Source: Novak93, Conversion of Units of Measurement, IEEE Trans Software Engineering SE-21n8(pp651-661).
A standardized quantitative measurement is based on an underlying attribute, a defined unit, a primary reference of example of the unit, some secondary references used in practice and some methods of measurement used to get these measurements. For example the idea of "the inertia of an object" is experienced when we try to move a heavy and light object, or compare the effort to brake a truck with that of a child's toy car. This idea is refined into the concept of inertial mass with a unit like the kilogram and standard ways of measuring it.
In America the National Institute of Standards (NIS) is in charge of these things [ http://www.nist.gov/ ] Their Information Technology Laboratory has been working on deriving the metrology of Information technology [ http://www.itl.nist.gov/ ]
The SI Units are described at [ units.html ] (with thanks to Vince Tilroe, February 17th 2005)
[click here if you can fill this hole]
Now if we multiply two lengths then we get an area - not another length. So the different dimensions are also related. Dimensional Analysis expresses all dimensions ( length, mass, weight, velocity, force, magnetic flux,...) in terms of a small number of fundamental dimensions - length, mass, time, and a few others.
A unit gives a mapping of numbers into dimensions. For instance using 'inch' as a unit of length means that 1.inch is a measure of length,... A set of units works exactly like a set of numbers under addition. If Length is a dimension and inch a unit of length Length./inch indicates numbers which represent the numbers of inches of length.
In general then a unit is map from abstract numbers into dimensions that preserves the operation of addition and the value of zero:
Next define product and quotient of two dimensions.
Notice that if n is a number and u a unit the n.u is the measure. But if c is some constant measure (say the speed of light or the acceleration due to gravity) then c./u is the number in units u.
Metric system units of volume (international):
.Dangerous_Bend A common mistake is to think that a light-year is a measure of time. It is a measure of distance - the distance covered by light in one year. At the atomic level the light_second is a similar useful unit of distance.
Do not confuse these with the forces that they cause due to the gravity of earth.... [ Force and weight ]
Angles are an abstraction form the ratios between the radius of a circular arc and it's length.
Physicists believe that the speed of light is a fundamental constant - a kind of universal speed limit for all matter. It is often symbolized by the letter "c":
If g is the acceleration due to gravity in units ug and m is a mass in units of um then the weight of a mass is the force exerted by gravity, or g*m in the units of ug.um. To simplify some expressions there are special units that take this into account:
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(with thanks to Vince Tilroe, February 17th 2005, who emailed me corrections)
. . . . . . . . . ( end of section SI Units) <<Contents | End>>
This means that the definition of unit implies that temperature is not a unit! Instead we can use a weaker form:
In the centigrade scale:
There are well known formulas for converting Fahrenheit to Centigrade and Fahrenheit to Centigrade:
For details see [ Temperature ] in the Wikipedia.
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Please add the definitions for any other currencies that you know about by clicking here: [click here if you can fill this hole]
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. . . . . . . . . ( end of section Dimensioned Numbers) <<Contents | End>>
Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.
The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.
For a complete listing of pages in this part of my site by topic see [ home.html ]
For a more rigorous description of the standard notations see