MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78 Journal homepage: https://content.sciendo.com The Algebraic Structure of Quantity Calculus II: Dimensional Analysis and Differential and Integral Calculus Álvaro P. Raposo1 1Department of Applied Mathematics, Universidad Politécnica de Madrid, Av. Juan de Herrera, 6, 28040, Madrid, Spain, [email protected] In a previous paper, the author has introduced and studied a new algebraic structure which accurately describes the algebra underlying quantity calculus. The present paper is a continuation of that one, which extends the purely algebraic study by adding two more ingredients: an order structure and a topology. The goal is to give a solid justification of dimensional analysis and differential and integral calculus with quantities. Keywords: Quantity calculus, algebraic structure, dimensional analysis, pi-theorem. 1. INTRODUCTION two drawbacks which show it not to be an accurate descrip- In a previous paper [1] the author has produced a new alge- tion of the actual algebraic properties of quantity calculus. braic structure, axiomatically defined, called algebraic space First, it is based on the concept of unit, as all the structure is of quantities, which resembles accurately the algebra struc- built upon a starting set of base units. In fact, quantities are ture of quantity calculus. In that paper there is a review of introduced as objects of the form a number times a unit (just past attemps of giving a mathematical framework to quan- as Maxwell did over 100 years ago [7]), thus identifying the tity calculus and of the motivation to introduce yet another, concepts of quantity and quantity value, which are well dif- different, algebraic structure which is now summarized here. ferentiated by the VIM. This is quite unsatisfactory, for this Despite in the standard approach to quantity calculus (e.g. the approach blurs the distinction between the system of quan- International Vocabulary of Metrology (VIM) [2]) the calcu- tities (such as the ISQ) and the system of units (such as the lus focuses on quantities and dimensions as central concepts, SI). This situation has been recognized by several authors in the mathematical approach to the algebraic structure underly- previous years (see de Boer [8] for an excellent review and ing quantity calculus is based on units as the main concept. discussion, [9] for a recent one) who have claimed, instead, In the recent analysis of Kitano [3], for instance, or the first for an algebraic structure defined axiomatically and based on papers on this model, [4], [5], [6], the model is set up starting the concept of quantity and of dimension rather than on the from a set of base units (u1,...,un) and expresses any quan- concept of unit, which should be secondary. tity q in terms of them as The second drawback is the use of real exponents in (1). They are not necessary. It is a remarkable fact that the laws of α r1 rn q = u1 ···un , (1) Physics involve dimensionful quantities (in contrast with di- mensionless quantities) only through three operations: prod- where α,r1,...,rn are real numbers. In this equation α is the numerical value of q with respect to this system of units, uct of quantities, product by a scalar, and addition of quanti- r1 rn ties of the same kind. Therefore, only integer exponents are while u1 ···un is the unit in which q is measured. The set r1 rn found in physical equations. Some simple examples seem to of all units, that is {u1 ···un : ri ∈ R, i = 1,...,n} has the structure of a vector space over R, written in multiplicative prove the previous statement wrong, for instance, the veloc- form. The set of quantities, thus, adopts the form of a family ity v of a body of mass m for which the kinetic energy T is of rays, each identified and spanned by a unit as in (1) by known is given by letting α run through the reals. The rays coincide in a point, 2T the zero of the algebraic structure. The numbers (r1,...,rn) v = , (2) are referred to as the dimensions of the quantity q. r m This algebraic structure has served its main purpose: giv- so the dimensionful quantity T/m has an exponent 0.5. How- ing a proof of Buckingham’s Pi Theorem. However, it has ever, the dimension of T/m is already a squared one: [T/m]= DOI: 10.2478/msr-2019-0012 70 MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78 −1 2 −1 (LT ) , so the use of the exponent 0.5 is only, say, acciden- fiber has its own zero element. The fiber dim 1D is called tal. The period t of a pendulum given its length l and the the set of dimensionless quantities or, more appropriately, acceleration of free fall, g, provides a similar example, for it quantities of dimension one. This fiber is, in addition to is given by the structure of vector space, an algebra over F and, in fact, l is naturally isomorphic with F by means of the map F → t = 2π , (3) −1 α α g dim 1D : 7→ 1Q, so we identify both whenever needed. s Notice that the structure of the fibers as one dimensional but, again, the quantity l/g has a squared dimension, [l/g]= vector spaces cannot be the ultimate model of quantity calcu- T2 , so the exponent 0.5 does not provide a dimension with lus, for it does not fit properly to quantized quantities. If a fractional exponent. A more critical example is a system of quantity is quantized, the corresponding fiber should not be units in which volume is taken as a base quantity and, e.g. the the whole vector space, but a discrete subset of it. However, liter as the corresponding base unit. In that case we should this issue will not be dealt with in this paper. admit fractional exponents for the derived units for length, A system of units is a section of the fiber bundle, that is, 1/3 2/3 (liter) , or surface area, (liter) . Nevertheless, experience a map σ : D → Q such that dim ◦ σ = idD . The system is in measurement science has shown, so far, that a suitable coherent if σ is a group homomorphism, and it is nonzero if choice of base quantities is possible which avoids such frac- σ(A) is not a zero for any dimension A. In[1] the conditions tional exponents to define units, and this is the case this paper for its existence are established. wants to support from the mathematical point of view. With In this paper we only consider spaces of quantities free of this goal in mind, the allowance of real exponents is unnec- zero divisors. Two advantages from such a choice are: (i) essarily oversized. It works, of course, since real numbers every nonzero quantity has an inverse and (ii) there exists a contain integer numbers, but it is not the algebraic structure nonzero coherent system of units. So we also assume without which best fits the quantity calculus. further notice that such a system is always available and all The paper [1] addresses these two issues and gives a way systems of units considered are nonzero and coherent. out by defining, in axiomatic form, a new algebraic structure When a coherent nonzero system of units σ is defined in a which resembles, more accurately, the calculus with quanti- space of quantities then we have the map ν : Q → F (where ties (i.e., only integer exponents allowed) and is completely we identify the fiber of dimensionless quantities with F) de- focused on quantities and dimensions. It starts with the def- fined by D inition of the group of dimensions as a finitely generated ν(q)= qσ(dim(q))−1, (4) free Abelian group. Its elements are denoted A,B... and are referred to as dimensions. The identity of the group is denoted which assigns to a quantity q its numerical value with respect σ 1D . to the system of units ; it is a homomorphism. Notice that σ In the quantity calculus real numbers play a prominent role the composition ◦ dim : Q → Q gives the unit in the fiber as the measurement of a quantity is nothing but its compari- of each quantity. Then, a quantity q is written in terms of a son with the unit of its kind, and the result is a real number. system of units as This number and the unit make the quantity value. In the def- q ν q σ dim q (5) inition of the algebraic structure, the role of real numbers is = ( ) ◦ ( ). F taken by an abstract field . Nevertheless, this abstract field With the units at hand paper [1] explores the way in which is going to get more specific along the sections of this paper, quantities and the operations among them are expressed in first as an ordered field in section 3, then as the real numbers terms of the units. Ways to create new spaces from old ones in sections 4 and 5. With these elements, a summary of the (subspace, tensor product, quotient space) are also explored Q definition given in [1] is as follows. A space of quantities and, finally, the tool for comparison is defined, the homomor- D F Q with group of dimensions over the field is a set , whose phism of spaces of quantities, and utilized to give a character- elements are called quantities, together with a surjective map ization theorem of these structures. Finally, it is demonstrated Q D A dim : → and three operations.