ON THE NONEXISTENCE OF UNITS IN ALGEBRAIC
SYMBOLS AND THE CONSEQUENT IDENTIFICATION
OF (dy/dx) WITH (dy/b)/(dx/a), AS ILLUSTRATED
BY ECONOMIC COMPARATIVE STATICS
Gregory L. Light
Department of Finance, Providence College
Providence, Rhode Island 02918, USA

Abstract

This short communication seeks to draw researchers' attention to the simple fact that algebraic symbols, while capable of carrying units externally, do not contain these units internally. As a result, a derivative such as $dy/dx\in \mathbb{R}$, when evaluated at a point $p$ in a domain of definition, is a scalar in a $1-$dimensional vector space $\left( x\right) $ bounded at $p$ that multiplies the tensor $\left( dx\right) \equiv \left( 1\right) $ and then applies $\left( dy/dx\right) $ to any vector $\left( x\right) $ to result in a scalar. Any vector space contains exactly two kinds of objects: vectors and scalars, with scalars closed as an algebraic field; hence the scalars cannot contain units or else they do not form a (closed) field. If $dy/dx$ is a unit-free pure number to begin with, then $x=1$ and $y=1$ are subject to arbitrary underlying unit specifications. As such, one can identify $(dy/dx)$ with $(dy/b)/(dx/a)$, with $a,b>0$ conveniently chosen.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 30
Issue: 3
Year: 2017

DOI: 10.12732/ijam.v30i3.6

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