```Date: Nov 5, 2012 2:59 PM
Author: Dave L. Renfro
Subject: Re: An Interesting Point

Peter Duveen wrote (in part):http://mathforum.org/kb/message.jspa?messageID=7918003> A while back, it occurred to me that there are operator numbers> which operate on other numbers, and since these seemed to be> different from other numbers, and it never occurred to me> before that there might be different types of numbers, each> with different properties. so 5 x 75 is an! operation performed> on 75, and thus, the "5" is a different type of number from 75.> It sort of stopped me dead in my tracks, since I had never> explored such an idea before. Wonder if you have ever explored> such an idea. I would also like to know a simple source whereby> I could review what you seem to indicate is "Indian" mathematics> where the expansion of the number system and the operations> involved are more carefully elaborated.I don't know if this (what I discuss below) is what you're talkingabout, but there are certainly different notions in mathematicsthat are represented by the numeral '5'.For one thing, there is '5' as a natural number, '5' as an integer(formally, an equivalence class of pairs of natural numbers),'5' as a rational number (formally, an equivalence class ofpairs of integers), '5' as a real number (formally, an equivalenceclass of Cauchy sequences of rational numbers), and '5' as acomplex number (formally, an equivalence class of pairs ofreal numbers). And yes, I'm aware that there are several waysof proceeding through these steps. I know of at least half adozen ways of getting the reals from the rationals, for example.The above are all essentially flavors of '5' as a number. Youcan also have '5' as a function, namely the constant functionf(x) = 5. In the well known (in mathematics) book "Rings ofContinuous Functions" by Gillman and Jerison, constant functionsare represented by bold face numerals. Thus, in that book,a bold face '5' means the constant function f(x) = 5, whereasa regular face '5' means the number. I believe Karl Mengeralso used numerals to represent constant functions in his 1940sand 1950s attempts to reform the teaching of elementary calculus.Functions represented by numerals can get more exotic thanthis. For instance, when solving differential equations usingalgebraic operator methods (google "differential equations"along with "D operator"), numerals now represent multiplicationby constant operators on sets of functions. For example, 'D + 5'represents the operation "d/dx + 5", which when you inputthe function f(x), outputs the function f'(x) + 5*f(x).In this setting, '5' would then represent a function whosedomain is a certain set of real-valued functions of one realvariable (all functions differentiable on a specified interval,for example) and whose range is a similar (but not necessarilythe same) set of functions, which is defined by "5 evaluatedat f(x)" is equal to 5*f(x).Dave L. Renfro
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