Our methods of measurement define who we are and what we value.
Ken Alder, The Measure of All Things
Units are taught infrequently now. Their blandness excites less than drying paint. Only the heart of a true pedant beats faster at the prospect of a good, long argument over the merits of his preferred unit system. They are also irrelevant in the sense that the universe operates in exactly the same way regardless of humanity's attempts to document it. Following Thomson, however, there is something reassuring about numbers. Drop a mass, m, on your foot and you worry. Drop nine grams and you don't.
In magnetism, unfortunately, you quickly encounter the problem of its unit systems. That's right: systems, plural. You can understand that the world has different unit systems for length or mass. One half talks of pounds, feet and inches while the other says grams, millimetres and metres*. People by their millions find the older 'imperial' or British system works for them (despite the occasional mishap). They conclude it not to be worth effort and expense changing.
Magnetism, you might think, would be different. It's only scientists and engineers who calculate it, and that it would be easy for them to concur about units. Having more than one set of units makes life hard for beginners in a subject which is quite difficult enough already, thank you very much. Sadly, scientists have spent at least seventy years disagreeing about the units for electromagnetism. Even were everyone to adopt the 'new' SI units the need for familiarity with the older CGS systems would remain in order to understand the large number of original works published in the latter. We'll spend a couple of paragraphs on the historical reasons for this schism because it also sheds light on the science. To this end, we also ignore all post-2019 developments since they do not explain how diversity arose.
The laws of Ampère and Coulomb
The trouble started with the definitions for the units of electric current and electric charge. Three simple, but fundamental, equations are crucial here. The first is Ampère's Force Law from which may be derived the electromechanical force, Fm, between two thin, parallel wires (see fig USA). The length of the wires, s, has to be much greater than their separation, a, and the wires must be thin in comparison with a. These conditions ensure that -
- The field produced by each wire decays at a rate proportional
to 1 / a,
- Field lines produced by wire 1 which intersect wire 2 do so at right angles both to wire 2 and a line joining 1 to 2 normally, and vice versa.
If s > 15a then the error will be below 1%. The Force Law predicts the force on these wires to be -
where the magnetic force constant, km or κm, is a constant whose value depends upon the unit system used to measure the currents, forces and distances. In other words, having decided how to define distance and force, you can then choose a km that defines how large your unit of current will be. The factor of 2 here is not intrinsic to Ampère's Force Law but arises through its application to the particular case of parallel wires. Some writers incorporate it into km, but most do not.
The second equation is Coulomb's Law which gives the electrostatic force, Fe, between two isolated particles, separated by a distance, r, and carrying electric charges Q1 and Q2 -
Where ke is a constant whose value depends upon the unit system used to measure the charges, forces and distances. In other words, having decided how to define distance and force, you can then choose a ke that defines how large your unit of charge will be. Or, at least you could do if you had not already decided upon km. Why? To understand this we need to introduce our third equation which expresses the fact that electric charge, Q, and current, I, are dependent quantities -
where T is the time, in seconds, for which a given current, I, flows. Equation USQ must hold regardless of the unit system chosen; one unit of charge being represented by the passage of one unit of current for one second. You should now be able to see that it is possible either to invent some convenient value for km and then calculate ke or else invent some convenient value for ke and then calculate km. You cannot choose both ke and km independently because that would violate Equation USQ.
Performing both Ampere's and Coulomb's experiments (Weber and Kohlrausch, 1856) gives you a value for the relationship between ke and km -
where c is the speed of light in a vacuum. This equation is considered further below.
That's the basic physics out of the way. However, the freedom to select an arbitrary value for either ke or km leads to the profusion of unit systems from which we suffer today. Consider next how the choices just described are made in the two major unit systems: the CGS and the SI.
The approach taken by the CGS systems
Every electromagnetic quantity may be defined with reference to the
fundamental units of Length, Mass, and Time.
James Clerk Maxwell, A Treatise on Electricity and Magnetism
In the unit system called CGS (for Centimetres Grams Seconds) force is measured in dynes: the force required to accelerate a mass of one gram at one centimetre per second squared.
|SI||force (mechanical)||F||newton||N||1.0×105 dyn|
|CGS||force (mechanical)||F||dyne||dyn||1.0×10-5 N|
However, the CGS system doesn't like to pick winners and losers in the km & ke game. Instead it defines two subsystems known as electromagnetic units (e.m.u. or CGSm) and electrostatic units (e.s.u. or CGSe).
The electromagnetic units subsystem
In the e.m.u. subsystem it is Monsieur Ampère who is the winner, but kmCGSm takes the value of 1.0. That is, two wires one centimetre apart carrying 1 e.m.u. of current experience a force of two dynes for each centimetre of their length. What is the value of this current in SI units? Rewriting Ampère's Force Law shows -
This is consistently in SI units so we must translate the CGS force and length before substituting them -
In CGS electromagnetic units this is sometimes called the abampere, or the biot or sometimes just 'one e.m.u. of current'.
Also important here is the speed of light in a vacuum, cCGS ≅ 2.998×1010 centimetres per second. Using equation USJ tells us that keCGSm is equal to cCGS2 ≅ 8.988×1020. Also, via equation USQ, -
Sometimes no name is given for this unit other than an 'e.m.u. unit of charge'. You can, with the assistance of figure USU, continue from here to derive values in the rest of the e.m.u. such as potential (1 abvolt = 1.0×10-8 volts), capacitance resistivity and so on. The abvolt is so small in comparison with customary potentials because the dyne is feeble. Trouble brews faster yet.
The electrostatic units subsystem
In the CGS e.s.u. it is Monsieur Coulomb who gets first prize and a value of keCGSe equal to 1.0. Coulomb's Law now looks like -
That means two charges of one electrostatic unit each, separated by one centimetre, experience a force of 1 dyne. How much is this quantity of charge in SI units: coulombs? First we re-arrange Coulomb's Law -
Substituting equation USJ -
This is consistently in SI so we must translate the 1 dyne and one centimetre before substituting -
Which simplifies to -
In CGS electrostatic units this is sometimes called the statcoulomb and sometimes just 'one e.s.u. of charge'.
|Q||coulombs||Q||≅ 3×109 statcoulombs|
|Q||statcoulombs||-||≅ 3.336×10-10 coulombs|
Using equation USJ in reverse tells us that kmCGSe is equal to 1/cCGS2 ≅ 1.113×10-21. Also, via equation USQ, -
|SI||current||I||ampere||A||≅ 3×109 statamperes|
|e.s.u.||current||I||statampere||-||≅ 3.336×10-10 amps|
Sometimes this quantity is called 'an esu per second of current'. Other e.s.u. units follow on. For example the statvolt is equal to about 299.8 volts.
Combining Equation USE with the result that 1 abamp = 10 amperes gives -
|e.s.u.||current||I||statampere||-||1 / cCGS abampere|
This curious factor of c is considered further below. It's worth pointing out here that the 'equivalence' shown between e.m.u. current and e.s.u. current does not extend to the units, which differ by that velocity: c. Ditto, of course, for the other quantities.
The e.m.u. and e.s.u. systems were principally the work of Gauss and Weber.
The Gaussian subsystem
The e.m.u. and e.s.u. subsystems described above are not adopted in their pure forms. Instead a composite system is devised called Gaussian Units. This takes a 'pick and mix' selection from the e.m.u. and the e.s.u. . From the former it takes the 'magnetic units' of field strength, flux, flux density, magnetization etc.. From the latter it takes the 'electric units' charge, current, permittivity etc.. See table USS below.
The Gaussian unit of magnetic flux density, the gauss, is defined using the Force Law -
Equation USF says that a wire 1 centimetre long, sitting in a flux density equal to one gauss and carrying a current of 1 abampere experiences a force equal to 1 dyne. What is this flux density in the SI? The SI defining relationship is -
Substituting the current, force and length values converted from their CGS definitions -
The Gaussian unit of magnetic field strength, the oersted, was defined as equal to unity (1.0) at the centre of a circular loop of thin wire, radius one centimetre, carrying 1/(2π) abampere of current. Modern definitions rest on Maxwell's equation for H, but, under steady current conditions, it all boils down to Ampère's circuital law (and it's more refined sibling the Biot-Savart equation). Let's convert the oersted into SI using the result for the field at the centre of a current loop:
where a is the radius of the loop.
|Quantity Name||ESU dimensions||EMU dimensions|
|Electric potential (voltage)||+1/2||+1/2||-1||+3/2||+1/2||-2|
|Electric flux density||-1/2||+1/2||-1||-3/2||+1/2||+0|
|Electric field strength||-1/2||+1/2||-1||+1/2||+1/2||-2|
|Magnetic flux density||-3/2||+1/2||+0||-1/2||+1/2||-1|
|Magnetic field strength||+1/2||+1/2||-2||-1/2||+1/2||-1|
A most important fact to grasp is that the Gaussian has no separate concepts of absolute and relative permeability. If you take the Gaussian flux density (in gausses) and divide by the field strength (in oersteds) then you still get a result which is called permeability. That result, however, is numerically equal to what in the SI is the relative permeability. So, for a vacuum, oersteds/gausses = 1.0 and for iron it might be 8000. The same calculation in the SI, using teslas/(amps per metre), would yield values lower by a factor of 4π×10-7 and are designated absolute. It follows that, Gaussian permeability being a dimensionless ratio (just like 'relative permeability'), the fundamental units for flux density and field strength are the same: cm-½ g½ s-1 .
Another trap for the unwary is the use of the symbol μ0 by some authors (eg Bozorth) to represent initial permeability. This is not, repeat not, the permeability of a vacuum as used by authors working in the SI. Oh, me 'ead :-(
The Gaussian system, adopted in 1881, was (despite its name) principally the work of Heinrich Hertz and Maxwell. Its success meant that the term 'CGS units' become a synonym for it.
The approach taken by the SI
I often say that when you can measure what you are speaking about, and
express it in numbers, you know something about it ...
William Thomson, Popular Lectures
In the SI, distance is measured in metres and force is measured in newtons (the force which when applied to a mass of one kilogram accelerates it at one metre per second squared). Now the important bit: Monsieur Ampère is again the winner and kmSI is given the value of 10-7. In the SI current is measured in amperes -
One ampere is that constant electric current which, if maintained in two straight parallel conductors of infinite length, of negligible cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10-7 newton per metre of length.
If you were wondering why the constant is 2×10-7 then the short answer is that this definition of the amp is compatible with earlier definitions of it. The ampere was in use long before the SI. However, you also have a field strength (H) of one ampere per metre at the centre of a circular loop of wire of diameter one metre when it carries one ampere of current. See the Biot-Savart page for more details. Anyway, the runner up, Monsieur Coulomb, has to define his unit of electric charge via Equation USQ and then equation USJ gives the electrostatic force constant -
In passing you might note that this means that two charges, of one coulomb each, placed one metre apart exert a force upon each other that exceeds the weight of two fully laden oil super-tankers. Do not try this at home. One ampere is a significant amount of current, yet it still produces a feeble magnetostatic force. One ampere second, i.e. one coulomb, produces an enormous electrostatic force. That is why Monsieur Ampère and Monsieur Coulomb do not see eye to eye and why science has had such a hard time reconciling all this Gallic discord.
It's easy to be confused by this stark definition of the ampere, in terms of the mechanical forces produced by it, into thinking that it is a derived unit from the base units of mass, length and time in the same way that joules, watts or pascals are. This indeed is how earlier systems treated current. The SI produces instead, depending on your point of view, either a stroke of genius or a fudge. it says "OK, we have said that you can tell how large an amp is in terms of force and length; but forget that we told you that. As far as the rest of the SI goes we treat the ampere as a fundamental unit which cannot be decomposed into more basic ones - in the same way that the second or the kilogram cannot".
Immediately, however, we hit a problem. Re-arranging ampere's force law slightly -
On the left we have amperes2 (A2). On the right we just have the old mechanical units. The equation is not dimensionally balanced. The workaround is to say that kmSI is not just a constant but a quantity with units of its own. Remembering that the length (s) and the separation (a) both have dimensions of metres, inspection of Equation USR tells us that the units of kmSI must be newtons per ampere squared (N A-2).
Another problem with the SI is that, after its introduction in 1960, a number of 'loose ends' were left concerning its application to magnetics. This led to the Kennelly and Sommerfeld subsystems of units. Each of these is based on a different model of the way magnetism works. The Kennelly subsystem makes use of the concept of 'magnetic poles' to model the interaction of materials with a field at an atomic level. The Sommerfeld system ascribes 'dipole moments' instead to circulating electric currents. The end result is that different definitions (and units) are used for magnetization.
On a brighter note, modern authors appear to favour the Sommerfeld system - as will these web pages.
The evolution of unit systems
The amp, the volt and ohm agree
to live together peacefully ...
(As not drawn by) Joseph Barbera, The Truce Hurts
Devising a unit system is difficult because of the conflicting requirements which a good one should meet -
- The units should be of a size that is appropriate for use in
science and engineering and in everyday ('customary') use by the wider
commercial and public community.
- The units should be defined in a manner that is both precise
and capable of being reproduced anywhere in the world.
- The units should be coherent. In a coherent system all the
subsidiary or derived units can be expressed as combinations of the
basic or fundamental units without the introduction of any
Compromise is unavoidable. First to adopt a customary metric system was France in 1799. Progress towards its adoption elsewhere has been distinctly less rapid now that those opposed to it are no longer decapitated. Anyway, the CGS system enjoyed wide acceptance until the electricity industry decided that the e.m.u. abvolt was too small for practical use and succeeded in introducing a value 108 times larger: our present value for the volt. The ohm and the amp were also re-scaled. Electricians were happy, but coherence with mechanical units was lost. That is the fundamental reason why kmSI takes the value it does: it's what's needed to make electrical units a 'sensible' size for the installers of electric heat and light.
In case anybody was still able to understand the state of the unit system it was decided in 1932 that what had been called the gauss would thereafter be known as the oersted. Only since then has the gauss been a unit of flux density.
In 1948 it was decided that the centimetre, the gram and the erg (the CGS unit of energy) should be replaced by the larger metre, kilogram and joule. This was called the MKSA system and it restored coherence (as foreseen shortly after 1901 by Giovanni Giorgi).
In 1960 the radian, steradian, kelvin and the candela were added to the MKSA system to form the International System of Units, abbreviated to SI (Système International d'Unités). Only minor tweaks to the SI followed, apart from the introduction in 1971 of the mole.
Why then, a hundred years after Giorgi's proposal, does everyone not use a coherent system? The reasons vary. They include, for example, a reluctance to specify flux density in teslas because Carl Friedrich Gauss was a greater intellect than Nikola Tesla. More importantly the SI permeability and permittivity are altogether nastier affairs compared to the CGS equivalents.
That, though, is a fundamental difference - because the SI includes the ampere as a base unit, which CGS does not. It also explains why the dimensions of magnetic induction in the SI are different from those of magnetizing force. People with knowledge of the B-field and H-field only in the SI tend to see them as physically different. That the two exist for the sake of computational convenience alone is more obvious in Gaussian Units. Some have argued (Larson) that the unit system is in a mess because physics is in a mess. They say that ascribing units to kmSI is not science but computational legerdemain. Perish the thought.
So, the answer to our original question "Why use different unit systems?" is not simple reluctance to change. It is, as Ken Alder says, about what you hold to be important. Pragmatists favour the SI's utilitarian approach to calculation, want all the factors of c kept out of sight, accept the embarrassing units for km and will never replace a 3 abampere fuse. OTOH, philosophically minded physicists want only the three base quantities of the Gaussian system to better reflect the underlying science. They dream of a visit to Woolworths where they enquire of the sales assistant
"Excuse me, have you any 600,000,000 ergs per second light bulbs?"
"Why, certainly, sir. We sell all our electrical goods by Gaussian units because in advanced texts on electrodynamics we find that the tensor calculus equations which unify the electric and magnetic fields are just so much clearer!"
"Great, I'll take 59,958,491,600 / c of them, please."
"While you're here, may I interest you in our 0.005 statvolt AA cells?"
Stuck in between these groups are the engineers; we must cope both with text books written by the latter and problems set by the former. What system should we adopt? It's anybody's gauss. Whatever your own view may be, don't expect a truce soon. However, fear not. 1799 has gone and the Bureau no longer knocks on your door should you make inadvertent mention of the pre-revolutionary. In fact, it explicitly recognises that, for good or ill, non-SI units will endure.
The ratio of the e.s.u. to the e.m.u.
This velocity, therefore, which indicates the relation between
electrostatic and electromagnetic phenomena, is a natural quantity of
definite magnitude, and the measurement of this quantity is one of the
most important researches in electricity.
James Clerk Maxwell, A Treatise on Electricity and Magnetism
Although not essential to understanding unit systems, this topic does relate to equation USJ above. Examine the first entry in table USS above: that for electric charge, and divide the electrostatic units by the electromagnetic units. You obtain a velocity: cm s-1. You obtain the same for current, electric flux density and magnetic field strength. For magnetic flux, magnetic flux density, voltage, and electric field strength you obtain the reciprocal of a velocity: cm-1 s. For capacitance and permittivity you obtain the square of a velocity: cm2 s-2. Finally, for permeability, inductance and resistance you obtain the reciprocal of the square of a velocity: cm-2 s2.
Fundamentally, what this boils down to is that a dimensional analysis of the ratio ke / km, via a rearrangement of Ampère's Force Law and Coulomb's Law, shows that it has units of velocity squared. This seems a bit curious since nothing in either Ampère's or Coulomb's experiments needs to move (apart from the electrons themselves). Even more curious is the experimental value of this ratio -
where c is the speed of light in a vacuum. This fact lent real proof to the proposal made many years previously by Leonard Euler that light had something to do with electromagnetism. It can be shown that the magnetic force is just the electrostatic force after adjustment by the relativistic Lorentz transform. You may already know that the speed of light features heavily in the theory of relativity too.
The constants keSI and kmSI are not used directly in the rest of SI electromagnetism. Instead new constants are defined as -
Introducing the factor of π at this early stage (called rationalisation) prevents it from infesting later formulæ as in the Gaussian system. Note that because kmSI is defined as exactly 10-7 that μ0 is also exact. Since 1983 the speed of light has been defined to be exactly 2.997 924 58×108 m s-1 . Therefore keSI and ε0 also acquire exact values. Although μ0 has retained the units of kmSI these are translated into the equivalent H m-1.
Trivia point: designation of the letter c is from the word celerity: quickness, or rapidity of motion. Its Latin root also gives accelerate.
An advantage of the SI is the ease with which dimensional calculations can be performed. To assist these table USF below lists the base units for a few quantities which are not listed elsewhere in these web pages.
|Angular momentum||L||joule seconds||J s||1||2||-1||0|
|Electro motive force||E||volt||V||1||2||-3||-1|
|Moment of inertia||I||kilogram metres2||kg m2||1||2||0||0|
|Permittivity||ε||farads per metre||F m-1||-1||-3||4||2|
|Torque||T||newton metres||N m||1||2||-2||0|
Example: Flux, mmf and reluctance are related via the equation
is the equation dimensionally correct?
On the right hand side we have Fm in amps, divided by Rm in per henry,
A / (A2 s2 kg-1 m-2) = A (A-2 s-2 kg m2) = kg m2 s-2 A-1
and these are the base units for the left hand side of the equation: flux in webers.
Dimensional calculations are not merely party tricks which enable you to check your working. When you can follow the process by which the higher level (derived) units are composed from the simpler, more intuitive base units then you obtain a better understanding of quantities like resistance or inductance. Even the strangest and most abstract quantities eventually boil down to metres, kilograms, seconds and amps. Is there anything abstract or hard to grasp about them? Drop a kilogram on your foot. These pages are particular about units because of this.
You could also impress your friends by mentioning your 10 million kilogram metre squared per second cubed per ampere squared resistor. Uhh ... well, maybe not.
Dimensional analysis, in the form we use today, was first proposed by
Turns of wire
As mentioned in the section on current, many engineers prefer that the number of turns is included explicitly in the units of magneto-motive force. They say, for example, that the coil in figure TMX carries '12 ampere turns' of MMF, or '12 A t'. Similarly, inductance factor has units of henries per turn squared.
Now, units come in one of two flavours: basic (like kilograms) or derived (like ohms). The former group, of which the SI has seven, are starting points for the units system, and cannot be decomposed into more fundamental ones. The latter group, on the other hand, is each made up from a combination of two or more of the basic units.
The question arises, then, into which category you fit the unit for turns. The established view is that turns aren't really part of either. Instead, they are regarded as having 'unit dimensions'. That is, they have the same units as the digit '1'. Informally, you say they are 'dimensionless'.
So, as far as dimensional calculations are concerned -
etc. And -
From which you see that, in a technical sense, turns are redundant. Their inclusion, or omission, whenever or wherever you like, makes no difference to the dimensional validity of an equation. You could, for example, specify MMF as 'amperes per turn squared' and be no more and no less correct than 'ampere turns'.
| Quantity |
|Magneto motive force||0||0||0||1||1|
|Electro motive force||1||2||-3||-1||-1|
"That's not very helpful.", you complain, "Yah, boo, sucks to the SI ! Let's just treat turns as an eighth basic unit alongside amps, seconds and the rest." Well, as figure USU shows, all the electric units are related in a kind of 'family tree'; with amperes as the common ancestor. A change to one quantity has a 'knock-on' effect with the units from which and to which it relates. If you go tinkering with MMF then you face a dilemma. Either you have to accept that, in at least one equation, you will have dimensional inconsistency, or you must make the same alteration to all the other electric quantities, substituting 'A t' wherever 'A' appears originally -
If you do this then the physicists will ask if you have gone mad. They don't care about turns. Whether a coil has four turns carrying three amps or three turns carrying four amps then it's all just '12 amps' to them. The magnetic field will be identical - so quit fussing. Consider the 'dual' case of a battery. No one is confused if it's described as having '9 volts'. No one imagines that each cell within it has an EMF of 9 volts. We accept that it's the overall potential that's being specified, without a special unit called the 'volt cell'. If engineers were consistent then they wouldn't need 'ampere turns' to talk about MMF.
We can only stutter "Yes, but ...", or else eschew the ampere, use Gaussian Units and the 'gilbert' for MMF instead. Dohhh!
Some other conversions
If you have a quantity expressed in CGS units then multiply by k to find the equivalent in SI units.
|CGS Unit||k||SI Unit|
|magnetic field strength||oersted||250/π||amp per metre|
|magnetic flux density||gauss||10-4||tesla|
|magnetization||abamps per cm||103||amps per metre|
|Energy product||gauss-oersted||7.962×10-3||J m-3|
Converting formulae between the two systems requires care. You may also see flux being specified in 'lines' - this is synonymous with maxwells.
Much data on susceptibility is still only available in CGS units. In CGS the susceptibility is defined as
Where μ is the permeability. If this surprises you then recall that CGS permeability also differs from SI permeability (by a factor of 4 π×10-7).
Where you have a susceptibility expressed in CGS units then multiply by the factor tabulated below to obtain the susceptibility in the corresponding SI units.
|CGS symbol||κ, χ||χρ|| χM|
|CGS units||1||cm3 g-1||cm3 mol-1|
|SI type||SI symbol||SI units|
|bulk||χ||1||4 π||4 π×10-3 ρ||4 π×10-6 ρ / M|
|mass||χρ||m3 kg-1||4 π / ρ||4 π×10-3||4 π×10-6 / M|
|molar||χM||m3 mol-1||4 π M / ρ||4 π×10-3 M||4 π×10-6|
where ρ is the density of the substance in kg m-3 and M is the molar mass in kg mol-1. For example, the element copper, Cu, has a molar mass of 0.06355 kg mol-1 and a density of 8940 kg m-3, so
Using units and symbols
Although the following tips on the usage of units and symbols are not specific to electromagnetism they gain importance where the range of exponents is large or the reader may be unfamiliar with the quantity.
Refer to the four parts comprising a measurement as follows:
|Quantity name|| Quantity
|Unit name|| Unit|
An interval of time might be 20 seconds, a mass may be forty kilograms, a resistance ten ohms, a temperature 600 kelvins and an energy ninety joules. The point is that we write all these unit names without a capital letter - even where the unit is named after a scientist.
Strictly speaking, the m, Ω and Wb above are symbols and not abbreviations. Thus you don't use a full stop after one (as you would an abbreviation) nor need you follow them by an s to denote a plural. Like other symbols you can put them in formulas. Where the unit name is that of a scientist you then capitalize the unit symbol, e.g. C = 1.8×10-6 F.
You need to include a space between the numerical value and the unit symbol. For example, L = 2.753 mH rather than L = 2.753mH. Also, L = 2.753 millihenries is better than L = 2.753 milli henries or L = 2.753 milli-henries.
Table headings and graph axis labels
You sometimes see tables headed like that on the left
|River name|| length,
d x 106 m
|River name|| length, d /
where the writer has tried to indicate that you need to multiply the values in the length column by one million before using them. Other writers use the same scheme to indicate that the data values have already been multiplied by the given factor. For river lengths it's perfectly obvious what the correct interpretation is but for many quantities in science and engineering the ambiguity may be fatal. Avoid it with the type of heading in USTb: This you read like a dimensionally consistent equation . So, for the Nile:
which we can re-arrange
A note on spelling
It's illiterate, Lewis.
Colin Dexter, "Ghost in the Machine"
|metre||meter||William the Conqueror|
In the age of the search engine your spelling acquires an importance beyond the purely academic. Here's how to irritate anyone from the USA or who despises pedantry -