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Re: An Interesting Point
Posted:
Nov 5, 2012 3:59 PM
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Is this number or syntax?
Bob Hansen
On Nov 5, 2012, at 2:59 PM, "Dave L. Renfro" <renfr1dl@cmich.edu> wrote:
> You
> can also have '5' as a function, namely the constant function
> f(x) = 5. In the well known (in mathematics) book "Rings of
> Continuous Functions" by Gillman and Jerison, constant functions
> are represented by bold face numerals. Thus, in that book,
> a bold face '5' means the constant function f(x) = 5, whereas
> a regular face '5' means the number. I believe Karl Menger
> also used numerals to represent constant functions in his 1940s
> and 1950s attempts to reform the teaching of elementary calculus.
>
> Functions represented by numerals can get more exotic than
> this. For instance, when solving differential equations using
> algebraic operator methods (google "differential equations"
> along with "D operator"), numerals now represent multiplication
> by constant operators on sets of functions. For example, 'D + 5'
> represents the operation "d/dx + 5", which when you input
> the function f(x), outputs the function f'(x) + 5*f(x).
> In this setting, '5' would then represent a function whose
> domain is a certain set of real-valued functions of one real
> variable (all functions differentiable on a specified interval,
> for example) and whose range is a similar (but not necessarily
> the same) set of functions, which is defined by "5 evaluated
> at f(x)" is equal to 5*f(x).
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