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Dimensional Units

Reality can be described (or defined) in terms of dimension units.  There are several ways to do this.  The most common method (used on this page) is to define a set of base (fundamental) units, and then describe everything else in terms of derived units.

Farther below is a large table of units (both derived and fundamental), but lets start with the Table of Fundamental Units:

 Dimension Unit Description Symbol Name Symbol Name T Time s second 9192631770 cycles of caesium-133 (133Ce) groundstate hyperfine splitting frequency (at 0 K) [1/86400 = 1/(24*60*60)] of an Earth day L Length m meter Distance traveled by light (in vacuum) in 1/299792458 of a second M Mass kg kilogram 1 liter (1 cubic decimeter = 1000 cubic centimeters) of pure water (H2O) at sea-level-pressure and at 4°C Q Electric Charge C coulomb charge of 6.24150913 • 1018 protons negative charge of 6.24150913 • 1018 electrons Θ Temperature Kelvin K 1/273.16 of the triple-point of water

This page considers electric charge (Q) as a fundamental unit, in contrast to the SI (a.k.a, "metric system") which considers electric current (I) as the fundamental unit. In other words, the SI considers electric charge to be a derived unit, while this page makes it a fundamental unit.

The SI considers the candella (cd) as the fundamental unit of luminous intensity. It is really just radiant intensity at a specific frequency (540 THz). Since all known light sources, except lasers, produce multiple frequencies, some form of weighting function, specific to human vision, should be employed. The SI doesn't define one, and several methods have been published, so this unit is quite murky and not considered as fundamental on this page.

The SI also considers the mole (mol) to be a dimensional unit. However, it is a pure (dimensionless) number, like a dozen or a hundred; it is a rather large number, approximately 6.0221408×1023. You may find it listed occasionally in the table below under the column of SI Units, but if you look at the corresponding Dimension Symbol, you should see the mole has no effect on the physical dimension.

Similarly, both plane angles and solid angles have no real physical dimension, but for a different reason. The way these are defined, the dimension of length (L) gets cancelled out. Thus a plane angle is listed with a length dimension of L1-1 because it is a length (L1) divided by a length, and a solid angle is listed as L2-2 because it is an area (L2) divided by an area.

In case you didn't already know and the above didn't make it clear, negative exponents in the dimensional symbol refers to division by that dimension. For example, meters per second (or miles per hour, etc.) has dimensional symbol L1T-1 because it is length divided by time (notice length has a positive exponent, while time has a negative exponent).

Usually, dimensions which do not appear in a unit of measurement are simply omitted. For consistency, I have included all the base dimensions described above for each unit in the table below. Those dimensions which are not present have an exponent of zero. For example, length is listed as L1T0M0Q0Θ0.

 3 Types of Symbols

One thing that confuses many beginning students of algebra and physics is that the same symbol can have very different meanings in different contexts.  Even professionals sometimes misunderstand the meaning of a symbol.  The main reason for this is there are many more concepts than there are letters of the Roman and Greek alphabets.  To prevent confusion (as much as possible), I will tell you this page has three (3) different types of symbols:

• Dimensional Symbols (1st column below) are described in the table above; briefly: L=Length, T=Time, M=Mass, Q=Charge, Θ=Temperature
• SI Unit Symbols (3rd column below) are defined by international standard; examples include: L=liter, T=Tesla or Tera, M=Mega, m=meter or milli, s = second, and many more!
• Math Symbols (5th column below) are commonly used in engineering and mathematical formulas (there is no official standard for all, but a few are defined by SI or ISO)

In general, all symbols should be considered case-sensitive.  In particular, the second type (SI Unit Symbols) can be confusing because the same symbol (for example m) could mean two different things in the same (SI Unit) context.  Fortunately for you, I have included the corresponding name of the SI Symbol in the 4th column; this should resolve any ambiguity.

The final Description column (in the table below) describes a dimensional unit in English text and often with SI Unit Symbols.  For example, a Watt-Hour may be described as "3600 J" which means "3600 joules" because J is the SI symbol for joule.  However, it is often informative to have a description in terms of engineering/math symbols.  When math symbols appear in the Description column, they will be enclosed in square brackets; for example, heat capacity (entropy) may be described as energy / temperature [E/T].  Note the E and T shown in square brackets are math symbols (not dimensional symbols nor SI symbols).  In summary, the description column uses English text with SI symbols in general, but also includes math symbols in square brackets for extra information.

 Colored Units

Under "SI Units" (columns 3 and 4 in the table below), most entries are "official" (explicitly listed or implicitly endorsed) by the SI system; these have a normal (white) background color.  Some of them are "recognized" or considered "acceptable" with the SI system; these have a yellow background (the most popular is the liter).  A few are not recognized (perhaps even forbidden) by the SI system, but are shown here because they are used by people anyway (for example kilowatt-hour, kW•hr), and some of these have no official SI unit; these have a light-red background.

To keep this page relatively short, I have included only SI Units (as much as possible).  There are other metric but non-SI units in use (such as currie, dyne, and cubic centimeter), and many other non-metric units in common use (such as miles and pounds).  These would also be shown with a light-red background if included (but that would greatly multiply the size of this page).

 Extra Subscripts

Several of the "Math Symbols" (5th column below) may include a subscript letter(s) to be consistent and precise.  Some of these will rarely be seen in other documentation, because of their limited scope; few papers span all possible dimensional units!  A good example is Einstein's famous equation, which is most often written: E = mc2.  The context of most documents which use that form implies that E is mass-energy (and not kinetic energy or something else), and c is light-speed (and not some other speed, like the speed of sound).  On this page, the same equation would be written with subscripts: EM = mc02.  The symbol EM is for mass-energy, and c0 is for light-speed.

Some subscripts are generic (often "i" and "n").  For example, the math symbol for molar concentration is ci when referring to an unknown, but might written as cH when referring to Hydrogen or cAr when referring to Argon.

Before we dive into the details, it is important to note that other dimensional units are possible.  Here are three good examples:
• Time could be defined in terms of length (or vice versa), according to Einstein/Lorentz relativity
• Mass could be defined in terms of length, according to Schwartzchild radius
• Electric charge could be defined in terms of the square root of mass times length (√M•L)
Anyway, below is a list of dimensional units, sorted in the order of (first) Length, Time, Mass, Charge, and (finally) Temperature...