General > General Technical Chat
Rational, Irrational, and Integer. The correct definition.
pcprogrammer:
Well in the other thread Peter Taylor does give a hint of where he lives 8)
In Lala Land :-DD
And that was already clear with his superb website that he made :palm:
TimFox:
Mathematics is something that happened to other people.
rsjsouza:
--- Quote from: gnuarm on December 21, 2022, 03:54:39 am ---Why are you making up your own math? I'm not following what you are trying to do.
--- End quote ---
It is to "decolonize math", as these vicious numbers are transphobes since they don't believe in trangenderism and categorize themselves in binary terms (rational/irrational). Burn the books! :-DD
Nominal Animal:
I've always found it comfortable how the classification of numbers could be used as a tool. They are not arbitrary, and they are useful.
You start with integers, then rationals, and then irrationals. You extend these into complex numbers, and you'll find integers and rationals and irrationals have interesting features. For example, consider \$e^{-2 i \pi x}\$, where \$x \in \mathbb{R}\$, and what it means for \$x\$ to be integer, rational, or irrational.
You start with integers, and extend these into vectors, and you get regular rectangular lattices. Integers correspond to lattice points, rationals to lines that cross an infinite number of lattice points, and irrationals to lines that intersect with a single lattice point. Examine the properties of such lattices, extending to non-rectangular lattices (say, face-centric cubic, body-centric cubic, hexagonal, and so on), and you've arrived at crystallography.
From rationals, you can represent both the numerator and the denominator with a polynomial, and you arrive at rational polynomials, which have a lot of practical applications.
You start with integers, think about rationals and divisibility, and arrive at prime numbers, prime number decomposition, and all sorts of discrete math.
You combine the idea of irrational numbers, prime numbers, and rational polynomials, and you arrive at ideas like elliptic polynomials, which can lead you to elliptic cryptography.
Everything else like discrete logarithms, factorization, and so on, used everyday in cryptography and computers and networks, is extensible from the first principles like this, and easily understood at the conceptual level, even though the actual mathematical machinery may be hard to fully understand and utilize. The space of such concepts has an infinite number (hah!) of dimensions, with each conceptual approach their own dimension, and the elements along that dimension are the concepts in which it is useful to divide the number or value into. For scalars, it is integers, rationals, and reals. For complex numbers, you have unity (\$\lvert z \rvert = 1\$, i.e. \$z = e^{i x}\$, \$x \in \mathbb{R}\$), roots of unity, pure reals, pure imaginary numbers, et cetera.
Instead of a vast continuous infinite-dimensional space of concepts we'd be hopelessly lost in, these give us reasonable and trustworthy stepping stones we can walk along. They may not lead to everywhere yet, but you can always build new ones to reach places not yet reached.
The only reason you'd want to redefine these concepts is if you insist on sitting at the singular starting point, with your eyes closed, hoping for an universe that was less complicated so you could fool yourself into believing that you understood it all, fully. So foolish.
T3sl4co1l:
--- Quote from: rsjsouza on December 21, 2022, 02:52:14 pm ---
--- Quote from: gnuarm on December 21, 2022, 03:54:39 am ---Why are you making up your own math? I'm not following what you are trying to do.
--- End quote ---
It is to "decolonize math", as these vicious numbers are transphobes since they don't believe in trangenderism and categorize themselves in binary terms (rational/irrational). Burn the books! :-DD
--- End quote ---
Got just a little chip on your shoulder there?
Tim
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