# Why is kg the standard unit for mass and not g in SI?

Why is $\mathrm{kg}$ the standard unit for mass and not $\mathrm{g}$?

I know that there is the kilogramme des Archives which is a kilogram and not a gram. But originally on April 7, 1795 the gram was defined as

The absolute weight of a volume of pure water equal to the cube of the hundredth part of the metre, and at the temperature of melting ice.

What is the reason that they switched to the $\mathrm{kg}$ when using the kilogramme des Archives? Perhaps it was easier to make and less sensitive to mistakes? Are there other reasons?

To clarify why I think this is weird: The other six standards, namely metre, second, ampere, Kelvin, mole and candela don't have a SI prefix when used as standard unit.

• It depends on what do you mean by 'standard.' Different kinds of units are used depending on the context, specially in physics. In the end a kilo is just a prefix to the unit, so the elementary unit is the gram. On the other hand, the widespread use of $\text{kg}$ as a unit in our daily life is probably due the fact that the order of magnitude of most of the things we deal with (including our own weight) is in the kilogram order. – hjhjhj57 Jul 20 '15 at 20:15
• Because using grams makes most commonly occurring masses into large (and long) numbers, which is inconvenient. – Conifold Jul 20 '15 at 21:35
• Some people use cgs (centimeter-gram-second) units and others use SI (meter-kilogram-second), so it's not really true that the kg is the basic unit for mass. The physical artifact used as a standard is presumably a kilogram rather than a gram for reasons of convenience and precision, e.g., corrosion or dust would be more significant on a smaller object. – Ben Crowell Jul 21 '15 at 2:52
• Depending on the field you're in, you can use cgs, mks, or any other system of units you choose. In Astrophysics, for example, cgs is more common. In particle physics, one uses neither this nor that, but rather a "natural" set of units. – Omry Jul 21 '15 at 8:25
• @Conifold That wouldn't explain though why it is the SI standard. In everyday people als talk more about hours (in the order of 1 ks), and weeks (in the order of 1 Ms), than in seconds itself. – wythagoras Jul 21 '15 at 19:27

Why is kg the standard unit for mass and not g?

Tongue in cheek answer: Because a foolish consistency is the hobgoblin of little minds?

More seriously, none of the immediate predecessors of the SI bothered to have all of their base units be consistent with the prefix-free units. Gauss proposed a millimeter-gram-second system in the 1830s. Maxwell and Thomson modified this to a centimeter-gram-second system in the 1860s. There was a lot of infighting over the electromagnetic units in those CGS systems. Giorgi proposed yet another system in 1901, the meter-kilogram-second-ampere system. This system is the immediate predecessor to the current International System.

What is the reason that they switched to the kg when using the kilogramme des Archives? Perhaps it was easier to make and less sensitive to mistakes? Are there other reasons?

You have it backwards. The original concept of mass by the French revolutionaries working on the metric system was the mass of a liter of water. This unit of mass was to be called the grave. French scientists worked on making this realizable (the mass of a volume water turned out not to form a good basis). The Republican government that followed the French Revolution thought this grave was too big for practical uses, so they invented the gramme as the mass of a milliliter of water. The work on the grave prototype continued, only now this would be called the kilogram prototype.

• Why grave ? ? ? – Pacerier Jul 30 '17 at 14:09
• @Pacerier -- Because the inventors were French, not English. The French grave comes from the Latin gravis, which means "heavy". (Note: Thanks to 1066, this is one of the two very distinct meanings of the English grave. The other meaning comes from the old English grafan, "to dig".) – David Hammen Jul 30 '17 at 14:25
• I still wonder why didn't they simply rename kg to something else that doesn't have a prefix so all fundamental units would have been unprefixed. – Calmarius Sep 25 '20 at 16:29

The kilogram is the base unit of mass because electrical engineers in the late 19th century chose a particular set of practical electrical units. Their practical units were a success, and we are still using them today: ohm, volt, and ampere. In 1881 the International Electrotechnical Commission (IEC) created two sets of units: a set of theoretical units, and a set of practical units. The theoretical electrical units, abampere, abvolt, abohm were coherent with the mechanical units cm, g, s. Coherence in this case primarily means that electrical energy and mechanical energy have identical units: $V\cdot I\cdot t = F \cdot L$. Unfortunately, the abvolt and abohm were inconveniently small. On the other hand, the practical electrical units, ampere, volt, and ohm, were not coherent with cm, g, s, nor with m, g, s. However, by coincidence they were coherent with m, kg, s. That is why the kilogram was chosen as the base unit of mass in the SI system, in 1960.

• So the engineers won again. – Pacerier Jul 30 '17 at 14:09
• @Pacerier As it should be :-). – Russell McMahon Jan 3 '20 at 3:39
• This is correct, although it leaves out a subtle point. I added an answer as a sort of addendum to this answer. The subtle point is that the non-mechanical units like the volt and the ampere are not coherent in the three-dimensional MKS system—only in the QES system are all practical units coherent. Only the purely mechanical practical units such as the watt and the joule are coherent in the three dimensional MKS, and that truly is a lucky accident. It is what makes it possible to include the practical units in a coherent four-dimensional MKSA system (whereas e.g. 'CGSA' would not work). – linguisticturn Jun 11 '20 at 12:30

The overall presentation is largely borrowed from here, but the actual facts come mostly from these sources: here, here and here.

Introduction

One has to be careful when saying that the practical units (the volt, the ampere, etc.) were 'coherent with' the meter-kilogram-second (MKS) system.

If by 'MKS' we mean a three-dimensional mass-length-time system, then the volt, the ampere, and the ohm were most definitely not coherent units in it. (However—and this will be the key—the product volt × ampere, which was named the watt, is a purely mechanical unit—of power—which was coherent in MKS. It is that fact—that the watt is coherent in MKS—that truly was a lucky accident.)

On the other hand, if by 'MKS' we really mean a four-dimensional, MKSX system, where 'X' is the unit of some non-mechanical electric quantity,1 then it is incorrect to say that it is an accident that the volt, the ampere, etc. were coherent units in such a system—of course they were, since one of them was chosen to be a base unit!

1Serious consideration was given to proposals where X was either the coulomb, or the ampere, or the ohm, or the volt. Eventually, metrological considerations turned out to favor the ampere.

The key here are the units that straddle both the electric and the mechanical domains—in particular, the watt. It is those units that would have made it impossible to extend the CGS system by e.g. adding the ampere to it as a fourth independent base unit: the watt is not equal to the erg per second, and so would not be a coherent derived unit in such a system. However, the watt does turn out to be kg × meter2/second3, and so the practical electric units can be integrated with the MKS system by adding a fourth independent base unit.

Discussion

At the time the 'practical' electric units were adopted (1873–1893), everyone as a matter of course assumed that a scientific system of units should be absolute, meaning that the base dimensions should be just the three mechanical ones: length, mass, and time. The abvolt, for example, is g1/2cm3/2/s2 when expressed in the base cgs units (see here). Now, there is indeed an absolute (i.e. a three-dimensional, length-mass-time) system is which the practical units are coherent, but it is not the meter-kilogram-second system. It is, rather, a system in which the base unit of length is 107 meters (called a quadrant, as it is very nearly one half of a meridian of the Earth), and the base unit of mass is 10-11 grams (an eleventh-gram): the quadrant-eleventh-gram-second (QES) system.

This can be derived from the following facts. The practical units were defined in 1873 as decimal multiples and submultiples of the 'electromagnetic' absolute cgs units, cgs-emu. We will somewhat anachronistically use the following names for the emu units: the 'abvolt' for the potential, the 'abampere' for the current, etc.1 When expressed in the base cgs units, the abvolt is g1/2cm3/2/s2, the abampere is g1/2cm1/2/s, and the abcoulomb is g1/2cm1/2 (see here, here, and here). On the other hand, the volt was defined as 108 abvolts, the ampere as 0.1 abamperes, and the coulomb as 0.1 abcoulombs (see the same three links). Now imagine we change the base units of lenghts, mass, and time by factors of M, L, and T, respectively. Then the base unit of potental will become (M g)1/2(L cm)3/2/(T s)2 = M1/2L3/2/T2 × g1/2cm3/2/s2 = M1/2L3/2/T2 abvolts. We want this new unit to be the volt, so we must have M1/2L3/2/T2 = 108. Similarly, if we want the new unit for current to be the ampere, we obtain that M1/2L1/2/T = 0.1, and if we want the new unit of charge to be the coulomb, we obtain that M1/2L1/2 = 0.1. We thus have a system of three equations with three unknwns. The solution is L = 109 (so the base unif of length should be 109 cm = 107 m), M = 10-11 (so the base unit of mass should be 10-11 g), and T = 1 (so the second remains the base unit of time).

1This naming convention, where the name of the emu unit is formed by adding a prefix 'ab-' (short for 'absolute') to the name of the corresponding practical unit, came only in 1903, three decades after the practical units were originally defined in terms of the absolute emu units. At that earlier time, the absolute cgs electric units themselves didn't have any special names. One just used 'e.m.u.' or 'C.G.S', as in 'a current of 5 e.m.u.' or '5 C.G.S. units of current' (or perhaps one could also use the base units, e.g. a current of 5 g1/2cm1/2/s) However, for convenience, we will use 'abvolt', 'abampere', etc. in what follows.

The way the meter-kilogram-second system enters the story is this. In addition to the purely electric and magnetic units such as the ohm, the volt, the ampere, etc., the practical system of units also had to include some purely mechanical units. This is because of relations such as voltage × current = power. In particular, the volt times the ampere gives a unit of power, which was in 1882 given a special name: the watt. Then the watt times the second gives a unit of energy, which was named the joule. Of course, these purely mechanical practical units were coherent in the QES system. However, they are in fact coherent in a whole family of systems. To see why that is so, recall that the dimensions of power are ML2/T3. It follows that if the watt is coherent in a system, it will also be coherent in any system obtained from the original system by simultaneously changing the base unit of length by a factor of L and the base unit of mass by a factor of M in such a way that ML2=1, i.e. in such a way that M=L-2. We are told that the watt is coherent in the QES system; thus, it will also be coherent in any system in which the base unit of length is L × 107 meters while the base unit of mass is L-2 × 10-11 grams. Picking L = 10-7 gives the meter and the kilogram. Moreover, it is easy to check that, if we insist that the new base units should be decimal multiples or submultiples of the meter and the gram, then the choice L = 10-7 is the only choice that produces base units of practical sizes. For example, if we pick L = 10-8, so the base unit of length is the decimeter, then the base unit of mass becomes 1016 × 10-11 grams = 105 grams = 100 kg, which is impractically large.

Probably many people noticed that the watt is coherent in the meter-kilogram-second system, but it was Giovanni Giorgi who really took note of it. He had the further insight—which was sort of iconoclastic at the time—that while the purely electric and magnetic units cannot be made coherent in the three-dimensional meter-kilogram-second system, they could be made coherent in a four-dimensional extension of that system. Thus he proposed, in 1901, to introduce a fourth base dimension, which would be purely electric or magnetic. In principle, this fourth independent dimension could be any electromagnetic quantity, but only four received serious consideration: electric charge, electric current, electric resistance, and electric potential. Eventually, electric current was chosen because it was most advantageous metrologically. Another selling point of Giorgi's system was that it made it possible to rationalize (i.e. remove the awkward factors of 4π from) Maxwell's equations without a corresponding redefinition of units by factors of (4π)1/2 (which is what happens when the Gaussian system is rationalized, giving the Lorentz-Heaviside system).

The Giorgi proposal (with the ampere as the fourth base unit) was adopted by the International Electrotechnical Commission in 1935 and by the CGPM in 1946; the CGMP later incorporated it into the SI system.

Summary

The fact that the kilogram rather than the gram is the base unit of mass in the SI is all the more remarkable given that, for about a century, the scientific community had been almost universally using the centimeter-gram-second system. Let me summarize the main reason why the CGS was abandoned and the meter-kilogram-second (MKS) was adopted. The main background facts to be aware of are that (a) by the end of the 19th century, the so-called 'practical system' of electric units had become nearly universaly accepted in practical applications of electricity such as telegraphy, and (b) this system of units included the volt and the ampere, and therefore also their product; but this product is a purely mechanical unit (of power), and if one multiplies that by the second, one gets another purely mechanical unit (of energy). In 1882, these two units were named, respectively, the watt and the joule. Now: the MKS is the unique system which has all three of the following characteristics (and which keeps the second as a unit of time): 1. the watt and the joule are coherent, 2. the base units of length and mass are decimal multiples of the meter and the gram (so that the system is 'properly metric'), and 3. the sizes of the base units of length and mass are convenient (more or less) for practical work. All this assumes that the second remains the base unit of time; but it is definitely true that any proposal to replace the second would have been dismissed out of hand. The non-mechanical units such as the volt, the ampere, etc. are not coherent in a three-dimensional MKS system, which is why a fourth independent dimension was added: the ampere became a new base unit, dimensionally independent from the meter, the kilogram, and the second.

• What I find odd is why they didn't go for the metric tonne. It makes way more sense together with the meter (a cubic meter of water is a metric tonne). So both units are used in the same fields (construction, bulk production, ...). Just as we use "kilo" now as a shorthand for kilogram, we could use "milli" as a shorthand for millitonne (the same amount). And it sounds nicer to weigh 90 millis than 90 kilos :D – sanderd17 Jul 17 '20 at 6:43
• @sanderd17 What you are referring to is called an MTS system. It was technically the only legal system in France between 1919 and 1961 (although it wasn't actually used much), and it was also official in the Soviet Union from 1933 to 1955. It had various named derived units, such as the sthene for force, the pieze for pressure, and thermie for heat energy. – linguisticturn Jul 17 '20 at 16:51
• @sanderd17 A key principle that the SI was supposed to follow was that the practical electric units (the volt, the ampere, the watt) should be coherent in it. And the watt simply isn't coherent in the MTS system. As I explained above, in order for the watt to be coherent, the base units must be L meters and 1/L^2 kg. If you want the base unit of length to be a decimal multiple or submultiple of the meter, your choices are (1 m, 1 kg), (1 dm, 100 kg), (1 cm, 10 000 kg), (1 mm, 10^6 kg), ... If you really want the metric ton, your base unit of length must be 1/(1000)^(1/2) = 3.1623… cm. – linguisticturn Jul 17 '20 at 16:52
• @sanderd17 Why didn't the MTS see more use in e.g. France? Well, note that the base unit of mass/weight is often on the order of a kilogram (both 0.5 kg and 4 kg are ''on the order' of 1 kg): the pound (~450 g); the Chinese catty (500 g-600 g); the Japanese kan (~3.75 kg); the Indian ser (~640 g). We may conclude that units in the 1 kg range are the most convenient for most kinds of everyday use. – linguisticturn Jul 17 '20 at 16:52
• @sanderd17 Much smaller and much bigger units do have their uses, but these are usually more specialized, so systems that are based on such units don't see wide adoption. The CGS system, for example, was widely used by scientists (and some fields still use it almost exclusively, e.g. astronomy), but even engineers didn't use it that much, not to speak of the public at large. And even scientists shied away from adopting Gauss's milligram-based system; using milligrams made sense for Gauss (who was mostly interested in masses of magnetic compass needles), but not for most other people. – linguisticturn Jul 17 '20 at 16:52