By George W. Hart

This publication bargains with the mathematical houses of dimensioned amounts, resembling size, mass, voltage, and viscosity.

Beginning with a cautious exam of the way one expresses the numerical result of a dimension and makes use of those ends up in next manipulations, the writer carefully constructs the idea of dimensioned numbers and discusses their algebraic constitution. the result's a unification of linear algebra and standard dimensional research that may be prolonged from the scalars to which the conventional research is perforce limited to multidimensional vectors of the kind usually encountered in engineering, platforms conception, economics, and different applications.

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**Example text**

A clear exposition is given in Lin and Segel [1988, p. 195]. The idea is to divide dimensioned variables in a dimensioned equation by similarly dimensioned parameters in order to obtain dimensionless variables. For example, the variable y representing the altitude of a satellite might be replaced by y / R, where the parameter R is the radius of the Earth. This can usually be done in a systematic manner to obtain an equivalent equation with only dimensionless variables, because all the parameters can be combined together to cancel.

44 1. ", etc. , Newton's law that aU, m) = f /m. An intrinsic expression relates dimensioned quantities themselves and is basis-independent, whereas an extrinsic relationship is meaningful only if the axes are labeled or the dimensioned constants are given. The formula given earlier for Fx,v(x, v) is intrinsic if we think of it as a function F(X, V, x, v) of four variables, but extrinsic if we choose constants X and V and then think of it as a function F(x, v) of two variables. To justify an intrinsic function in an analysis, one explains how the terms in the formula are meaningfully combined, whereas to justify an extrinsic function one explains the measurement or empirical basis for the function and/or constants.

So the real numbers are isomorphically embedded in the dimensioned scalars as the elements of the form (r,O), somewhat analogously to how they are embedded in the complex numbers, but this is a vector O. The fundamental operations can be defined constructively in terms of the field and vector operations. One can easily check that the following operations satisfy our axioms. Definition. J. un e ne, i Vl -;- V2. (rl,Vl)· (r2,v2) = (rlr2,vl +V2). ,Vl-V2), , undefined, if r2 =I- 0; if r2 = O. (~,~ )(r2,v2) -_ { (r~2, r2vl), if V2 =0, .