there are also 160 ounces of mass in 10 avoirdupois pounds,
this equated a fluid ounce with the volume of
one avoirdupois ounce of water at 62°F.)
The Imperial gallon was later redefined in metric terms as 4.54609 L,
making the number of milliliters in a fluid ounce exactly equal to
4546.09/160 = 28.4130625, as advertised.
A nice metrological opportunity was missed by British lawmakers in 1985
when they enacted a final metric equivalence for the Imperial gallon.
From 1976 to 1985, the official British equivalence had been 4.546092 L
to the gallon.
For a while, the Canadians had already been using a rounded-down equivalent
of 4.54609 L, on which the British decided to align themselves.
If both countries had chosen the nearest multiple of 16
(namely, 4.54608 L)
the Imperial ounce would be exactly 28.413 mL for all eternity.
We wouldn't be stuck with four extra decimals
and a ludicrous conversion factor that nobody will ever care to memorize. Too bad.
Since 1963, the Imperial gallon has ben defined purely in metric terms.
The 1963 definition merely translate a metrological renovation of the 1824 naive
definition (volume of 10 pounds of water at 62°F )
according to which an Imperial gallon ought to be equal to the
space occupied by 10 lb of distilled water of density 0.998859 g/mL,
weighed in air of density 0.001217 g/mL against weights of density 8.136 g/mL.
4.545964591 L from 1963 to 1976.
4.546092 L from 1976 to 1985.
4.54609 L since 1985.
4.53608 L future reform (?) to enforce: 1 fl oz = 28.413 mL
( A. B. of Saint George, UT.
2000-05-02)
5 Different Gallons...
How many milliliters [ml or mL] in a gallon?
The US gallon is the Winchester gallon,
now defined as exactly equal to 231 cubic inches
(this odd value comes from rounding up the
volume of a cylindrical measure 7 inches in diameter and 6 inches in height,
which dates back to the days of the
Magna Carta).
Since 1959, the inch is exactly 25.4 mm.
Therefore, there are exactly 3785.411784 ml
in a US gallon.
If the British Gallon is meant, the answer is 4546.09 ml, also an exact value
according to the 1985 British "Weights and Measures Act" (in 1963,
the British Parliament had decided to redefine all British units in metric terms).
There are about 277.42 cubic inches in this modern
Imperial gallon.
Originally (in 1819),
the Imperial gallon was meant to be the volume occupied by
10 pounds of water at 62°F.
It was thus intermediate in value between the two British units it replaced in 1824,
namely the corn gallon of 272¼ cubic inches (4461.378174 ml)
and the ale gallon of 282 cubic inches (4621.152048 ml).
The old British wine gallon of 231 cu in survives as the
US gallon (see above).
Finally, a US dry gallon is defined as 1/8 of a US bushel
(or Winchester bushel, see below) and is
thus exactly equal to 268.8025 cu in (4404.88377086 ml).
This unit was once known in England as the Winchester corn gallon.
(Gérard Michon.
2000-11-2)
Bushels and Gallons
A US bushel (bu) is defined to be exactly 2150.42 cubic inches.
How many bushels in a cylindrical container
74 inches in diameter and 50 inches deep?
Explain the "curious" numerical result...
With ludicrous precision: 100.000007969708869510499316219846+ bu.
There are very nearly 100 bushels in such a container! Here's why:
The US system of capacity is based on the Winchester system whose two basic units are
the gallon for liquids and the bushel for dry goods.
The ancient Celtic city of Winchester was once an important Roman community,
and it became the capital of England in the 9th century, when
the kings of Wessex ruled the country.
It seems
that the Winchester bushel was originally equivalent to
4 Roman modii (or 4/3 of a Roman cubic foot).
On the other hand, there does not seem to be any direct link between Roman
measures and the Winchester gallon for liquids.
In the Roman system, the congius was the basis for liquid measures;
There were 8 congii to the amphora
(defined as precisely one Roman cubic foot)
and the culleus of 20 amphorae was the largest liquid unit.
For dry goods, the basic unit was the sextarius
(so named because it was 1/6 of a congius).
The modius ("peck" » 8.8 L)
of 16 sextarii was the largest dry measure unit.
In other words, there were 3 modii to the amphora,
but the modius was not used at all for liquids.
Unlike larger units, the submultiples of the sextarius were used for
both liquids and dry goods:
hemina (1/2 of a sextarius), quartarius (1/4),
cyathus (1/12),
cochlear ("spoonful"; 48 cochlearia to the sextarius).
Note that the Roman talent
was the mass of an amphora of water and was divided into 80 librae (Roman pounds).
Henry VII [Tudor] reigned from 1485 to 1509.
In 1495, the Winchester bushel was legally defined as the capacity of
actual standard bushels bearing his seal and kept at the Exchequer.
In 1696, these were measured to be about 2145.6 cubic inches,
under the supervision of members of the British House of Commons
who were discussing some excise duty on malt.
It was then suggested that the bushel itself be defined as
a simple circular measure roughly equivalent to this.
This was enacted in
1701
(during the reign of William III of Orange)
when the Winchester bushel was legally redefined,
under the name of corn bushel,
as the capacity of "any round measure with a plain and even bottom,
being 18½ inches wide throughout and 8 inches deep"
(there would have been exactly 100 of these in the above container).
This volume was later rounded from 2150.420171...
down to exactly 2150.42 cubic inches,
which is how the so-called malt bushel has been normally defined since
at least 1795.
(We couldn't determine the exact point at which the older
cylindrical definition of this bushel faded from view.
Please, tell us
whatever you may know. Thanks.)
The same thing happened to the US gallon, which is a descendant of the old
Winchester wine gallon, a cylindrical measure
from the days of the Magna Carta: 7" in diameter and 6" deep,
or about 230.90706 cubic inches.
This capacity was statutorily rounded to 231 cubic inches in 1707,
by Anne Stuart
(it was thus once known as the Queen Anne wine gallon).
Both Winchester units are thus tied to the inch and have, in effect,
been redefined every time the inch was.
The current units of capacity are based on the 1959 international inch,
which is now forever defined in metric terms
(1" = 25.4 mm).
The US adopted the Winchester system for capacities in 1836, using the above equivalences.
The British had adopted the competing Imperial system in 1824,
on the totally different basis of an Imperial gallon
then introduced as the volume occupied by 10 lb of water at 62°F
(later redefined in metric terms, as exactly equal to
4.54609 L) and an Imperial bushel equal to
exactly 8 of these gallons (36.36872 L).
Nowadays, agricultural goods are no longer sold by volume.
Instead, weight equivalents of the bushel are used for various commodities:
60 lb to the US bushel for wheat and potatoes,
56 lb for rye,
53 lb for tomatoes,
48 lb for barley,
32 lb for oats,
20 lb for spinach,
etc.
jlj3394
(2001-01-15)
Kegs and Barrels
How many 12 oz beers are in a keg?
The US government defines (for tax purposes and such) a barrel of beer as exactly equal
to 31 US gallons (these are Winchester gallons of exactly 231 cubic inches,
not the Imperial gallons used in the UK).
The US brewing industry calls a [full] keg a quantity of beer equal to half of such
a barrel, namely 15.5 gallons (half a keg is called a "pony-keg" and equals
7.75 US gallons).
A US gallon being divided into 128 oz, the above implies that a keg equals 1984 oz,
or 165 and 1/3 times a "12 oz beer".
The 12 oz size (can or bottle) is most commonly
sold in "packs" of 6 or 12 ("6 pack" or "12 pack"), but retail packs of
18, 20, 24 or 30 are also widely available. Traditionally, a case of beer consists
of 24 cans or 24 bottles.
There are thus almost 7 cases of beer
(which would be 168 cans) to the keg.
The above (modern) US "barrel of beer" has nothing to do with the
international barrel (of oil), which is used to measure crude oil and is defined
to be exactly equal to 42 US gallons, or 9702 cubic inches (158.987294928 liters).
This unit is best abbreviated "bo" (barrel of oil) to distinguish
it from the many other types of "barrels" which are all abbreviated "bbl".
It is acceptable to use metric prefixes with the symbol "bo", but not with "bbl",
which is far too ambiguous...
Besides the international barrel (42 US gallons) and the above US barrel of beer
(31 US gallons), there's also a US barrel of wine
(most commonly 31.5 US gallon) and a "barrel bulk" of 5 cubic feet.
The US "dry barrel" is 7056 cubic inches; it was so defined in 1912
as the US "apple barrel" (it's thus almost exactly equal to 105 "dry quarts",
or 105/32 US bushels).
The "barrel of cranberries" is 5826 cubic inches.
All this covers only modern US usage...
The Imperial system formerly used in the UK included
a larger barrel ("dry barrel" or "barrel of beer") of 36 Imperial gallons
(163.65924 L).
Worse, the "barrel" is also used as a measure of mass, which comes in
several flavors as well: The "barrel of cement"
(4 bags) is 376 lb (376 avoirdupois pounds, or about 170.55 kg).
The "barrel" used in the US for pork, beef or fish is 200 lb (90.718474 kg),
whereas a "barrel of flour" is only 196 lb (88.90410452 kg)...
Mass, "Weight"
(C. B. of Philadelphia, PA.
2000-10-25)
Is there a [unit of] measurement smaller than a milligram?
Here's a list of the smaller official units of mass in "concrete" terms:
gram (g): A paper clip.
milligram (mg): Cubic millimeter of water. Mass of a typical ant.
microgram or gamma: Dust mite (dermatophagoides pteronyssinus).
nanogram (ng)
picogram (pg): A typical bacterium (Escherichia coli).
femtogram (fg)
attogram (ag): A typical virus, or 20 prions.
zeptogram (zg, 10^{-21}g):
3 gold atoms, or 33 water molecules.
yoctogram (yg, 10^{-24}g):
60% of a hydrogen atom.
The zeptogram and yoctogram have been officially recognized
by the CGPM since 1991.
An atom of hydrogen is about 1.66 yg.
An electron is about 0.00091 yg.
This is roughly equal to the next unit down
the list (namely, yg/1000 or 10^{-27}g),
which doesn't yet have an official name.
(2012-11-07)
Solar mass :
1.98855(24) 10^{30} kg
[ CODATA 2010 ]
The unit of mass in the astronomical system of units.
Thus, the ratio of the mass of the Sun to that of the Earth
(atmosphere included) is known with excellent
precision, namely: 332946.0438(8).
Although the Sun loses
millions of tons per second,
it will take more than 2000 years
for this to affect the least significant digit of that last ratio.
This is good enough to use the changing mass of the Sun
as a very practical unit which allows the mass of
large celestial bodies in the solar system
to be expressed with much more precision than SI units (kilograms)
would allow, using the values of their relative
gravitational constants, as defined above.
Body
Mass
Reciprocal
Sun
1
1
Jupiter
9.5479194 (74) 10^{-4}
1047.3486(8)
Earth + Moon
3.040432685(9) 10^{-6}
328900.5558(11)
Earth
3.003489661(7) 10^{-6}
332946.0438 (8)
Moon
3.69430242(46) 10^{-8}
27068710 (34)
The Earth is 81.30059(1) times as massive as the Moon.
("Biker" of Jerome, ID.
2000-10-09)
What is a slug, in the [engineering] weight measurement system?
The slug is a unit of mass.
The word was coined in a 1902 textbook by the British physicist
A.M. Worthington
to designate the British engineer's unit of mass, which appeared
in engineering calculations late in the 19th century.
The slug is defined as the mass which would accelerate
at a rate of 1 ft/s^{2} under a force of one pound-force (lbf).
Since 1 lbf is the force exerted on a mass of one pound by a
standard gravitational field (of exactly 9.80665 meters per square second),
a slug is thus exactly equal to 196133/6096 pounds
(about 32.1740485564 lb or 14.593902937206 kg).
It's worth making a few technical points about this:
The slug is the unit of mass in a coherent system called either
"British engineering system" or "English gravitational system".
On the other hand, the Imperial
(formerly "English") unit of mass is the pound (lb),
which is now defined in metric terms
(0.45359237 kg exactly)._{ }
The "metric equivalent" of the slug
is the hyl of exactly 9.80665 kg
which is the unit of mass of the so-called "metric-technical system".
The hyl is also called "metric slug" or
designated by the German acronym TME
(Technische Mass Einheit ).
A mass of one hyl gets accelerated at a rate of one meter per square second
by a force of one kilogram-force (namely, 9.80665 N)._{ }
The SI unit of mass is the kilogram,
not the gram or the hyl._{ }
Both the pound and the slug are units of mass.
The latter weighs about 32 times as much as the former,
even on the surface of the moon.
On the moon, however the weight of a pound-mass
(lb or lbm) is only about one sixth of a pound-force (lbf).
(2007-05-13) _{ }
Surviving customary units of mass, in the electronic age.
What are the units of mass available on modern electronic balances?
The customary units listed below are mostly kept alive by
gold traders.
A common feature of electronic analytical and/or precision balances
is the ability to use various customary units of mass.
Copying each other over the years
(often misspelling "baht" and/or "mesghal") manufacturers have picked from the
following limited catalog of units, which caters to all international traders.
In East Asia, the catty is to the tael
(TL)
what the pound (lb) is to the ounce (oz).
There are 16 taels to the catty...
The Taiwanese tael
(37½ g) thus corresponds to a catty
of 600 g, whereas the "tael of Singapore"
(defined as 1/12 lb or 4/3 oz)
corresponds to a catty of 4/3 lb
(about 604.79 g).
1000 grams (g) to the kilogram (kg). 7000 grains (gn) to the
avoidupois pound (lb). Note that the abbreviation "gr" is best shunned (as
it could stand for either grams or grains).
(*)
A "newton-mass" unit was (improperly) introduced by some instrument
makers as the mass (about 102 g)
on which a standard gravitational field
of 9.80665 m/s^{2}
would exert a force of exactly 1 N (1 newton).
This is yet another offspring of the ongoing confusion between mass and weight
(the latter being the force exerted by gravity on a given mass).
(**)
This is not the avoirdupois drachm (symbol dm.)
which is the smaller unit of only
1/16 oz (1.7718451953125 g) still used for loading ammunition in the US.
The unit found on electronic scales is the troy dram
(symbol: dr. or 3)
which belongs to the deprecated apothecaries' weight system,
(illegal for trade in the UK since 1985).
It's equal to 3.8879346 g :
60 grains (1/8 ozt) or 3 scruples.
The troy system and the apothecaries' weight system are
fully compatible (units with the same names have the same values)
but some units are unused in either system.
By contrast, the troy and avoirdupois systems are incompatible;
they only have one unit in common, the grain (gn).
Some popular
conversion tables make a clear distinction between the
"drachm" (dm) and the "dram" (dr):
16 dm to the ounce and 8 dr to the ounce...
Unfortunately, this misguided "clarification" is inconsistent with
ancient usage: Actually, the former unit (1/16 oz) belongs
exclusively to the avoirdupois system and the latter (1/8 ozt) to the troy system,
while each unit may be called by either name (drachm or dram)
within its own system.
(2007-06-03)
Royal French weights:
18827.15 grains to the kg.
The ancient livre de Charlemagne and the
poids de marc system.
From the early definitions of the kilogram
survives only an exact equivalence between the old French units of mass
poids de marc and the metric system;
there are exactly
18827.15 French grains to the kilogram.
Once it had been realized that a definition of the kilogram as the mass of a
cubic decimeter of water was not satisfying
(by the metrological standards of the late 18th century)
the kilogram was evaluated using the best system of
weights then available.
In pre-revolutionary France, that was based on a
famous artifact known as the pile de Charlemagne,
which is still preserved in the
Musée National des Techniques in Paris, France.
In spite of its name, the "Pile de Charlemagne" was built in the 14^{th}
century—half a millenium after the days of Emperor Charlemagne.
It consists of 13 copper weights in the form of
truncated cones (larger base on top).
Except for the smallest one, all of
those are hollow, so that the next smaller weight may fit in it snugly.
The largest weight has a handle and a lid and serves as a box for the whole
thing, which stands at a height of about 9cm, with a top diameter of about 15.5cm,
and a lower diameter around 14cm (for a volume of about
1.54L, and a mass of about 12.2376 kg).
The nominal mass of this revered standard was exactly 50 marcs
(of 4608 French grains each). The royal French
livre (better known as
livre poids de marc )
was once defined as 1/25 of the "pile de Charlemagne" mass.
When the kilogram (then called the grave)
was first defined (on August 1, 1793), it was equated to
18841 grains of
the above poids de marc system,
from a single measurement by
Lavoisier and
Haüy.
Early in 1799, an accurate equivalence of
18827.15 grains
to the kilogram was established...
That last measurement was due to the French academician
Louis
Lefèvre-Gineau (1751-1829)
and the Italian engineer
Jean-Valentin
Fabbroni (1752-1822)
[elected to France's Corps Législatif in
1809].
Defining the kilogram as the mass of a cubic decimeter of water at 4°C
(close to the densest point at 3.984°C)
they weighed a hollow brass cylinder of known dimensions,
first in the air, then in water at 4°C.
The new determination was enacted on May 30, 1799, and it became the
final legal
equivalence between the kilogram and the "old" French units.
Because of that, the metric equivalent of the French
grain no longer depends on
the actual mass of the "pile de Charlemagne".
However, we may remark that,
at 9216 grains to
the livre , the
Pile de Charlemagne has a nominal mass of
230400 / 18827.15 or about 12.237646165...kg
whereas its actual mass has been
measured to be 12.2376429 kg.
Interestingly, the 0.27 ppm difference translates into 0.005 grains/kg,
which goes to show that the above 18th century determination (which settled,
once and for all, the conversion factor between grains
and kilograms) was indeed
fully accurate.
Since the newer equivalence was quite different from Lavoisier's original one
(13.85 grains is about 3/4 of a gram)
the standard weights that had been sent to all departmental
chef-lieus
had to be recalled. New ones were made.
The same thing did not happen for prototypes of the meter because the
difference with revised standards of length was considered acceptable.
The old French system of 18 onces to the
livre had been introduced in the wake of Charlemagne's
monetary reform.
The once
was understood to be exactly
the same as the Roman uncia
but there were 18 of those to
Charlemagne's livre
(French pound, poids de marc )
as opposed to 12 unciae to the
libra (Roman pound).
So, a French pound was exactly
1½ Roman pounds.
The above thus provides a paper trail
to what may be construed as a "legal" value of the ancient Roman
pound in metric terms, namely:
1 Roman pound (libra) =
12 onces (of 512 grains)
= 0.326337231... kg
(J. W. of Tustin, CA.
2001-02-07) Biblical Units
How many pounds was a talent?
How many ounces was a shekel?
A talent was the mass of a cubic foot of water.
The exact value of the talent thus depended on what foot was in use
in a specific part of the world at a certain period in history.
If there was such a thing as a modern Imperial talent
(based on water at 62°F)
it would be about 62.288 lb (or 28.25 kg).
The Roman talent was also defined as 80 Roman pounds
("librae", plural of "libra").
The above value of the libra,
from the days of Charlemagne, makes the Roman talent equal to about
26.107 kg.
Incidentally, this would imply a value
of about 0.2969 m for the Roman foot
(water at 62°F has a density of 10 lb per Imperial gallon).
For some obscure reason,
a foot whose length is derived backwards
from a given value of the talent is called a geometric foot.
The ancient Sumerian talent is estimated at about
28.8 kg (about 63.5 lb)
from the mass of surviving standard weights (basalt statuettes in the form of
sleeping ducks).
Outside of Rome, the talent was normally divided into 60 minas;
a mina (or maneh)
was thus roughly equal to a modern avoirdupois pound.
The shekel was always some submultiple of this mina:
The Babylonian shekel was 1/60 mina, the Phoenician shekel was 1/25 mina,
the Egyptian shekel was 1/100 mina,
whereas the "modern" Palestinian or Syrian shekel is 1/50 of a mina.
Solomon's mina of gold (1 Kings 10:17) was divided into
100 units (unnamed in the Hebrew text of 2 Chr. 9:16) not necessarily
related to the Biblical shekel of the sanctuary
(bishekel hachodesh)
whose value ought to be determined by the last words of Ezekiel 45:12.
Unfortunately, Bible scholars have been advocating
at least
two contradictory renditions of that verse, namely:
50 shekels to a mina
(Septuagint, according to Walther Zimmerli):
"[...] 5 shekels are to be 5, and 10 shekels are to be 10,
and 50 shekels are to amount to a mina with you."_{ }
60 shekels to a mina (King James and other English versions, also
supported by Rabbi Nosson Scherman, in the Stone Edition Tanach):
"[...] 20 shekels, 25 shekels, and 15 shekels shall be your mina."
The latter may have exhorted traders to check their minas against smaller
standard weights... If you know for sure, please
tell me.
There are many different kinds of tons.
In the US, you're most likely to encounter the short ton
(2000 lb, or about 907.185 kg) unless you're primarily concerned with ships,
for which the displacement ton and the gross ton are in fact units of
mass both equivalent to the British long ton of 160 stones
(2240 lb, or about 1016 kg).
The long ton is retained in this international context
because it's almost exactly equal to the mass of a cubic meter of seawater.
This is a prime example of crossbreeding between the metric and Imperial systems.
Another example of interbreeding between the metric system and the Imperial system
(and the troy system)
involves a much smaller "ton", the assay ton,
which is slighly more than an ounce.
It's defined to make 1 milligram
per assay ton equivalent to one troy ounce (ozt) per ton.
There are 2 or 3 kinds of assay tons, depending on which reference "ton" is used.
The most common one seems to be the short assay ton
of 29.1666... g, which corresponds to the ton of 2000 lb.
A troy ounce (ozt)
per ton is a milligram (mg) per assay ton.
Ton, in lb
Assay ton, in g
short ton
2000 lb
175 / 6
29.16666... g
long ton
2240 lb
98 / 3
32.66666... g
troy ton
2016 lb
147 / 5
29.4 g
Other types of tons include the very important metric ton
(better spelled tonne, which corresponds to 1000 kg or about 2204.62 lb)
and the totally unimportant and unused troy ton of 144 stones
(2450 lbt = 2016 lb = 914.44221792 kg).
The pound is understood to be the common
avoirdupois pound ("lb" or "avdp lb") of exactly 0.45359237 kg
(a 1959 international statute now defines the pound in metric terms).
For the record, the troy pound (lbt) was officially abandoned on
January 6, 1879 (175 lbt = 144 lb).
However, the troy ounce (ozt) is still widely used for
precious metals.
As if this were not bad enough, a few units of volume
are also called tons:
This includes, most notably, the international register ton
of 100 cubic feet (2831.6846592 L).
Of lesser importance is the British water ton of (exactly) 224 Imperial gallons,
which originally corresponded to the volume occupied by a long ton (2240 lb)
of distilled water at 62°F, when the Imperial gallon was still defined in like
terms as a "10 pound gallon".
(Under the modern definition of the Imperial
gallon, in metric terms, the British water ton is exactly 1018.32416 L.)
On the other hand, the unit variously called shipping ton, freight ton
or marine ton is 40 cubic feet (1132.67386368 L),
which happens to be equal to the so-called
ton of timber (of 480 board feet).
There's also
a fluid ton of 32 cubic feet (906.139090944 L),
a corn ton of 32 bushels
(which means exactly 1127.65024534016 L in the US and 1163.79904 L in the UK),
and a British tun, spelled with a "U", of two pipes or 252 Imperial gallons
(1145.61468 L).
I just wanted to drop you a line to tell you that I enjoyed your treatise on the "ton"(s).
[above]
For completeness, it would be interesting if you were to also mention and/or
describe the origination/relation of the "refrigeration ton" and/or the "explosion ton" units.
Regards, Darren Finck
Thanks for the kind words, Darren.
First a general remark:
The adjective "extensive" qualifies (loosely speaking) physical quantities for
which the measure of the whole is the sum of the measures of the parts.
Volume and mass are examples of extensive quantities
(pressure and temperature are not).
Choosing some "stuff" of reference, like water under normal conditions, establishes
a "conversion factor" (coefficient of proportionality)
between any pair of extensive quantities and/or
the units which measure them.
New "practical" units may thus be created ad nauseam,
including many flavors of tons which correspond to various
extensive properties of a ton of "stuff".
This is how some of the "tons" mentioned above as units
of mass gave rise to units of volume
(a volume of one ton being the volume occupied under standard
conditions by a mass of one ton of water).
This is also how a unit of mass may become a unit of force
(the corresponding weight in a standard gravitational field,
equal to 9.80665 m/s^{2 }).
In particular, the ton of thrust is a unit of force equal to the standard
weight of a metric ton/tonne, namely 9806.65 N.
[The newton (N) is the SI unit of force.
Applying for 1 second a force of 1 N
to a mass of 1 kg, initially at rest, will make it move at a speed of 1 m/s.]
The ton unit pertaining to nuclear explosions
is a unit of energy equal to 1000 000 000
thermochemical calories (of exactly 4.184 J)
and is thus exactly equal to 4184 000 000 joules.
(The kiloton and megaton are a thousand and a million times as large.)
Detonating
1000 kg of TNT
(227.134 g/mol) yields only 64% of such a ton:
C_{7}H_{5}N_{3}O_{6}
® 6 CO
+ ^{5}/_{2} H_{2}
+ ^{3}/_{2} N_{2}
+ C + 608.8 kJ
The carbon (C) produced appears as black smoke.
Some residues may subsequently burn in air to give
more energy (393.51 kJ per mole of carbon,
241.826 kJ per mole of hydrogen gas,
282.98 kJ per mole of CO).
The total heat
of combustion of TNT is thus about 3305 kJ/mol, which
translates into 3½ of the above tons of energy
for 1000 kg of TNT (227.13 g/mol)...
What's wrong?
Well, to optimize the energy of the initial blast, an oxidizer
(ammonium nitrate = AN) must be added to TNT
to form a balanced high explosive, called amatol.
The optimal proportion for a given total weight is 78.7% AN and 21.3% TNT,
matching the stoichiometry of the following reaction.
(A slight excess of AN seems better for dynamic reasons,
so the usual mix is 80/20.)
This yields 1.0174 tons of energy when 1000 kg of the mix are detonated,
which justifies quantitatively the term "ton of TNT "
commonly used for the above ton of energy,
although "ton of amatol" would have been more proper...
Other types of "tons" are used to measure energy in a more peaceful context:
Burning a ton of crude oil releases
about 10 times as much energy as exploding a ton of TNT/amatol.
On the other hand, the best grade of coal (anthracite) is supposed
to be about 30% less efficient than oil.
Burning pure carbon completely into carbon dioxide would
release about 393.51 kJ/mol, which is more than 7800 cal/g
(a mole of carbon is 12 g).
However, actual coal can be much less efficient;
see below.
This gave rise to two other "ton" units for measuring energy,
the ton oil equivalent (toe) and the ton coal equivalent (tce):
1 tce = 0.7 toe.
Both refer to metric tons (1000 kg) but, unlike the ton of TNT,
they are usually defined as round multiples of the IT calorie
(International Steam Table calorie
of exactly 4.1868 J instead of 4.184 J):
1 tce = 29 307 600 000 J
1 toe = 41 868 000 000 J
Natural gas is an important source of energy as well, so that the toe has also
been given the following equivalences in term of gas quantities,
using the different units of measurement preferred in various regions of the Globe
(these values are, unfortunately, slightly incompatible with each other
and with the above):
USA : 42900 cubic feet (about 1214.8 cubic meters).
Europe : 1270 cubic meters.
Japan : 0.855 metric tons of LNG ("Liquefied Natural Gas").
Now, the ton of refrigeration
or ton of cooling is a unit of power
(which can't be compared with any of the above units of energy).
It was first defined as the power released by a ton (2000 lb) of water when it freezes
in one day (86400 seconds) or, conversely,
the power absorbed by a ton of ice which melts in a day.
This would be about 3502.6 W (watts),
but the ton of cooling is now conventionally defined as
exactly 12000 Btu/h (about 3516.852842 W),
based on the rounded value of 144 Btu/lb for the latent heat of fusion of water.
In the United Sates,
air conditioning units
are now rated using the Btu of cooling, which is a unit of power simply
equal to a Btu per hour
(about 0.293 W, more precisely 0.2930710701722222...W).
The labeling of A/C units is in terms of thousands of Btu [per hour]
(typically: 024, 030, 036, 042, 048, or 060), but
betrays its origin in terms of tons of cooling
(2, 2½, 3, 3½, 4, or 5 tons of refrigeration).
The electrical energy fed to the motor of an A/C unit may allow the transfer of a greater
energy "uphill", from cold to hot.
The ratio of these two energies is called the coefficient of performance (COP),
which is normally much more than 100%.
This would be clear if refrigeration and electrical powers were both expressed
in the same units (W/W), but this fact is obscured in the US,
where the so-called EER (Energy Efficiency Ratio) is used instead:
An EER of 10 means 10 Btu/h/W,
(a COP of about 2.93 W/W, or 293%).
An EER of 15 is 439.6%.
Last, and probably least, we're told that the "ton" is also an informal
British unit of speed equal to 100 mph (160.9344 km/h or 44.704 m/s).
[Colloquially, in the UK, a ton can be 100 times
as large as any commonly understood unit.]