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Topic: An Interesting Point
Replies: 16   Last Post: Nov 7, 2012 11:14 AM

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kirby urner

Posts: 1,788
Registered: 11/29/05
Re: An Interesting Point
Posted: Nov 7, 2012 12:18 AM
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Hey Peter --

What you're talking about with 5* being an operation and like a
different kind of number, reminds me of lambda calculus and partial
functions.

We see 5 * 75 as a binary operation with * as the operator. As a
function, that's mul(5, 75). But then mul(5) by itself may be seen as
a function that returns another function ready for one argument, which
it will multiply by 5. by5 = mul 5 is sort of how that looks, such
that by5 75 gives 375.

The idea that a partially filled in function is itself a function
expecting the remainder of the arguments, is the idea of "currying".
A partially fed function is still a function. This concept is
embedded in the functional programming notations (languages), Haskell
a flagship.

> let mul x y = x * y

> mul5 = mul 5
> mul5 75

375

is legal syntax, interactive use of the Haskell "calculator" (aka
"chat window").

The first line defines what appears to be a function mul that expects
two elements. These get multiplied for an answer.

The subsequent line "currys" mul by feeding it a first argument and
giving a name to the result.

The result is likewise a function, expecting one argument, and in
applying it to 75, we get back 375.

Other things you were saying reminded me of the N < Z < Q < R < C
progression, which I think is fine to share right from the start.

N = Natural numbers, go ahead and say N + 0 = W if you want Whole
Numbers as its own set
Z = Integers which includes negatives reflected around 0 i.e.
negative - positive number line (best to show from many angles, not
always negative to the left (becomes misinformation / unconscious bias
when repeated so consistently without reasoning))
Q = (p/q) such that p, q are members of Z i.e. Rational Numbers
R = Reals which includes the idea of numbers being approached
algorithmically e.g. pi, or continued randomly (with randomness
algorithms)
C = Complex Numbers with their special rules begetting rotation in the
Argand Plane (Argand among others so it that way)

Since fractals provide such wonderful eye candy, I'm happy to start
with C and the Mandelbrot set and work inwards some days, towards the
more primitive N.

A starting point (one of many):

http://www.flickr.com/photos/kirbyurner/8161402982/in/set-72157625646071793

(didn't take much Python to get this in ASCII art -- I'm sure Haskell
could do / has done the same)

The above list gives a sense of "number types" but is far from
exhausting the list of "math objects" which include entities such as
vectors, quaternions, whatever players in whatever language games.
Polyhedrons, polynomials...

Kirby



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